Foundations of Nonequilibrium Statistical Mechanics in Extended State Space

Abstract

The review provides a pedagogical but comprehensive introduction to the foundations of a recently proposed statistical mechanics ($\mu$NEQT) of a stable nonequilibrium thermodynamic body, which may be either isolated or interacting. It is an extension of the well-established equilibrium statistical mechanics by considering microstates $\left\{{\mathfrak{m}}_k\right\}$ in an extended state space in which macrostates (obtained by ensemble averaging $\widehat{A}$) are uniquely specified so they share many properties of stable equilibrium macrostates. The extension requires an appropriate extended state space, three distinct infinitessimals $d_{\alpha }=(d,d_{\mathrm{e}},d_{\mathrm{i}})$ operating on various quantities q during a process, and the concept of reduction. The mechanical process quantities (no stochasticity) like macrowork are given by $\widehat{A}{d}_{\alpha }\mathsf{q}$, but the stochastic quantities ${\widehat{C}}_{\alpha }\mathsf{q}$ like macroheat emerge from the commutator ${\widehat{C}}_{\alpha }$ of $d_{\alpha }$ and $\widehat{A}$. Under the very common assumptions of quasi-additivity and quasi-independence, exchange microquantities $d_{\mathrm{e}}$q${}_k$ such as exchange microwork and microheat become nonfluctuating over $\left\{{\mathfrak{m}}_k\right\}$ as will be explained, a fact that does not seem to have been appreciated so far in diverse branches of modern statistical thermodynamics (fluctuation theorems, quantum thermodynamics, stochastic thermodynamics, etc.) that all use exchange quantities. In contrast, dq${}_k$ and $d_{\mathrm{i}}$q${}_k$ are always fluctuating. There is no analog of the first law for a microstate as the latter is a purely mechanical construct. The second law emerges as a consequence of the stability of the system, and cannot be violated unless stability is abandoned. There is also an important thermodynamic identity $d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W$$\ge 0$ with important physical implications as it generalizes the well-known result of Count Rumford and the Gouy-Stodola theorem of classical thermodynamics. The $\mu$NEQT has far-reaching consequences with new results, and presents a new understanding of thermodynamics even of an isolated system at the microstate level, which has been an unsolved problem. We end the review by applying it to three different problems of fundamental interest.

1. Introduction

Thermodynamics is the study of physical systems in nature that eventually evolve in time to stationary macrostates, in which any disturbance generates restoring forces to bring them back to the stationary macrostates [1,2,3], which makes them stable macrostates, usually called equilibrium (EQ) macrostate ${\mathfrak{M}}_{\mathrm{eq}}$, satisfying certain stability conditions. Any disturbance to modify macrostates of the system invariably results in nonequilibrium (NEQ) processes so that they abound in nature and obey the well-established second law [4,5,6,7]. The law is also obeyed by biological systems [8,9]. However, NEQ processes are not well-understood, as the corresponding thermodynamics (NEQT) is not yet fully developed, despite it having a long history of various competing schools [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], among which are the most widely known schools of local equilibrium thermodynamics, rational thermodynamics, extended thermodynamics, and GENERIC thermodynamics [21,29]. They mostly deal with the time evolution of macroscopic quantities only; the latter emerge as instantaneous averages over microstates in a more fundamental and statistical approach, and are used to characterize any thermodynamic process and the resulting nonnegative entropy generation ${\Delta }_{\mathrm{i}}S\ge 0$, as first proposed by Clausius [30,31]. In contrast, the equilibrium (EQ) thermodynamics (EQT) in which ${\Delta }_{\mathrm{i}}S\equiv 0$ is based on the original ideas of Carnot, Clapeyron, Clausius, Thomson, Maxwell, and many others [3,12,32,33,34,35,36,37,38,39,40,41,42,43,44,45], and has by now been firmly established in statistical physics, thanks to Boltzmann [46,47] and Gibbs [48], who established that classical EQ thermodynamics is a direct consequence of the EQ statistical mechanics [33,34,37,38] that deals directly with microstates $\left\{{\mathfrak{m}}_k\right\}$ of the Hamiltonian $\mathcal{H}$ of the system, and their equilibrium probabilities $\left\{p_k^{\mathrm{eq}}\right\}$ that together specify the EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$. In contrast, EQT deals directly with ${\mathfrak{M}}_{\mathrm{eq}}$ without any need to know $\left\{{\mathfrak{m}}_k\right\}$ and $\left\{p_k^{\mathrm{eq}}\right\}$.

In general, the collection $\left\{{\mathfrak{m}}_k,p_k\right\}$ of microstates and their probabilities is used in a statistical description of a macrostate $\mathfrak{M}$ of the system $\Sigma$ that may be isolated or interacting with a medium $\tilde{\Sigma }$, as shown in Figure 1. The same microstate set $\left\{{\mathfrak{m}}_k\right\}$ determines different macrostates depending on the probabilities $p_k=p_k\left(\mathfrak{M}\right)$ with which ${\mathfrak{m}}_k$ appears in $\mathfrak{M}$. As $\mathcal{H}$ is by definition deterministic, ${\mathfrak{m}}_k$ is also deterministic. Thus, it is independent of $p_k$, but is specified by its energies $E_k$ and the parameters defining $\mathcal{H}$. Because of this, $\left\{{\mathfrak{m}}_k\right\}$ and $\left\{E_k\right\}$ are the same for any of its possible macrostates $\mathfrak{M}$ including ${\mathfrak{M}}_{\mathrm{eq}}$. This allows $\left\{E_k\right\}$ to be treated as purely mechanical, which is then supplemented by $\left\{p_k\right\}$ to add stochasticity to the mechanical system. Such a description has proven very useful in EQ statistical mechanics [33,34], where the concepts of the entropy $S_{\mathrm{eq}}=S\left({\mathfrak{M}}_{\mathrm{eq}}\right)$ that was first introduced by Clausius [30,31] as a state function of ${\mathfrak{M}}_{\mathrm{eq}}$, and the temperature T are the new concepts that play a central role in the resulting EQ thermodynamics of $\Sigma$. As such, it is very common to use them to distinguish a thermodynamic system from a mechanical system by recognizing that the concept of heat (a consequence of a particular commutator as described later) is novel to thermodynamics but is not applicable to a mechanical system, which is traditionally taken to be described by a purely conservative Hamiltonian $\mathcal{H}$. We use $\mathbf{X}\doteq (N,E,V,\cdots )$ to collectively denote the number of particles N, their energy E, the volume V occupied by them, etc., as representing the common thermodynamic extensive state variables that determine ${\mathfrak{M}}_{\mathrm{eq}}\equiv \mathfrak{M}\left(\mathbf{X}\right)$ in the state space ${\mathfrak{S}}_{\mathbf{X}}$ spanned by $\mathbf{X}$. We call them observables. As observables, these variables can be controlled from the outside of the system. We will allow $\mathbf{X}=\mathbf{X}\left(t\right)$ to have time dependence in this work; here, t denotes the time. For the moment, we suppress the suffix “eq” for notational simplicity unless necessary as we are dealing with ${\mathfrak{S}}_{\mathbf{X}}$. To be useful, S and T must uniquely refer to the thermodynamic state $\mathfrak{M}\left(\mathbf{X}\right)$. This unique relationship is what is meant by S being a state function of $\mathbf{X}$, which when inverted gives E a state function of $\zeta \doteq (S,\mathbf{w}),\mathbf{w}\doteq \mathbf{X}\setminus E$, where $\setminus E$ stands for deleting E from the set preceding it. Being functions of $\mathfrak{M}$ of $\Sigma$, S and T, an intensive field, must be interrelated in some fashion such as

$$1/T=\partial S/\partial E,$$

(1)

(see Equation (129)) so only one of them can be treated as a primitive concept, which we take to be the entropy. The goal of NEQT is to then specify it in terms of $\mathbf{X}$ in the state space ${\mathfrak{S}}_{\mathbf{X}}$. In this respect, having S a state function considerably simplifies the study as we then deal with ${\mathfrak{M}}_{\mathrm{eq}}$. When this cannot be done, we must go beyond ${\mathfrak{S}}_{\mathbf{X}}$ to an extended state space in which the NEQ entropy also becomes a state function, which is the central theme of this review. In this space, a uniform global temperature of the body is defined as its unique field by the above derivative in the extended state space. Thus, our goal will be to identify the NEQ entropy in this space.

Although S plays important roles in diverse fields ranging from classical thermodynamics of Clausius [3,10,12,13,17,20,24,25,30,33,39,40,41,42,43,44,46,47,48,49,50,51,52,53,54,55,56,57,58], quantum mechanics and uncertainty [59,60,61], black holes [62,63,64], coding and computation [65,66,67], to information technology [68,69,70,71,72], it does not seem to have a standard definition in all cases, even though it is well-defined under EQ conditions, as extensively discussed in the literature; see, for example, [46,47,48,73,74,75,76,77,78]. As $S_{\mathrm{eq}}$ is uniquely determined by $\left\{p_k^{\mathrm{eq}}\right\}$ as a state function, $p_k^{\mathrm{eq}}$’s must be unique functions in ${\mathfrak{S}}_{\mathbf{X}}$, as is well-known [33]. Requiring this uniqueness will be a guiding force in our endeavor to formulate the NEQ statistical mechanics. Whether S has any physical significance in a NEQ macrostate $\mathfrak{M}$ has been a topic of extensive debate; see for example [73,74,75,76,77] and references therein. The  problem arises because it is not clear if, and how, $\mathfrak{M}$ can be uniquely identified. Because of the lack of uniqueness, introducing $S\left(\mathfrak{M}\right)$ as a state function becomes nontrivial. The same concern also applies to $\left\{p_k\right\}$.

Recently, we have been able to extend the classical concept of Clausius entropy from EQ states to NEQ states where irreversible entropy is generated [75,76,77,78]. That approach is an outgrowth of an earlier review [79] in this journal about a possible source of stochasticity that is required in a thermodynamic system, even though its mechanics is completely deterministic due to its Hamiltonian dynamics so that heat and temperature have no mechanical analogs. Not appreciating that the source of stochasticity is independent from the deterministic (mechanical) aspect has been a source of bitter debate between Boltzmann, Zermelo, Poincare, and many others ([45,56,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96], and references cited in there). The dispute required Boltzmann to propose the ideas of molecular chaos and of the ergodicity hypothesis [91] that have played a major role in EQ statistical mechanics. We discuss these important ansatze in [92,93,94,95] with an emphasis on Kac’s ring model [97,98] in more detail, where we find that they are not fulfilled in a deterministic dynamics. We infer, as is commonly believed, that one needs a stochastic dynamics for the ansatz to be satisfied. Both these ideas can only be supported by a stochastic dynamics as discussed in these references [99,100,101,102].

It is clear that we need to supplement a purely mechanical approach by supplementing it with stochasticity. We accomplish treating both aspects separately but unifying them together and enabling uniqueness by using an extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ spanned by extensive state variables (compactly denoted by $\mathbf{Z}=\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)\cup \xi \left(t\right)$ as an extension of $\mathbf{X}\left(t\right)$ in this review) to obtain a state function S. In general, $\mathbf{Z}$ includes the observables but possibly some more independent variables, compactly denoted by $\xi \left(t\right)$ required for an NEQ situation, as will become clear later. The additional state variable $\xi$, when properly chosen as will be described later, allows for a unique description of the macrostate $\mathfrak{M}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$. Once such a state space has been uncovered for $\mathfrak{M}\left(\mathbf{Z}\right)$, its entropy $S\left(\mathfrak{M}\right)$ also become a state function$S\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$. This again requires its $p_k$’s to be unique functions in ${\mathfrak{S}}_{\mathbf{Z}}$, just as $p_k^{\mathrm{eq}}$ are in ${\mathfrak{S}}_{\mathbf{X}}$. Thus, the identification of an appropriate ${\mathfrak{S}}_{\mathbf{Z}}$ immediately solves the problem of obtaining a unique statistical mechanics of an NEQ system as it directly leads to $p_k$ as a unique function of ${\mathfrak{m}}_k$ and $\mathfrak{M}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$.

In order for such an approach to work, and in particular for $S\left(\mathbf{Z}\right)$, which itself is a system quantity, it is crucial that we deal with only system-intrinsic (SI) quantities (they are determined by the system), and not medium-intrinsic (MI) quantities (they are primarily determined by the medium) for the simple reason that utilizing $\left\{{\mathfrak{m}}_k\right\}$ requires their specification by the Hamiltonian of the system and so require SI-quantities for its specification. (We will use body to refer to $\Sigma ,\tilde{\Sigma }$, and ${\Sigma }_0$, and BI-quantities to refer to quantities of a body.) As will become clear in the following, these quantities capture the internal processes going on within the system. They cannot be fully captured by the MI-quantities, even though they have been traditionally used in thermodynamics, for the simple reason that they retain the memory of the medium and can depend on the system only weakly. Thus, they will require additional steps to study internal processes. There has been a long debate about the relevance and significance of the two kinds of quantities that ensued from a very different perspective [103,104,105,106,107], but did not capture the importance these quantities acquire in our approach.

The SI-quantities allow us to develop our NEQ statistical mechanics, which for brevity is identified as the $\mu$NEQT, with $\mu$ referring to the microstates $\left\{{\mathfrak{m}}_k\right\}$, in which we directly capture internal processes that are responsible for irreversibility. As the collection $\left\{{\mathfrak{m}}_k\right\}$ is the central object in the $\mu$NEQT, the latter deals with quantities such as $\left\{E_k,p_k\right\}$. At the microstate level, there are fluctuations that are essential in a statistical treatment, and are properly captured in the $\mu$NEQT through the fluctuations in $E_k$ and $p_k$ over ${\mathfrak{m}}_k$. In contrast, the use of the MI-quantities does not directly describe $\left\{{\mathfrak{m}}_k\right\}$ so it cannot properly yield a statistical mechanical description of an NEQ process in a system. This is one of our most important conclusions. In particular, an important consequence of the $\mu$NEQT as will be shown later is that MI-quantities, after  reduction (being averaged over the microstates of the medium) under commonly accepted conditions of quasi-additivity and quasi-independence, do not exhibit any fluctuations. This explains why they are not suitable in developing the statistical mechanics. We call the resulting version of the microstate NEQT the $\mathring{\mu }$NEQT; the circle on $\mu$ is a reminder for the use of “exchange” microquantities derived from the MI-quantities in its formulation. The most prominent are the exchange (also called external) microwork $d_{\mathrm{e}}{W}_k=d_{\mathrm{e}}W,\forall k$, and the exchange (also called external) microheat $d_{\mathrm{e}}{Q}_k=d_{\mathrm{e}}Q,\forall k$, thus, explicitly exhibiting that they have no fluctuations. Because of this, it does not directly capture internal processes at the microstate level, which require additional steps to describe irreversibility as mentioned above. The  corresponding macroscopic NEQT from the two approaches are called the MNEQT and the $\mathring{\mathrm{M}}\mathrm{NEQT}$, respectively; here, M stands for the macroscopic description in terms of macrostates, the circle again having the same connotation as above. There are no fluctuations in these theories, as is well-known. The $\mathring{\mathrm{M}}\mathrm{NEQT}$ is the standard formulation of classical thermodynamics and has been discussed extensively by many prominent scientists [13,18,33,39,41,42,51,108], some including internal variables that play an important role in our approach.

It should be obvious from the above discussion that we need to make a clear distinction between fluctuating (Fl) and nonfluctuating (NFl) quantities. In  addition, we also recognize that there are many other macrostates in ${\mathfrak{S}}_{\mathbf{Z}}$ for which neither S nor the corresponding $p_k$’s are unique functions in ${\mathfrak{S}}_{\mathbf{Z}}$, so S must be treated independently of $\mathbf{Z}$. Our previous work did not consider such states, but they will be considered in this review. For this purpose, we will find it convenient to introduce the following state variable sets:

$$S,E,\mathbf{w}\doteq \mathbf{X}\setminus E,\mathbf{W}\doteq \mathbf{Z}\setminus E,\mathbf{Z},\zeta =(S,\mathbf{W}),\chi =(S,\mathbf{Z}),$$

(2)

and the corresponding state spaces ${\mathfrak{S}}_S,{\mathfrak{S}}_E,{\mathfrak{S}}_{\mathbf{W}},{\mathfrak{S}}_{\mathbf{Z}},{\mathfrak{S}}_{\zeta }\doteq {\mathfrak{S}}_S\cup {\mathfrak{S}}_{\mathbf{W}}$, and  ${\mathfrak{S}}_{\varnothing }\doteq {\mathfrak{S}}_S\cup {\mathfrak{S}}_{\mathbf{Z}}$, where the suffix denotes the variable set forming the state space.

We should emphasize that internal variables also appear in mechanical systems. A simple example is that of two particles in a system, whose interior is hidden in the lab from us so that we cannot see where the particles are inside the system. From outside the system, we can only be aware of the position of the center of mass by observing its motion in the lab. However, there is no way to determine their separation within the system. This separation and the corresponding relative motion are examples of the internal variable and its motion, and play a role in the dynamics of the mechanical system. Thus, it should not come as a surprise that such internal variables will also be relevant in a thermodynamic system. Indeed, we will see later in Section 14 that this relative motion becomes the source of “microfriction”, resulting in friction, when we treat the system in thermodynamics.

To appreciate at a more fundamental level the distinction between a mechanical and a thermodynamic system, we first realize that both systems are usually separated from their surroundings $\tilde{\Sigma }$ by some clear partitions, the most common being the walls between them; see Figure 1. We collectively call them containers or walls that contain the system [109]. In this review, we find it convenient to not include the container as part of the system, but use it to determine the boundary conditions for the equations of motion or as defining parameters in the Hamiltonian $\mathcal{H}$ of the system $\Sigma$. As $\mathcal{H}$ plays the role of E, the parameters $\mathbf{w}$ and $\mathbf{W}$ are obtained by taking out E from $\mathbf{X}$ and $\mathbf{Z}$, respectively:

$$\mathbf{w}=(V,\cdots ),\mathbf{W}=(V,\cdots ,\xi );$$

(3)

where ⋯ refers to the rest of the elements in $\mathbf{X}$ besides V [110]. As will become evident below, these parameters denote the work parameter in the Hamiltonian, which we will denote by $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right)$ or $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$, respectively. For simplicity in the following, we will always use $\mathbf{W}$ for the work parameter to refer to both cases and express the Hamiltonian as $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$. The parameter can be varied in a process with a concomitant change in $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ due to the work done by the system. This is in accordance with the work–energy theorem of mechanics that states that the change in the energy is due to work alone. Also, $\mathbf{x}=\mathbf{x}\left(t\right)\doteq \left\{\mathbf{r}\left(t\right),\mathbf{p}\left(t\right)\right\}$ is the dynamical variable and denotes the collection of coordinates $\left\{\mathbf{r}\left(t\right)\right\}$ and momenta $\left\{\mathbf{p}\left(t\right)\right\}$ of the N particles in the phase space $\Gamma \left(\mathbf{x}\left|\mathbf{W}\right.\right)$ of $\Sigma$ [110]. As internal variables play no role in EQ, $\mathbf{W}=\mathbf{w}$ in EQ. For any $\mathbf{x}\in \Gamma \left(\mathbf{x}\left|\mathbf{W}\right.\right)$, the  deterministic energy of $\Sigma$ in the state specified by $\mathbf{x}$ is $E_{\mathbf{x}}\left(\mathbf{W}\right)=\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$, which need not be constant. However, there is no stochasticity so there is no concept of heat. Thus, the Hamiltonian itself cannot explain the fundamental difference between the two systems [79].

We elaborate further. Mechanics is a branch of the physical science to study the deterministic behavior of the system in the presence of known forces and radiation in time. The central concept is that of energy whose changes are governed by deterministic Hamiltonian equation of motion in $\Gamma \left(\mathbf{x}\left|\mathbf{w}\right.\right)$ with deterministic boundary conditions such as at the walls confining the system $\Sigma$ (see Figure 1a) that generate deterministic wall potentials acting on the particles. Accordingly, a point $\mathbf{x}\left(t\right)\in \Gamma \left(\mathbf{x}\left|\mathbf{w}\right.\right)$ is uniquely determined by $\mathbf{x}\left(t_0\right)$ at some reference time $t_0$. A central aspect of the equation is that it uniquely determines the properties of the system in the future ($t>t_0$) as well as in the past ($t<t_0$) [111,112]. We will assume in this review that the fundamental weak nuclear forces are not included in our discussion [113]. A movie of such a deterministic process in the future, when run backward for the past, will appear just as natural with no hint of the direction of the time flow. Thus, starting from $\mathbf{x}$, which also identifies a microstate [110], at  t, the state undergoes a unique state transformation

$$\mathbf{x}⟼{\mathbf{x}}^{\prime }$$

(4)

in the same interval $\Delta t$ for any $\Delta t$. If we now consider an ensemble of the same mechanical system, each prepared in the state $\mathbf{x}$ at $t=0$, then at $t=\Delta t$, each system will be in the same state ${\mathbf{x}}^{\prime }$. In the language of probability theory [114], we say that ${\mathbf{x}}^{\prime }$ follows with certainty from $\mathbf{x}$ in $\Delta t\gtrless 0$. (This will be useful later to associate the concept of a constant entropy to a mechanical system but not heat.)

But the above invariance is contrary to our daily experience as a rule [115,116,117,118,119,120,121,122,123,124]. For example, the initial state ${\mathbf{x}}_0$ may be when all the gas particles are confined to a small portion of the container [109] located at the center of the container. We are not interested in particle momenta. As the gas expands spontaneously, it occupies the entire volume uniformly. However, once the gas has occupied the entire volume in the state ${\mathbf{x}}_1$, the reverse evolution is not seen in nature. Similarly, the  cream mixed in a cup of coffee does not ever unmix on its own. The smoke from a burning piece of wood only spreads out in the room, but never confines itself on its own. If we run the movies in any of these cases backward, we immediately realize that the backward movies do not represent physical phenomena that are consistent with our daily experience.

This lack of time-reversal invariance of the equations of motion is a natural fact of daily life where we deal with macroscopic systems [125] that eventually evolve in time to ${\mathfrak{M}}_{\mathrm{eq}}$. This is at the root of the second law of thermodynamics, and can be easily explained as follows. It happens here that each member of the above ensemble that was initially prepared in the same state $\mathbf{x}$ evolves during a fixed interval $\Delta t$ into different states ${\mathbf{x}}^{\left(n\right)}={\mathbf{x}}^{\prime },$${\mathbf{x}}^{″},$$x^{‴}$$,\cdots$ for different $n=1,2,3,\cdots$

$$\mathbf{x}⟶{\mathbf{x}}^{\left(n\right)},n=1,2,3,\cdots ,$$

(5)

then the certainty implied in Equation (4) is lost so that most often it would happen that the states of different members after $\Delta t$ would have no discernible pattern for ${\mathbf{x}}^{\left(n\right)}$ and appear haphazard for the members. The result is ([126], pp. 1–14) a loss of physical determinism [127]. Thus, the mapping

$$\mathbf{x}⤏{\mathbf{x}}^{\prime }$$

(6)

in (5) between $\mathbf{x}$ and one of its evolved states ${\mathbf{x}}^{\prime }$ is one-to-many, and the mapping becomes unpredictable, i.e., stochastic [114]. One possible explanation of the loss of certainty at the level of states lies in the presence of stochasticity in the system due to the uncontrollable interactions with the surroundings, as discussed elsewhere [79] and elaborated later in Section 7. This is the foundations of classical probability theory by Laplace, and used to formulate the idea of density matrix by Landau [59,128] and von Neumann [129]. In this case, the mapping (6) cannot be reversed, and we cannot perform time-reversal of the evolution anymore. It is the success of a probabilistic approach to nonequilibrium thermodynamics that prompted Maxwell [50] and Boltzmann [130,131] to promote the “ergodic hypothesis” to achieve EQ. One of our aims in this review is to follow the consequences of this stochasticity in the dynamics such as in the Brownian motion [132] and Langevin’s equation [133], and  extend the concept of ergodicity to a special class of NEQ states [134] that has been identified as internal equilibrium states; see Definition 9.

1.1. Scope of the Review

It should be obvious that the scope of the NEQ statistical mechanics, the  $\mu$NEQT, is more general than that of the equilibrium statistical mechanics, to be denoted here simply as the $\mu$EQT in short, in that the attempts are now mostly to deal with the most general time evolution of microscopic quantities in the former. The instantaneous averages of these quantities over microstates $\left\{{\mathfrak{m}}_k\right\}$ are used to specify the instantaneous macrostates $\mathfrak{M}$ required to characterize any thermodynamic process$\mathcal{P}$ in time in the MNEQT. Thus, the tasks in the $\mu$NEQT and the MNEQT are more difficult and their foundations less developed, which justifies the motivation of this review. The exception is the validity of the first law in terms of exchange (or external) work and heat between $\Sigma$ and $\tilde{\Sigma }$ in thermodynamics, which plays a central role in the $\mathring{\mathrm{M}}\mathrm{NEQT}$. These MI-quantities are determined uniquely by $\tilde{\Sigma }$ regardless of $\mathcal{P}$ being reversible or irreversible, and are easily identified under generally acceptable conditions such as $\tilde{\Sigma }$ always being in EQ, quasi-additivity and quasi-independence; see later. Some of the approaches in the $\mathring{\mathrm{M}}\mathrm{NEQT}$ employ the enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$ [18,21,42,108]. Being associated with an EQ $\tilde{\Sigma }$, the MI-quantities including the exchange (or external) entropy carry no information about irreversibilities going on within the system. In contrast, the MNEQT based on the use of the SI-quantities include, by definition, these irreversible contributions so they are directly obtainable. One of our goals, besides laying down the foundations of the $\mu$NEQT, is to justify the MNEQT from the $\mu$NEQT.

A system in EQ always has its observables uniformly distributed throughout the system so it is uniform in ${\mathfrak{S}}_{\mathbf{X}}$ [33]. In  contrast, an NEQ system is not uniform and requires additional information about the nonuniformity to uniquely specify its states, which is provided by a proper choice of internal variables in $\xi$. The set $\xi$ allows us to treat $\Sigma$ as uniform in the state space ${\mathfrak{S}}_{\mathbf{Z}}$ (see Section 5.7) so that there is a unique thermodynamic temperature and other fields for the entire system even though it is still nonuniform in ${\mathfrak{S}}_{\mathbf{X}}$. This is very useful to obtain a proper thermodynamics of the system. For example, the  single thermodynamic temperature T even for a nonuniform system satisfies Clausius’s theorem that heat flows from hot to cold. This is what makes the $\mu$NEQT in the extended space ${\mathfrak{S}}_{\mathbf{Z}}$ so useful and desirable.

Various microquantities associated with $\Sigma$ (having microstates $\left\{{\mathfrak{m}}_k\right\}$), $\tilde{\Sigma }$ (having microstates $\left\{{\tilde{\mathfrak{m}}}_{\tilde{k}}\right\}$), and ${\Sigma }_0$ (having microstates $\left\{{\mathfrak{m}}_{0k_0}\right\}$) carry the suffix $k,\tilde{k}$, and $k_0$, respectively. However, we are only interested in microquantities associated with $\left\{{\mathfrak{m}}_k\right\}$ as our focus is on $\Sigma$. This means that microquantities of $\tilde{\Sigma }$ and ${\Sigma }_0$ must be manipulated so that they can be associated with $\left\{{\mathfrak{m}}_k\right\}$. To accomplish this, we introduce the principle of reduction, which accounts for the correlation introduced by mutual interactions between $\Sigma$ and $\tilde{\Sigma }$. Under commonly accepted conditions about $\tilde{\Sigma }$, the principle shows that the effect of $\tilde{\Sigma }$ on $\Sigma$ can be incorporated by treating its microquantities in the form of exchange (or external) quantities having no fluctuations. This is what makes the MI-quantities play such an important role in classical thermodynamics, but makes them unsuitable to extract fluctuations in a statistical theory.

Our goal here is to provide a comprehensive and self-contained introduction to our recently developed NEQ statistical mechanics ($\mu$NEQT), in which we study deterministic time evolution of individual microstates in $\left\{{\mathfrak{m}}_k\right\}$ along Hamiltonian trajectories in $\left\{{\gamma }_k\right\}$ during $\mathcal{P}$. When quantities associated with these trajectories are averaged over them using their probabilities, the result is the MNEQT, an extension of the equilibrium thermodynamics to describe NEQ processes. This consistency with the MNEQT is not only a check on the validity of the $\mu$NEQT, but also a justification of the MNEQT by the $\mu$NEQT. The use of the SI-quantities in the $\mu$NEQT allows for directly obtaining quantities such as ${\Delta }_{\mathrm{i}}S$ after averaging. Thus, the  $\mu$NEQT is an extension of the EQ statistical mechanics [33,34], the $\mu$EQT, that was originally developed by Boltzmann [46,47] and Gibbs [48], and limited to ${\Delta }_{\mathrm{i}}S\equiv 0$.

We will follow deterministic trajectories $\left\{{\gamma }_k\right\}$ during $\mathcal{P}$ between two macrostates ${\mathfrak{M}}_{\mathrm{in}}$ and ${\mathfrak{M}}_{\mathrm{fin}}$. Only the latter determine the trajectories so they are the same for all processes $\mathcal{P}$ between them. This makes $\left\{{\gamma }_k\right\}$ independent of the trajectory probabilities $\left\{p_{{\gamma }_k}\right\}$ controlling various $\mathcal{P}$’s, which is similar to $\left\{{\mathfrak{m}}_k\right\}$ being independent of the microstate probabilities $\left\{p_k\right\}$. The extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ is chosen appropriately to uniquely specify $\left\{{\mathfrak{m}}_k\right\}$ and $\left\{{\gamma }_k\right\}$ in it. This uniqueness is an important aspect of the $\mu$NEQT and the MNEQT as it is missing in other contemporary NEQT theories [10,12,13,17,18,19,20,21,24,25,26,27,28,99,135,136,137,138,139,140,141,142,143,144,145,146,147]. The instantaneous $E_k$ along ${\gamma }_k$ can only change mechanically due to the variation in $\mathbf{W}$. This variation is responsible for the net change $\left\{\Delta E_k\right\}$ along $\left\{{\gamma }_k\right\}$, and is only determined by ${\mathfrak{M}}_{\mathrm{in}}$ and ${\mathfrak{M}}_{\mathrm{fin}}$ and not by $\left\{p_{{\gamma }_k}\right\}$ as noted above. To complete the formulation of the $\mu$NEQT, we determine the unique $\left\{p_{{\gamma }_k}\left(\mathcal{P}\right)\right\}$ for any $\mathcal{P}$ in ${\mathfrak{S}}_{\mathbf{Z}}$, which is another exceptional aspect of the $\mu$NEQT. This way, the deterministic aspect of a process (the mechanical work) has been separated from the stochastic aspect (the heat) in thermodynamics in a unique way in the $\mu$NEQT for any $\mathcal{P}$, NEQ or not. With the unique probabilities in hand, all calculation can be carried out exactly in the $\mu$NEQT, once ${\mathfrak{S}}_{\mathbf{Z}}$ has been identified. In the $\mathring{\mu }$NEQT, the trajectory probabilities need to be determined using additional steps such as using the master equation [54], Fokker–Planck equation [37,102], etc., which are phenomenological.

Being deterministic, microquantities associated with $\left\{{\mathfrak{m}}_k\right\}$ or $\left\{{\gamma }_k\right\}$ are not constrained by the second law, which is a macroscopic law based on stochasticity. This is not surprising, as the Hamiltonian dynamics has nothing to say about the second law. For the MNEQT, we need to determine various thermodynamic averages over $\left\{{\gamma }_k\right\}$ using $\left\{p_{{\gamma }_k}\right\}$. Thus, the development of the $\mu$NEQT is carried out in two independent stages. First we determine mechanical quantities as if the system is a mechanical one following Hamiltonian dynamics. Its stochastic aspects are captured by $\left\{p_{{\gamma }_k}\left(\mathcal{P}\right)\right\}$, which determine not only mechanical averages such as work but also the stochastic averages such as heat and entropy. It is the latter that finds itself manifested in the second law for appropriate choices of $\left\{p_k\right\}$ and $\left\{p_{{\gamma }_k}\right\}$. By simply modifying the second stage, we are able to investigate the catastrophic consequences of violating the second law. This proves the usefulness of our approach. With $\left\{\Delta E_k\right\}$ and $\left\{p_{{\gamma }_k}\left(\mathcal{P}\right)\right\}$ in hand, we now have a complete NEQ statistical mechanics to describe any process $\mathcal{P}$. The division in the two distinct and independent stages is of central importance to the $\mu$NEQT and the MNEQT [148,149,150,151,152,153,154,155,156,157].

We have successfully applied the $\mu$NEQT recently to study free expansion [154], to provide a correct application of microwork and microheat [155,156] in the various modern fluctuation theorems [26,158,159], and to describe viscous dissipation [157] associated with the dynamics of a Brownian particle (BP) [115,132,133,140] in its medium by developing an alternative to the stochastic Langevin description [38,99]. The above applications clearly show the usefulness of the $\mu$NEQT. However, our previous studies were mostly limited to microworks; microheats were not treated as extensively. One of our major incentives here is to overcome this limitation to determine the $\mu$NEQT for which the central requirement is the unique microstate probability $p_k$ in the state space ${\mathfrak{S}}_{\mathbf{Z}}$. This ensures that $\mathfrak{M}\left(\mathbf{Z}\right)$ and $S\left(\mathbf{Z}\right)$ are uniquely identified in ${\mathfrak{S}}_{\mathbf{Z}}$. Such macrostates are said to be in internal equilibrium (IEQ) in ${\mathfrak{S}}_{\mathbf{Z}}$ and written as ${\mathfrak{M}}_{\mathrm{ieq}}$ or $\mathfrak{M}\left(\mathbf{Z}\right)$, as opposed to EQ macrostates ${\mathfrak{M}}_{\mathrm{eq}}=\mathfrak{M}\left(\mathbf{X}\right)$ in ${\mathfrak{S}}_{\mathbf{X}}$. The  unique entropy $S\left(\mathbf{Z}\right)$ has the maximum possible value for a given $\mathbf{Z}$ so it has no memory of where the microstate has come from. Once $\mathfrak{M}$ becomes uniquely specified as $\mathfrak{M}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, it satisfies the extension of the ergodic hypothesis for ${\mathfrak{M}}_{\mathrm{ieq}}$; see Section 14 for an example.

But the applications so far of the $\mu$NEQT have provided only a piecewise and incomplete description of the $\mu$NEQT [148,149,150,151,152,153,154,155,156,157] that was restricted in scope to highlight its NEQ aspects in the limited context. This comprehensive review aims to overcome this limitation and provide a complete introduction to the foundation of the $\mu$NEQT by assimilating and extending together the previous results and by including missing details and newer aspects that emerge from the use of the SI-quantities in the extended state space ${\mathfrak{S}}_{\mathbf{Z}}$, where ${\mathfrak{m}}_k$ and $\mathfrak{M}$ are uniquely specified in an IEQ macrostate ${\mathfrak{M}}_{\mathrm{ieq}}$ just as they are uniquely specified for an EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ in the EQ state space ${\mathfrak{S}}_{\mathbf{X}}$. The $\mu$NEQT has met with success, as we will describe in this review, so it is desirable to introduce it to a wider class of readers.

Due to its microscopic SI-nature, the $\mu$NEQT provides a more detailed description of fluctuations in a thermodynamic process that are hidden in the MNEQT. For this reason, therefore, the former is highly desirable from both a theoretical and experimental point of view. It is an extension of the MNEQT [77,78,134,148,149,152,153,160] to the microstate level, which brings about a very close parallel with $\mu$EQT [32,33,34,36].

A microstate ${\mathfrak{m}}_k,k=1,2,\cdots$, carries an index k; the set $\left\{{\mathfrak{m}}_k\right\}$ forms a countable set and is specified by its energy set $\left\{E_k\left(\mathbf{W}\right)\right\}$; however, we will usually suppress $\mathbf{W}$ in ${\mathfrak{m}}_k$ and $E_k\left(\mathbf{W}\right)$, unless necessary. In a macrostate $\mathfrak{M}$, ${\mathfrak{m}}_k$’s appear with a probability $p_k\left(\mathfrak{M}\right)$; see Section 7 for details. For simplicity, we will also not explicitly show the argument $\mathfrak{M}$ in $p_k$; the dependence is always implicit. In the rest of the review, all quantities pertaining to $\mathfrak{M}$ are identified as macroquantities, while those pertaining to ${\mathfrak{m}}_k$ are identified as microquantities that always have the microstate index k of ${\mathfrak{m}}_k$ or of ${\mathbf{x}}_k$ in $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{W}\right)$; see Definition 4. After statistical averaging over microstates using their probabilities $p_k$ (see Equation (12) for its proper definition), we obtain quantities without k or $\mathbf{x}$.

A microquantity associated with ${\mathfrak{m}}_k$ will always carry the index k (see later). A macrostate $\mathfrak{M}$ and a macroquantity associated with it do not carry the index k so it is always easy to distinguish the two kinds of quantities. We will continue to use “quantity” to stand for both microquantity and macroquantity, unless clarity is needed.

1.2. System-Intrinsic and Medium-Intrinsic Thermodynamics

As the medium is always taken to be in EQ, its properties do not change even if the system is out of equilibrium. This has made the choice of MI-description ($\mathring{\mathrm{M}}\mathrm{NEQT}$) very convenient to formulate classical thermodynamics [13,18,33,39,41,42,51,108], in which one uses the exchange macroheat ${\Delta }_{\mathrm{e}}Q=T_0{\Delta }_{\mathrm{e}}S$ in terms of the exchange entropy (see Equation (46)) and the exchange macrowork ${\Delta }_{\mathrm{e}}W$ (see Equation (135c)) such as ${\Delta }_{\mathrm{e}}W=P_0{\Delta }_{\mathrm{e}}V=P_0\Delta V$ for the PV-macrowork; see Equation (94) for the first law as an example. Here, $T_0$ and $P_0$ are the temperature and pressure of the medium (see Figure 1), which remain the same for all possible states of the system. This has made the $\mathring{\mathrm{M}}\mathrm{NEQT}$ a highly desirable thermodynamic theory as it is applicable in all cases. The main problem with this theory is that it is not directly applicable to an isolated system in Figure 1b for which exchange quantities are identically zero, but which provides the most cogent formulation of the second law $\Delta S_0\ge 0$; see Equation (213) in Proposition 3. It is useful only for an interacting system in Figure 1a for which the second law is stated indirectly in terms of irreversible entropy generation ${\Delta }_{\mathrm{i}}S\ge 0$; see Equation (67c). Indeed, all irreversible quantities including irreversible macrowork are indirectly determined.

In contrast, the MNEQT provides an SI-description involving quantities associated with the system alone so it is applicable to both systems in Figure 1 by explicitly taking into account the EQ properties of the medium, when it is present. All irreversible quantities including macroworks and macroheats are contained in this approach so they are determined directly in the MNEQT.

We elaborate on the distinction between the MNEQT and the $\mathring{\mathrm{M}}\mathrm{NEQT}$. The  exchange quantities $d_{\mathrm{e}}Z$ require the system $\Sigma$ to be embedded in a medium $\tilde{\Sigma }$ (see Figure 1a) and are controlled by $\tilde{\Sigma }$ [154] so that $d_{\mathrm{e}}Z=-d_{\mathrm{e}}\tilde{Z}$ (see Section 2) and  are easy to handle and measure, as $\tilde{\Sigma }$ is normally taken to be in equilibrium with no irreversibility ($d_{\mathrm{i}}\tilde{Z}=0$) so that $d_{\mathrm{e}}\tilde{Z}=d\tilde{Z}$. Thus, the exchange quantities do not directly provide any information about $d_{\mathrm{i}}Z$ and any irreversibility as mentioned above. As an example, the lost macrowork due to irreversibility in the $\mathring{\mathrm{M}}\mathrm{NEQT}$ is defined as

$${\mathring{d}}_{\mathrm{lost}}W={\mathring{d}}_{\mathrm{rev}}W-{\mathring{d}}_{\mathrm{irr}}W\ge 0,$$

where various ${\mathring{d}}_{\mathrm{rev}}W$ and ${\mathring{d}}_{\mathrm{irr}}W$ refer to the exchange macroworks along two distinct processes: a reversible and an irreversible. We have used a new notation $\mathring{d}$ to ensure that any $\mathring{d}W$ is not confused with $dW$ in the MNEQT. It is easy to see that ${\mathring{d}}_{\mathrm{lost}}W$ is precisely the irreversible macrowork $d_{\mathrm{i}}W$, which is determined by the actual process.

Similar distinctions can also be noted between the $\mathring{\mu }$NEQT and the $\mu$NEQT; they differ at least in the following important ways, with sweeping consequences, as we will see:

A.

The internal microwork ${\Delta }_{\mathrm{i}}{W}_k$ has no analog in the former because it uses the following questionable conjecture:

$${\Delta }_{\mathrm{e}}{W}_k\stackrel{?}{=}-\Delta E_k,$$

(7)

(see Section 15) which is often used in fluctuation theorems [99,135,136,137,138,139,140,141,142,143,144,145,146,147]; the use of $\stackrel{?}{=}$ is a reminder of its possible questionable nature, which is justified later in Theorem 7. In these fluctuation theorems, one begins with the conventional form of the first law $dE=d_{e}Q-d_{e}W$ in terms of exchange macroquantities, but identifies

$$d_{e}W\stackrel{?}{=}-{\sum }_k{p}_{k}d{E}_k,d_{e}Q\stackrel{?}{=}{\sum }_k{E}_{k}d{p}_k.$$

As a consequence of the above identification, no distinction can be made between fluctuating microwork

$$d{W}_k\equiv -d{E}_k,$$

which is an identity in accordance with the work–energy theorem (see Theorem 6) and nonfluctuating exchange microwork

$$d_e{W}_k=d_{e}W,\forall k;$$

see Theorem 7. The distinction is always maintained in the latter, in which we also show (see Section 10.1) why the above identification cannot be rigorously justified. Similar conclusions as above are obtained by replacing infinitesimal $d_{\alpha }$ by accumulation ${\Delta }_{\alpha }$, properly defined in Section 13 along a process $\mathcal{P}$.

B.

Consequently, the microforce imbalance ($\mu$FI) that results in fluctuating ${\Delta }_{\mathrm{i}}{W}_k=-{\Delta }_{\mathrm{i}}{E}_k$, a ubiquitous quantity, is absent in the former in that ${\Delta }_{\mathrm{i}}{W}_k=\Delta W_k-{\Delta }_{\mathrm{e}}{W}_k\equiv 0$ but is always present (${\Delta }_{\mathrm{i}}{W}_k\ne 0$) in the latter.

C.

The former results in a first law of thermodynamics ($\Delta E_k={\Delta }_{\mathrm{e}}{Q}_k-{\Delta }_{\mathrm{e}}{W}_k$) for each ${\mathfrak{m}}_k$, while the latter has it hold ($\Delta E={\Delta }_{\mathrm{e}}Q-{\Delta }_{\mathrm{e}}W$) only for a $\mathfrak{M}$; however, see Equation (243).

D.

The lost or dissipated macrowork ${\Delta }_{\mathrm{lost}}W$ measured by the average ${\Delta }_{\mathrm{i}}{W}_k$ should be absent in the former due to its above conjecture, but is always present in the latter.

E.

The exchange microwork ${\Delta }_{\mathrm{e}}{W}_k$ depends on the entire trajectory ${\gamma }_k$ in the former to make it fluctuating over ${\gamma }_k$, while in the latter, $\Delta W_k$ depends only on the terminal microstates of ${\gamma }_k$, and ${\Delta }_{\mathrm{e}}{W}_k\equiv {\Delta }_{\mathrm{e}}W$ is nonfluctuating (it is the same for all ${\gamma }_k$’s).

1.3. Main Results

The review emphasizes the very close parallel with EQ statistical mechanics ($\mu$EQT) that is clearly seen in the microstate probabilities and the existence of IEQ partition functions for ${\mathfrak{M}}_{\mathrm{ieq}}$. There are also major differences mainly in new concepts, some of which are very counter-intuitive, such as ubiquitous $d_{\mathrm{i}}{E}_k$, microforce imbalance ($\mu$FI) and internal microwork $d_{\mathrm{i}}{W}_k$ resulting from it, etc., for any macrostates including ${\mathfrak{M}}_{\mathrm{eq}}$ that have not been appreciated so far. They have been introduced previously [77,78,150,156,157] but now receive detailed explanation here. For example, it is a well-known fact that $d_{\mathrm{i}}E=0$ [12] (see Equation (53a)) for any $\mathfrak{M}$; yet $d_{\mathrm{i}}{E}_k$ is fluctuating and so can be different from zero, its average. The presentation here is simple enough to reach even an untrained reader. To accomplish this goal, we only focus on some examples borrowed from undergraduate physics so that a reader will not be lost; however, it does require an open mind to learn new concepts that are counter-intuitive and perplexing, as it is very hard to shake off old preconceptions.

Remark 1. 

As μEQT only deals with EQ processes, the second law plays no role here. However, the situation in the μNEQT is different, where we deal with NEQ processes. As the second law does not operate at the microstate level, our development of the μNEQT is not limited by this law. To make contact with thermodynamics, however, we will have to impose it at the level of macrostate. By investigating the internal inconsistencies that emerge if the second law is violated, we are able to conclude that the law cannot violated for a stable system. This is one of the most important benefits of our approach.

Throughout this review, we work in the enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$ so we include at least one internal variable $\xi$ as a prototype to make our discussion more realistic, as will become clear in Section 4 and Section 14. The main emphasis here will be to demonstrate the ubiquitous nature of internal changes such as $d_{\mathrm{i}}{E}_k$, a new concept whose existence has not been previously appreciated in various fluctuation theorems [26,158,159]. Not recognizing its existence has resulted in the conjecture $d_{\mathrm{e}}{W}_k=-d{E}_k=-$$d_{\mathrm{e}}{E}_k$ (see Equation (7)), used extensively in the $\mathring{\mu }$NEQT. This is contrary to a central result of the $\mu$NEQT; see Theorem 6. It is the microforce imbalance ($\mu$FI) between the internal and external microforces, a hitherto unrecognized purely mechanical concept at the microstate level in EQ and NEQ thermodynamics, that generates $d_{\mathrm{i}}{E}_k$ and is present in all processes, whether they are thermodynamic or not, as we will demonstrate. This is the most important outcome of the our approach; see Proposition 2. It emphasizes the importance of SI-quantities (such as in $d{E}_k=-d{W}_k$) that are very different from the MI-quantities (such as in $d_{\mathrm{e}}{E}_k=d{\tilde{W}}_k$) for any ${\gamma }_k$, even if the trajectory belongs to a reversible process. The use of generalized work $dW=-d{E}_{\mathrm{m}}$ in Equation (234a) as isentropic change allows us to calculate microscopic work (microwork) $d{W}_k$, which changes $E_k$ but not $p_k$. This is because ${\mathfrak{m}}_k$, whose concept is independent of $p_k$, uniquely determines $E_k$ for a fixed work set $\mathbf{W}$; see Definition 5. Therefore, $d{E}_k$ is uniquely determined by $d\mathbf{W}$ and does not have any contribution from the change in $p_k$. On the other hand, the generalized heat $dQ$ allows us to introduce microscopic heat (microheat) $d{Q}_k$, which does not change $E_k$ but changes $p_k$. The above mutually exclusive nature of $d{W}_k$ and $d{Q}_k$ proves to be a great simplification and allows us to treat $d{W}_k$ and $d{Q}_k$ as purely a mechanical and a stochastic concept, respectively, in the development of the $\mu$NEQT. In addition, as $d{E}_k$ does not have any contribution from $d{p}_k$, it has no microheat contribution, so there is no first law for ${\mathfrak{m}}_k$ in the $\mu$NEQT.

As $E_k$ is fluctuating, $d{E}_k=-d{W}_k$ is also fluctuating and is uniquely determined as $d{E}_k\doteq E_k(\mathbf{W}+d\mathbf{W})-E_k\left(\mathbf{W}\right)$ for ${\mathfrak{m}}_k$; the (slow or fast) nature of the process is irrelevant. The latter only controls $p_k$. This provides a simplification in evaluating the cumulative change $\Delta W_k$, which is independent of the nature of $\mathcal{P}$ between two macrostates; see Remark 71 and the discussion following it. The  fluctuating microwork $d{W}_k$ is different from $\Delta {\tilde{W}}_k=\Delta \tilde{W}$, which is the microwork done by the working medium on ${\mathfrak{m}}_k$ after reduction, and which depends strongly on the nature of $\mathcal{P}$ but is the same for all microstates for a given $\mathcal{P}$.

The most important new results that emerge in the $\mu$NEQT are the following:

• a clear separation of different kinds of work and heat and their fluctuations that emerge from $d_{\alpha }$;
• additional thermodynamic forces for irreversibility due to internal variables;
• stochasticity resulting from a nonvanishing commutator ${\widehat{C}}_{\alpha }\doteq d_{\alpha }\widehat{A}-\widehat{A}{d}_{\alpha }$;
• exchange microquantities are nonfluctuating, which makes them useless for directly obtaining fluctuations and irreversibility;
• the fundamental identity ${\Delta }_{\mathrm{i}}W={\Delta }_{\mathrm{i}}Q$ between irreversible macrowork and macroheat generalizing the result of Count Rumford and the Gouy-Stodola theorem;
• the origin of work dissipation ${\Delta }_{\mathrm{i}}W>0$ in an irreversible process;
• the uniqueness of macrostates and microstate probabilities in the enlarged state space for $\mathfrak{M}\left(\mathbf{Z}\right)$ determined by the experimental setup;
• the $\mu$NEQT justifies the MNEQT as the $\mu$EQT justifies the EQT.

1.4. Layout

The layout of the paper is the following. In the next section, we introduce our notation, definitions, and new concepts, which may be unfamiliar to many readers but are justified in the following sections. We describe here our basic approach that a thermodynamic description is equivalent to treating microquantities as purely mechanical without any consideration of stochasticity, to be followed by bringing in microstate probabilities to determine macroquantities, just as in EQ statistical mechanics. Microstate probabilities are not truly microquantities as they are not independent of each other. The stochasticity adds the dimension of entropy, without which we only have a mechanical description of an NEQ body in ${\mathfrak{S}}_{\mathbf{Z}}$. An arbitrary macrostate ${\mathfrak{M}}_{\mathrm{arb}}$ is divided into an EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ and an NEQ macrostate ${\mathfrak{M}}_{\mathrm{neq}}$; the latter is further divided into an IEQ (internal equilibrium) macrostate ${\mathfrak{M}}_{\mathrm{ieq}}$ and an NIEQ (non-internal equilibrium) macrostate ${\mathfrak{M}}_{\mathrm{nieq}}$. The IEQ macrostates share all the properties of EQ macrostates, except that the former have nonvanishing irreversible entropy generation ${\Delta }_{\mathrm{i}}S>0$. The principle of reduction is also introduced here. In Section 3, we discuss the mathematical properties of and manipulations with the linear operators $d_{\alpha }$, and give some examples for clarification. The origin of internal variables is explained in Section 4, where we show that they also emerge in mechanical descriptions so that they are not unique to thermodynamics. This explains why we need the enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$ for microscopic mechanical descriptions as well. We finally present the fundamentals of the $\mu$NEQT in Section 5. This is a very important section, where we present various axioms and requirements of the $\mu$NEQT. We then discuss stochasticity to derive a very general formulation of the entropy in terms of $\left\{p_k\right\}$, which is then used to obtain the unique form of $\left\{p_k\right\}$ for ${\mathfrak{M}}_{\mathrm{ieq}}$. An important and surprising aspect of the $\mu$NEQT is obtained in the equality of internal microwork (a mechanical microquantity) and microheat (a stochastic microquantity) even though they have distinct origins. At this stage, we have a complete and unique NEQ statistical mechanics (the $\mu$NEQT) in ${\mathfrak{S}}_{\mathbf{Z}}$. We identify SI-macroquantities and use them to derive the MNEQT for ${\mathfrak{M}}_{\mathrm{ieq}}$ exemplified by the Gibbs fundamental relation in ${\mathfrak{S}}_{\mathbf{Z}}$, which is then generalized to obtain the Gibbs fundamental relations for ${\mathfrak{M}}_{\mathrm{nieq}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$.

In Section 6 and Section 7, we begin to introduce the mechanical and stochastic aspects of the $\mu$NEQT, respectively. In Section 6, we use $\mathbf{W}$ to identify microforces that operate in the mechanical formulation of the body so they are also present in its thermodynamic formulation. We use them to introduce the concept of microforce imbalance in Section 6.4, which captures the mechanical disparity between $\Sigma$ and $\tilde{\Sigma }$. The imbalance is responsible for the internal microwork. In Section 6.5, we derive the extension of the work–energy theorem of mechanics in ${\mathfrak{S}}_{\mathbf{Z}}$. In Section 7, we revisit a previous proposal for the origin of stochasticity and extend it further by discussing the effect of correlations between $\Sigma$ and $\tilde{\Sigma }$, and introducing the principle of reduction in Section 7.2. We then discuss quasi-independence in Section 7.3, and the simplification it brings about in thermodynamic considerations after reduction, especially with respect to the effects produced by $\tilde{\Sigma }$ on $\Sigma$, which is discussed in Section 7.4 and Section 7.5. The discussion, which forms a very important part of the review, shows why classical thermodynamics works so well.

In Section 8, we discuss the properties of the unique entropy $S_{\mathrm{ieq}}$ for ${\mathfrak{M}}_{\mathrm{ieq}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$, and discuss its approximate formulation as a flat distribution that is commonly used in EQ statistical mechanics. This distribution neglects any fluctuations in the entropy, which are always present in the body. Despite this, it correctly gives the entropy so it can always be used to determine it as it simplifies the calculation. We show that the entropy additivity requires quasi-independence in Section 8.1 so the latter should not be confused with the principle of additivity for $\mathbf{W}$. Using this flat distribution, we provide a simple proof of the second law for ${\mathfrak{M}}_{\mathrm{ieq}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ in Section 8.3 by simply counting the number of distinct microstates as the system evolves in time, which can only increase with time; see Theorem 8. This direct proof is supplemented by Theorem 9 in Section 8.4 that the law is simply a direct consequence of the stability of the system so it does not need to be included as an additional part of Axiom 2 in the $\mu$NEQT; see Section 5). In Section 9, we show that a violation [161] of the second law results in internally inconsistent thermodynamics for stable physical systems, and cannot be taken seriously (see Conclusion 7), even though thermodynamic instabilities arise in approximate calculations such as van der Waals equations or mean field, but are always removed from consideration; see Remark 58. Therefore, we will always assume that we are dealing with a stable system for which the law is always valid, as noted in Section 1, except in Section 9. In Section 10, we initiate the formulation of the $\mu$NEQT by focusing on the two most important concepts, those of generalized or BI-microwork and microheat for ${\Sigma }_{\mathrm{b}}$. We show that various micro- and macroheats emerge from the nonvanishing commutator ${\widehat{C}}_{\alpha }$ introduced in Equation (229). For a fuller understanding, we first revisit in Section 10.1 the ensemble average of a fluctuating state variable, and its change in a process $\mathcal{P}$. We show that for $Z\in \mathbf{Z}$ such as E belonging to ${\mathfrak{S}}_{\mathbf{Z}}$, its change $dZ$ consists of two independent process contributions in orthogonal state spaces ${\mathfrak{S}}_{\mathbf{Z}}$ and ${\mathfrak{S}}_S$, a mechanical one $d{Z}_{\mathrm{m}}$ at fixed $\left\{p_k\right\}$ in ${\mathfrak{S}}_{\mathbf{Z}}$, and a stochastic one $d{Z}_{\mathrm{s}}$ at fixed $\left\{Z_k\right\}$ in ${\mathfrak{S}}_S$. Thus, $dZ\doteq d〈Z〉\in {\mathfrak{S}}_{\chi }$. In contrast, the stochastic state variable $S\in {\mathfrak{S}}_S$ has only stochastic contributions belonging to ${\mathfrak{S}}_S$. For E, $d{E}_{\mathrm{m}}$ represents the negative of the generalized macrowork $dW$, and $d{E}_{\mathrm{s}}$ the generalized macroheat $dQ$ in the body. Their statistical interpretation is covered in Section 10.2, where we show that $d{W}_k$ is purely mechanical, and $d{Q}_k$ purely stochastic. In Section 11, we discuss how $d_{\mathrm{e}}{p}_k$ and $d_{\mathrm{i}}{p}_k$ are determined, and how they determine the forms of various microworks, microheats, and microentropies. We also give a general proof of the identity $d_{\mathrm{i}}E\equiv 0$, even if $d_{\mathrm{i}}{E}_k\ne 0,\forall k$. This now completes the formulation of the unique NEQ statistical mechanics ($\mu$NEQT) in ${\mathfrak{S}}_{\mathbf{Z}}$.

The only thing remaining for a complete formulation of the $\mu$NEQT is to identify the choice of ${\mathfrak{S}}_{\mathbf{Z}}$, which is discussed in Section 12. This is a very important section that describes how the choice of ${\mathfrak{S}}_{\mathbf{Z}}$ is dictated by the way an experiment is performed, which must not come as a surprise for an NEQ process. This is because the observation and relaxation times play important roles here. By ordering various internal variables with their relaxation times in decreasing order, we show that only those internal variables have to considered whose relaxation times are greater than the observation time to uniquely specify the macrostate in ${\mathfrak{S}}_{\mathbf{Z}}$. We show how the unique microstate probability is identified. We consider the possibilities of fluctuating (Fl) and nonfluctuating (NFl) work parameter $\mathbf{W}$. It will be convenient to take the parameters to be fixed so that they are the same for all microstates. We introduce the Legendre transform $E_k^{\mathrm{L}}$ of the microenergy $E_k$, which proves to be very useful in expressing $p_k$. The discussion justifies that once ${\mathfrak{S}}_{\mathbf{Z}}$ has been identified in which $\mathfrak{M}$ becomes uniquely specified, the microstate probabilities are also uniquely specified. No auxiliary step is required to determine $p_k$. This is what makes the $\mu$NEQT so useful. The discussion is easily extended to consider a microstate that is not unique in ${\mathfrak{S}}_{\mathbf{Z}}$.

So far, we have provided a complete formulation of the $\mu$NEQT for any ${\mathfrak{M}}_{\mathrm{arb}}$ at each instant. To proceed further to extend the $\mu$NEQT for any process, we need to introduce a trajectory ensemble and determination of various path and process quantities, which is taken over in Section 13. We show that different trajectory quantities have different trajectory probabilities (path microprobabilities), which has not been appreciated so far. This finally provides a complete description of the $\mu$NEQT for any process.

We now turn to some of the applications of the $\mu$NEQT in the next three sections. In Section 14, we use it to describe the origin of microfricton at the microstate level. A new NEQ work fluctuation theorem is derived in Section 15 between any two arbitrary macrostates. In Section 16, we use the $\mu$NEQT to study the quantum and classical free expansion using our work fluctuation theorem. The final section provides a brief discussion of our conclusions and a summary.

2. Notation, Definitions and New Concepts

Before proceeding further, it is useful to introduce in this section our notation to describe various systems and their behavior, and new concepts for their understanding without much or any explanation (that will be offered later in the review where we discuss them). We also give various definitions and briefly discuss new concepts such as various forms of NEQ work and heat that need to be carefully distinguished for a precise formulation of the $\mu$NEQT. Various important concepts are highlighted in the form of Remarks to draw the attention of the reader. It is the hope that a reader can always come back here to be refreshed in case of confusion. In this sense, this section plays an important role in the review for the purpose of bookkeeping.

2.1. Systems and State Variables

Definition 1. 

A system Σ is a collection of material particles and radiation enclosed in a region of space defined by some parameter $\mathbf{W}$, and its Hamiltonian dynamics is determined by the Hamiltonian $\mathcal{H}\left(\mathbf{x}\left|\mathbf{W}\right.\right)$. A system can be embedded in a medium $\tilde{\Sigma }$, which is extremely large compared to it, and with which it interacts. The combined system ${\Sigma }_0$ formed by Σ and $\tilde{\Sigma }$ is commonly treated as an isolated system.

Remark 2. 

For convenience, we take the parameter $\mathbf{W}$ to be fixed so that it is the same for all microstates, even though it is not hard to take it to be unfixed so that it changes over the microstates. We will refer to them as nonfluctuating and fluctuating, respectively.

We draw attention to Figure 1 to introduce the notation. The  review mostly deals with statistical mechanics of macroscopically large systems $\Sigma$; however, we will also digress a bit to discuss small systems. In both cases, $\Sigma$ is extremely small compared to the medium $\tilde{\Sigma }$; see Figure 1b. The medium $\tilde{\Sigma }$ consists of two parts: a work source ${\tilde{\Sigma }}_{\mathrm{w}}$ and a heat source ${\tilde{\Sigma }}_{\mathrm{h}}$, both of which can interact with $\Sigma$ directly but not with each other. This separation allows us to study work and heat exchanges between $\Sigma$ and $\tilde{\Sigma }$ separately. We will continue to use $\tilde{\Sigma }={\tilde{\Sigma }}_{\mathrm{w}}\cup {\tilde{\Sigma }}_{\mathrm{h}}$ to refer to both of them together. The collection ${\Sigma }_0=\Sigma \cup \tilde{\Sigma }$ forms an isolated system, which we assume to be stationary. We remark that the concept of an isolated system in a laboratory is an important approximation [79,162] but extremely useful as no such system locally exists in reality. We need to always keep this in mind.

The system in Figure 1a is an isolated system, which we may not be able to divide into a medium and a system. Each medium in Figure 2, although not interacting with each other, has a similar relationship with $\Sigma$. In case they were mutually interacting, they can be treated as a single medium. The collection ${\Sigma }_0=\Sigma \cup {\tilde{\Sigma }}_1\cup {\tilde{\Sigma }}_2$ forms an isolated system. In the following, we will mostly focus on Figure 1 to introduce the notation, which can be easily extended to Figure 2 or to an extension with several mediums.

Definition 2. 

Observables $\mathbf{X}=(E,V,N,\cdots )$ of a system are extensive quantities that can be controlled from outside the system, and internal variables  $\xi =({\xi }_1,{\xi }_2,{\xi }_3,\cdots )$ are extensive quantities that cannot be controlled from outside the system. Their collection $\mathbf{Z}=\mathbf{X}\cup \xi$ is called the set of extensive state variables of Σ forming ${\mathfrak{S}}_{\mathbf{Z}}$, which we may simply write as $\mathfrak{S}$ when no confusion will arise. The set $\mathbf{W}$ or a subset of it may be fixed or may fluctuate over all microstates of Σ.

Definition 3. 

A system-intrinsic (SI) quantity is a quantity that pertains to the system Σ alone and can be used to characterize the system. A medium-intrinsic (MI) quantity is a quantity that is determined by the medium $\tilde{\Sigma }$ alone and can be used to characterize it and also the exchanges between Σ and $\tilde{\Sigma }$. No external exchange is allowed for ${\Sigma }_0$.

We use a suffix 0 to denote all quantities pertaining to ${\Sigma }_0$, a tilde $\left(\tilde{}\right)$ for all quantities pertaining to $\tilde{\Sigma }$, and no suffix for all quantities pertaining to $\Sigma$, even if it is isolated. Thus, the set of observables is denoted by ${\mathbf{X}}_0,\tilde{\mathbf{X}}$, and $\mathbf{X}$, respectively, and the set of state variables by ${\mathbf{Z}}_0,\tilde{\mathbf{Z}}$, and $\mathbf{Z}$, respectively, in the state space ${\mathfrak{S}}_{\mathbf{Z}}$; the set of internal variables are ${\xi }_0,\tilde{\xi }$, and $\xi$, respectively.

Remark 3. 

We will use the term “body” to refer to any of  $\Sigma ,\tilde{\Sigma },$ and ${\Sigma }_0$ in this review and use ${\Sigma }_b$ to denote it. However, to avoid notational complication, we will use the notation suitable for Σ for ${\Sigma }_b$ if no confusion would arise in the context. The mechanical aspect of a body is described by its Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$, and we refer to all quantities pertaining to it as body-intrinsic (BI), which includes SI, MI, and ISI (for the isolated system) as the case may be.

The discussion below is mostly for a body ${\Sigma }_{\mathrm{b}}$, but the notation is suited for a system. Thus, it covers the three systems $\Sigma ,\tilde{\Sigma }$, and ${\Sigma }_0$, unless mentioned otherwise.

Definition 4. 

A microstate of $\mathcal{H}\left(\mathbf{x}\left|\mathbf{W}\right.\right)$ represents the instantaneous deterministic state of ${\Sigma }_b$. The quantum microstates are specified by a set of good quantum numbers, which we usually denote by k as a single quantum number for simplicity; we take $k\in \mathbb{N},\mathbb{N}$ denoting the set of natural numbers. In the classical case, we use a small cell $\delta {\mathbf{x}}_k$ of volume $h^{3N}$ around ${\mathbf{x}}_k=\mathbf{x}$ as the microstate ${\mathfrak{m}}_k$ [163]; the collection $\left\{\delta {\mathbf{x}}_k\right\}$ covers the entire phase space Γ. A microstate ${\mathfrak{m}}_k$ appears with probability $p_k$ that is central for statistical mechanics.

Below, we clarify the definition further.

2.2. Microstates and Macrostates

Remark 4. 

In order to obtain a microscopic understanding of thermodynamics, we need to focus on the countable set of microstates ${\left\{{\mathfrak{m}}_k\right\}}_{k=1,2,\cdots }$. Then

$$E_k=\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{W}\right)$$

(8)

denotes the microenergy of ${\mathfrak{m}}_k$. In explicit form, the microenergy for ${\mathfrak{m}}_k$ of Σ will be expressed as $E_k\left(\mathbf{W}\right)$, for ${\tilde{\mathfrak{m}}}_{\tilde{k}}$ of $\tilde{\Sigma }$ it will be expressed as ${\tilde{E}}_{\tilde{k}}\left(\tilde{\mathbf{w}}\right)$ (see Equation (28a)), and for ${\mathfrak{m}}_{0k_0}$ of ${\Sigma }_0$ it will be expressed as $E_{0k_0}(\mathbf{W},\tilde{\mathbf{w}})$ (see Equation (28b)).

Remark 5. 

For clarity and ease of presentation, we will assume each microstate to be nondegenerate, i.e., a singlet. Extending the discussion to degenerate microstates is trivial, as discussed in Section 15.

We now identify microstates $\left\{{\mathfrak{m}}_k\right\}$. In quantum mechanics, they refer to the countable microstates of the Hamiltonian of a bounded body, with k denoting the set of quantum numbers. In classical mechanics, they are usually identified as follows. We will normally employ a discretization of the classical phase space $\Gamma$ of a bounded system by dividing it into countable nonoverlapping cells $\delta {\mathbf{x}}_k$, centered at ${\mathbf{x}}_k$ and of some small size, commonly taken to be ${\left(2\pi \hslash \right)}^{3N}$. The cells cover the entire phase space $\Gamma$. To  account for the identical nature of the particles, the number of cells and the volume of the phase space are assumed to be divided by $N!$ to count distinct microstates ${\mathfrak{m}}_k\doteq \delta {\mathbf{x}}_k$, indexed by k $=1,2,\cdots$; the center of $\delta {\mathbf{x}}_k$ is at ${\mathbf{x}}_k$. The energy and probability of these cells are denoted by $\left\{E_k,p_k\right\}$ in which $E_k\left(\mathbf{W}\right)$ is a function of $\mathbf{W}$. The microstates obey deterministic evolution of the Hamiltonian $\mathcal{H}\left(\mathbf{x}\left|\mathbf{W}\right.\right)$ of the body. For ${\Sigma }_{\mathrm{b}}=\tilde{\Sigma }$, $\left\{{\tilde{\mathfrak{m}}}_{\tilde{k}}\right\}$ appear with probabilities $\left\{{\tilde{p}}_{\tilde{k}}\right\}$; for ${\Sigma }_{\mathrm{b}}={\Sigma }_0$, $\left\{{\mathfrak{m}}_{0,k_0}\right\}$ appear with probabilities $\left\{p_{0k_0}\right\}$.

With the discretization, we will use the same symbol $\Gamma$ to denote the space occupied by microstates ${\mathfrak{m}}_k$.

Claim 1. 

It is through the changes in microstate probabilities that a thermodynamic process $\mathcal{P}$ gets its stochastic nature. In contrast, constant $p_k$’s describe a mechanical process, which is deterministic.

A thermodynamic process $\mathcal{P}$ between any two arbitrary states (we will instead use $\mathring{\mathcal{P}}$ to denote a process between two equilibrium (EQ) terminal microstates) is understood in the context of the MNEQT [12,51] as a temporal sequence of macrostates $\mathfrak{M}\left(t\right)$ of the body which keep changing during $\mathcal{P}$ due to changes in $\left\{{\mathfrak{m}}_k\left(t\right)\right\}$ and/or $\left\{p_k\left(t\right)\right\}$. The rate of time variation (fast or slow compared to the equilibration time ${\tau }_{\mathrm{eq}}$) determines the (reversible or irreversible) nature of $\mathcal{P}$.

Definition 5. 

At the microscopic level, the state of ${\Sigma }_b$ is specified by microstates set $\left\{{\mathfrak{m}}_k\right\}$, their energy set $\left\{E_k\right\}$, and their probability set $\left\{p_k\right\}$. For the same set $\left\{{\mathfrak{m}}_k,E_k\right\}$, different choices of $\left\{p_k\right\}$ describe different macrostates $\mathfrak{M}$ (see Definition 6), one of which, ${\mathfrak{M}}_{eq}$, corresponding to $\left\{p_k^{eq}\right\}$, specifies an EQ macrostate having the maximum entropy; all other states have smaller entropies and are called nonequilibrium (NEQ) macrostates.

It is important to draw attention to the following important distinction between the Hamiltonian $\mathcal{H}$ required for a microstate and the average energy E of a macrostate. While the thermodynamic energy accounts for the stochasticity through microstate probabilities, the use of the Hamiltonian is going to be restricted to a particular microstate. In other words, the Hamiltonian depends on $\mathbf{x}$ and $\mathbf{W}$ but the energy depends on the entropy S and $\mathbf{W}$. The energy $E_k$ of ${\mathfrak{m}}_k$, on the other hand, depends only on $\mathbf{W}$ and denotes the value of $\mathcal{H}$ associated with ${\mathfrak{m}}_k$; see Equation (8). In the following, we will always treat Hamiltonians and microstate energies as equivalent descriptions, which does not depend on knowing $\left\{p_k\right\}$; the average energies depend on $\left\{p_k\right\}$ for their definition; see Equation (12) with $\mathsf{q}=E$ and ${\mathsf{q}}_k=E_k$.

Definition 6. 

A macrostate $\mathfrak{M}$ in $\mathfrak{S}$ is a collection $\left\{{\mathfrak{m}}_k,p_k\right\}$ of microstates ${\mathfrak{m}}_k$ and their probabilities $p_k,k=1,2,\cdots$ for a ${\Sigma }_b$. Quantities that are the same for all microstates are called macroquantities as they refer to the macrostates $\mathfrak{M}$. Quantities that refer to microstates are called microquantities, and  carry the suffix k when associated with the microstate ${\mathfrak{m}}_k$ such as ${\mathbf{X}}_k$ or ${\mathbf{Z}}_k$, which are the microanalogs of $\mathbf{X}$ or $\mathbf{Z}$, respectively; however, see Remark 14. We will simply use “quantity” to refer to both of these quantities in short.

For example, we will refer to $d{W}_k$ as the microwork; similarly, we will refer to $d{\tilde{W}}_k$ as the external microwork, $d_{\mathrm{e}}{W}_k$ as the exchange microwork, and $d_{\mathrm{i}}{W}_k$ as the internal microwork. The corresponding macroworks are denoted by $dW,d\tilde{W},d_{\mathrm{e}}W$, and $d_{\mathrm{i}}W$. We thus see that there are various possible notions of works in NEQT.

A macrostate $\mathfrak{M}$ is usually described by the state variable $\mathbf{Z}$ in thermodynamics but functions of $\mathbf{Z}$ can also be used to characterize $\mathfrak{M}$. They are all macroquantities. In statistical mechanics, microstates of the Hamiltonian are used to describe $\mathfrak{M}$ at the microstate level.

Remark 6. 

Microquantities can be divided into two kinds: pure and mixed. A pure microquantity such as $E_k$ is determined solely by ${\mathfrak{m}}_k$ but not by $\mathfrak{M}$. A mixed microquantity such as microheat and microentropy is one that is also determined by $\mathfrak{M}$. With this caveat in mind, we will call both kinds microquantities.

We find the shorthand notation [12,13,51]

$$d_{\alpha }=(d,d_{\mathrm{e}},d_{\mathrm{i}})$$

(9)

quite useful in the following for the various infinitesimal contributions. Thus, $d_{\alpha }{E}_k=d{E}_k,d_{\mathrm{e}}{E}_k,d_{\mathrm{i}}{E}_k$ will refer to microenergy change, exchange microenergy change, and internal microenergy change, respectively. We similarly use $d_{\alpha }{Q}_k=d{Q}_k,d_{\mathrm{e}}{Q}_k,d_{\mathrm{i}}{Q}_k$ for various forms of microheats, and $d_{\alpha }{S}_k=d{S}_k,d_{\mathrm{e}}{S}_k,d_{\mathrm{i}}{S}_k$ for various forms of microentropies; see Equation (27a) and Remark 14. In particular, the random variable dq should not be confused with the differential of q, which may not even be defined; see Remark 20. We will refer to $d{Q}_k$ and $d{S}_k$ as microheat and microentropy, respectively. The corresponding macroquantities are denoted by $d_{\alpha }E,d_{\alpha }Q$, and $d_{\alpha }S$, respectively, without the index k. The following notation generalizes the physics of various infinitesimals and their relationship.

2.3. Micro–Macro Variables

Notation 1. 

We introduce the sets of state variables

$${\chi }_k\doteq \left\{S_k,{\mathbf{Z}}_k\right\},\chi \doteq \left\{S,\mathbf{Z}\right\},{\zeta }_k\doteq \left\{S_k,{\mathbf{W}}_k\right\},\zeta \doteq (S,\mathbf{W}),$$

(10a)

and infinitesimals

$$d_{\alpha }{\theta }_k\doteq \left\{d_{\alpha }{\chi }_k,d_{\alpha }{W}_k,d_{\alpha }{Q}_k\right\},d_{\alpha }\theta \doteq \left\{d_{\alpha }\chi ,d_{\alpha }W,d_{\alpha }Q\right\}.$$

(10b)

Notation 2. 

We introduce a compact notation $\left[q\right]$ for the collection $\left\{q_k,q\right\}$:

$$\left[q\right]\doteq \left\{q_k,q\right\}.$$

(11a)

and $\left[d_{\alpha }q\right]$ to cover all of the following quantities:

$$\left[d_{\alpha }q\right]\in \left[d_{\alpha }\theta \right]\doteq \left\{\left[d_{\alpha }\chi \right],\left[d_{\alpha }W\right],\left[d_{\alpha }Q\right]\right\}.$$

(11b)

Thus, $\left[\chi \right]\doteq \left\{{\chi }_k,\chi \right\}\in {\mathfrak{S}}_{\chi },$$\left[\zeta \right]\doteq \left\{{\zeta }_k,\zeta \right\}\in {\mathfrak{S}}_{\zeta },\left[d_{\alpha }\chi \right]\doteq \left\{d_{\alpha }{\chi }_k,d_{\alpha }\chi \right\}\in {\mathfrak{S}}_{\chi }$, etc. For specificity, we use ${\chi }_k^j$ and ${\chi }^j$ to refer to the jth element of ${\chi }_k$ and $\chi$, respectively. Similarly, we use ${\zeta }_k^j,{\zeta }^j$ for the jth element of ${\zeta }_k$ and $\zeta$, respectively, and $d_{\alpha }{\theta }_k^j,d_{\alpha }{\theta }^j$ for the jth element of $d_{\alpha }{\theta }_k$ and $d_{\alpha }\theta$, respectively.

2.4. Random Variable and Average

Remark 7. 

In the language of probability theory, $\mathfrak{M}\left(t\right)$ can be thought of as a random variable with outcomes ${\mathfrak{m}}_k$ with probability $p_k\left(t\right)$. A microquantity ${\mathsf{q}}_k$ associated with ${\mathfrak{m}}_k$ appears with probability $p_k\left(t\right)$ at time t. Thus, ${\mathsf{q}}_k$ denotes an outcome of a random variable q, and usually forms a fluctuating (Fl) microquantity.

Definition 7. 

The ensemble average for $\left\{{\mathsf{q}}_k\right\}$ or of the random variable  q  is defined by

$$\widehat{A}\mathsf{q}\left(t\right)=\mathsf{q}\left(t\right)\mathrm{or}\overline{\mathsf{q}}\left(t\right)\mathrm{or}〈\mathsf{q}〉\left(t\right)\equiv \sum _k{p}_k\left(t\right){\mathsf{q}}_k$$

(12)

for a countable set $\left\{p_k\right\}\left(t\right)$ that satisfies the sum rule

$${\sum }_k{p}_k\left(t\right)=1$$

(13)

due to the conservation of probability. We can also extend Equation (12) to q for which ${\mathsf{q}}_k=\mathsf{q},\forall k$. We have used $\widehat{A}$ to denote the above averaging operator in Equation (12).

In thermodynamics, it is customary to use the simpler notation $\mathsf{q}$ for $〈\mathsf{q}〉=\widehat{A}\mathsf{q}$, which we will also follow in this review, such as $E,S$, etc., for the average energy, entropy, etc. However, we will also use the notation $\widehat{A}\mathsf{q},\overline{\mathsf{q}}$ or $〈\mathsf{q}〉$, when clarity is needed, as we will see in Section 10 that such a convention can lead to confusion if care is not exercised. We wish to emphasize that $\widehat{A}\mathsf{q}=\mathsf{q}$ does not imply that $\widehat{A}=1$, except when ${\mathsf{q}}_k=\mathsf{q},\forall k$.

Remark 8. 

To avoid confusion with the notation $d_{\alpha }{\chi }_k$, which can either mean $d_{\alpha }\left({\chi }_k\right)$ as $d_{\alpha }$ acting on ${\chi }_k$, or ${\left(d_{\alpha }\left(\chi \right)\right)}_k$ denoting the microquantity associated with $d_{\alpha }\left(\chi \right)$, we will continue to use $d_{\alpha }{\chi }_k$ for the former, and $d_{\alpha }{\overline{\chi }}_k$ for the latter, where $\chi =\overline{\chi }$ stands for the macroquantity associated with ${\chi }_k$; see Section 10.1 for details. However, we will simply use $d_{\alpha }{\chi }_k$ in $d_{\alpha }{\theta }_k$ to simplify the notation, but we will always use the specific notation when clarity is needed.

In this review, we will not consider a constant random variable. Hence, a random variable will always have fluctuating outcomes.

Notation 3. 

We use modern notation [13,51] and its extension (see Figure 1), which will be extremely useful to understand the usefulness of our novel approach. Any infinitesimal and extensive ${\Sigma }_b$-intrinsic quantity dq$\left(t\right)$ (see Equation (11b)) during an arbitrary infinitesimal process $d\mathcal{P}$ can be partitioned as

$$dq\left(t\right)\equiv d_{\mathrm{e}}q\left(t\right)+d_{\mathrm{i}}q\left(t\right),$$

(14a)

where $d_{\mathrm{e}}$q$\left(t\right)$ is the change caused by exchange (“e”) with the surroundings such as the medium and $d_{\mathrm{i}}$q$\left(t\right)$ is its change due to internal or irreversible (“i”) processes going on within ${\Sigma }_b$. As mentioned earlier, the term external quantity will also be used for an exchange quantity to emphasize its external nature in this review. The partition also applies to the outcome $d{\mathsf{q}}_k$ as follows:

$$d{q}_k\left(t\right)\equiv d_{\mathrm{e}}{q}_k\left(t\right)+d_{\mathrm{i}}{q}_k\left(t\right),$$

(14b)

As an example, we have (see Equation (27a) for the definition of $S_k$)

$$d{E}_k=d_{\mathrm{e}}{E}_k+d_{\mathrm{i}}{E}_k,d{S}_k=d_{\mathrm{e}}{S}_k+d_{\mathrm{i}}{S}_k$$

(15)

for ${\Sigma }_{\mathrm{b}}$; here $d_{\mathrm{i}}{E}_k$ or $d_{\mathrm{i}}{S}_k$ does not have to vanish or have a particular sign even though $d_{\mathrm{i}}E=0$ (see Equation (53a)) or $d_{\mathrm{i}}S\ge 0$ (see Equation (67c)). We see that the linear operators $d_{\alpha }$ satisfy

$$d\equiv d_{\mathrm{e}}+d_{\mathrm{i}}.$$

(16)

Claim 2. 

An extensive quantity of ${\Sigma }_b$ is additive over its various macroscopic parts, but the energy E is usually quasi-additive; see Section 5.6.

For the sake of clarity, we will take V as a symbolic representation of $\mathbf{X}$, and a single $\xi$ as an internal variable in many examples. Then, $\mathbf{w}=\left(V\right),\mathbf{W}=(V,\xi )$, and $\mathbf{Z}=(E,V,\xi )$.

2.5. Different States in NEQT

Definition 8. 

An equilibrium (EQ) macrostate is a uniform macrostate having the maximum possible entropy in ${\mathfrak{S}}_{\mathbf{X}}$.

Definition 9. 

A nonequilibrium macrostate can be classified into two classes:

(a)

Internal-equilibrium macrostate (IEQ): The nonequilibrium entropy $S(\mathbf{X},t)$ for such a macrostate is a state function $S\left(\mathbf{Z}\right)$ in the larger nonequilibrium state space ${\mathfrak{S}}_{\mathbf{Z}}$ spanned by $\mathbf{Z}$; ${\mathfrak{S}}_{\mathbf{X}}$ is a proper subspace of ${\mathfrak{S}}_{\mathbf{Z}}$: ${\mathfrak{S}}_{\mathbf{X}}\subset {\mathfrak{S}}_{\mathbf{Z}}$. As there is no explicit time dependence, there is no memory of the initial macrostate in IEQ macrostates.

(b)

Non-internal-equilibrium macrostate (NIEQ): The nonequilibrium entropy for such a macrostate is not a state function of the state variable $\mathbf{Z}$. Accordingly, we denote it by $S(\mathbf{Z},t)$ with an explicit time dependence. The explicit time dependence gives rise to memory effects in these NEQ macrostates that lie outside the nonequilibrium state space ${\mathfrak{S}}_{\mathbf{Z}}$. An NIEQ macrostate in ${\mathfrak{S}}_{\mathbf{Z}}$ becomes an IEQ macrostate in a larger state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }},{\mathbf{Z}}^{\prime }\supset \mathbf{Z}$, with a proper choice of ${\mathbf{Z}}^{\prime }$.

Definition 10. 

An arbitrary macrostate ${\mathfrak{M}}_{arb}$ of a system refers to all possible thermodynamic states, which include EQ macrostates, and NEQ macrostates with and without the memory of the initial macrostate. From now on, we denote an arbitrary macrostate by $\mathfrak{M}$, NEQ macrostates by ${\mathfrak{M}}_{neq}$, EQ macrostates by ${\mathfrak{M}}_{eq}$, and IEQ macrostates by ${\mathfrak{M}}_{ieq}$.

Different choices of $\left\{p_k\right\}$ for the same set $\left\{{\mathfrak{m}}_k,E_k\right\}$ describe different macrostates for a given $\mathbf{W}$, one of which corresponding to $\left\{p_k^{\mathrm{eq}}\right\}$ uniquely specifies the EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$; all other states are called NEQ macrostates ${\mathfrak{M}}_{n\mathrm{eq}}$. Among  ${\mathfrak{M}}_{\mathrm{neq}}$ are some special macrostates ${\mathfrak{M}}_{\mathrm{ieq}}$ that are said to be in internal equilibrium (IEQ); the rest are nonIEQ macrostates ${\mathfrak{M}}_{\mathrm{nieq}}$. An arbitrary macrostate $\mathfrak{M}$ refers to either an EQ or an NEQ macrostate; the latter can be either ${\mathfrak{M}}_{\mathrm{ieq}}$ or ${\mathfrak{M}}_{\mathrm{nieq}}$.

2.6. Mechanical Description

Claim 3. 

There are two distinct approaches to handling state variable $\mathbf{W}$ for a macrostate $\left\{{\mathfrak{m}}_k,p_k\right\}$ of ${\Sigma }_b$; see Remark 2 and Definition 2.

• Nonfluctuating (NFl) approach: It can be treated as a nonfluctuating (fixed) parameter in the Hamiltonian of ${\Sigma }_b$ so that it is the same for all of its microstates. If we alter $\mathbf{W}$, it changes the same way for all ${\mathfrak{m}}_k$’s. We say that $\mathbf{W}$ is a NFl-parameter over ${\mathfrak{m}}_k$’s. This results in fluctuating generalized microforce

  $${\mathbf{F}}_{\mathrm{w}k}\doteq -\partial E_k/\partial \mathbf{W}$$

  (17a)

  over ${\mathfrak{m}}_k$’s, with its ensemble average (see Equation (12)), given by the generalized macroforce

  $${\mathbf{F}}_{\mathrm{w}}\doteq {\sum }_k{p}_k{\mathbf{F}}_{\mathrm{w}k}=-\partial E/\partial \mathbf{W}.$$

  (17b)

  Even though ${\mathbf{F}}_{\mathrm{w}k}$ is a microvariable, we find it useful conceptually to think of it as the outcome of a random variable ${\mathbb{F}}_{\mathrm{w}}$ on ${\mathfrak{m}}_k$. We use the notation $\left\{\mathbf{W},{\mathbf{F}}_{\mathrm{w}k}\right\}$ to compactly refer to this case.
• Fluctuating (Fl) approach: Alternatively, we let $\mathbf{W}$ fluctuate over ${\mathfrak{m}}_k$’s and think of it conceptually as a random variable $\mathbb{W}$ with outcomes $\left\{{\mathbf{W}}_k\right\}$, even though ${\mathbf{W}}_k$ is a microvariable. To be consistent with the NFl-approach (see below), we require that ${\mathbf{F}}_w$ becomes nonfluctuating (fixed) defined by

  $$\forall k,-\partial E_k/\partial {\mathbf{W}}_k=-\partial E/\partial \mathbf{W}={\mathbf{F}}_{\mathrm{w}}.$$

  (18)
  In this view, the macroforce ${\mathbf{F}}_w$ is fixed (so it is the same for all macrostates) with the result that ${\mathfrak{m}}_k\left({\mathbf{W}}_k\right)$ is determined by the fluctuating random variable $\mathbb{W}$ over ${\mathfrak{m}}_k$’s, with its average (see Equation (112)) given by

  $$\mathbf{W}\doteq {\sum }_k{p}_k{\mathbf{W}}_k.$$

  (19)
  We use the notation $\left\{{\mathbf{W}}_k,{\mathbf{F}}_w\right\}$ to compactly refer to this case.

The same two approaches apply as well if we replace $\mathbf{W}$ by $\mathbf{w}$, and ${\mathbf{F}}_{\mathrm{w}}$ by ${\mathbf{f}}_{\mathrm{w}}$ in the above equations.

Claim 4. 

The presence of a parameter in the Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ of the body brings forth the Legendre-transformed Hamiltonian ${\mathcal{H}}^L\left(\left.\mathbf{x}\right|{\mathbf{W}}^L\right)$ as the most important quantity to consider, where ${\mathbf{W}}_w^L$ is the work parameter in ${\mathcal{H}}^L$; see Section 6.3.

Claim 5. 

The nonfluctuating (NFl) parameter $\mathbf{W}$ results in fluctuating (Fl) microfield $\left\{{\mathbf{F}}_{wk}\right\}$ that plays the role of ${\mathbf{W}}^L$, and fluctuating $\left\{{\mathbf{W}}_k\right\}$ results in a NFl workfield ${\mathbf{F}}_w$ that plays the role of ${\mathbf{W}}^L$. As noted in Remark 2, we find it convenient to take $\mathbf{W}$ and ${\mathbf{F}}_w$ as the parameters, respectively, as will become clear later in the review.

We provide an intuitive understanding of the two approaches. For the NFl-$\mathbf{W}$, we use the microstates $\left\{{\mathfrak{m}}_k\right\}$ of $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ so that every microstate is specified by the same $\mathbf{W}$. If we use the same Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ for the Fl-$\mathbf{W}$ case, this will require considering different Hamiltonian $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|{\mathbf{W}}_k\right)$ for different microstates so that their slopes are all equal to ($-{\mathbf{F}}_{\mathrm{w}}$); see Equation (18). This is quite cumbersome. It is well-known that in this case, it is most convenient to consider ${\mathcal{H}}^{\mathrm{L}}\left(\left.\mathbf{x}\right|{\mathbf{F}}_{\mathrm{w}}\right)$ with ${\mathbf{W}}^{\mathrm{L}}={\mathbf{F}}_{\mathrm{w}}$ so that every microstate is specified by the same ${\mathbf{W}}^{\mathrm{L}}$, which plays the role of the work-parameter in ${\mathcal{H}}^{\mathrm{L}}\left(\left.\mathbf{x}\right|{\mathbf{F}}_{\mathrm{w}}\right)$; see Section 6.3.

Remark 9. 

We now explain the concept of consistency noted above. Consider $E_k\left(\mathbf{W}\right)$ for some microstate ${\mathfrak{m}}_k$ in the NFl-approach, and determine ${\mathbf{F}}_{wk}$ at some $\mathbf{W}$. Using the variation $d\mathbf{W}$, we determine the change $d{E}_k^{NFl}=-{\mathbf{F}}_{wk}·d\mathbf{W}$. In the Fl-approach, we choose that particular value ${\mathbf{W}}_k$ at which $E_k$ has the NFl slope ${\mathbf{F}}_w$ as shown in Equation (18). We emphasize that only the particular $\left\{{\mathbf{W}}_k\right\}$ is considered that satisfies Equation (18). We then determine the variation $d{\mathbf{W}}_k$ so that $d{E}_k^{Fl}=-{\mathbf{F}}_w·d{\mathbf{W}}_k$, as follows from Equations (17a), has exactly the same value as $d{E}_k^{NFl}$. Therefore, we do not have to distinguish between $d{E}_k^{NFl}$ and $d{E}_k^{Fl}$, and use the simpler notation $d{E}_k$ for both of them. As a consequence, we can make the following:

Claim 6. 

We have the same microwork in both approaches:

$$d{W}_k={\mathbf{F}}_{wk}·d\mathbf{W}={\mathbf{F}}_w·d{\mathbf{W}}_k=-d{E}_k.$$

(20)

Remark 10. 

In the NFl approach, we introduce the Legendre transform

$$E_k^{L,NFl}\left({\mathbf{F}}_{wk}\right)=E_k\left(\mathbf{W}\right)+{\mathbf{F}}_{wk}·\mathbf{W},$$

(21a)

as a function of ${\mathbf{F}}_{wk}$ with

$$\mathbf{W}=\partial E_k^{NFl}\left({\mathbf{F}}_{wk}\right)/\partial {\mathbf{F}}_{wk}.$$

(21b)

In the Fl approach, we introduce the Legendre transform

$$E_k^{L,Fl}\left({\mathbf{F}}_w\right)=E_k\left({\mathbf{W}}_k\right)+{\mathbf{F}}_w·{\mathbf{W}}_k,$$

(22a)

as a function of ${\mathbf{F}}_w$ with

$${\mathbf{W}}_k=\partial E_k^{NFl}\left({\mathbf{F}}_w\right)/\partial {\mathbf{F}}_w.$$

(22b)

We see that the above definitions of the Legendre transform $E_k^{\mathrm{L}}$ of $E_k$ in the two approaches can be compactly denoted by

$$E_k^{\mathrm{L}}\left(\mathbf{b}\right)=E_k\left(\mathbf{a}\right)+\Phi (\mathbf{a},\mathbf{b})$$

(23a)

in terms of a scalar function

$$\Phi (\mathbf{a},\mathbf{b})\doteq \mathbf{a}·\mathbf{b};$$

(23b)

see also Section 6.3. It is clear from Equation (23a) that it is sufficient to investigate the behavior of $E_k$; the behavior of $E_k^{\mathrm{L}}$ is easily obtained from it. Therefore, we will mostly focus on $E_k$ in the review.

Remark 11. 

As microstates ${\mathfrak{m}}_k$ play the central and important role in our approach involving the Hamiltonian, the microstate energies $\left\{E_k\right\}$ represent the outcomes of a random variable E  over the microstates. Thus, we always deal with a fluctuating microstate energy. Consequently, the corresponding “macroforce”

$$f_{\mathrm{s}}\doteq -\partial E/\partial S=-T$$

(24)

(see Equation (1) or equivalently Equation (129)) always appears as a NFl-parameter for ${\Sigma }_b$, which can be combined with ${\mathbf{f}}_w$ and ${\mathbf{F}}_w$ as

$$\mathbf{f}\doteq \left\{f_s,{\mathbf{f}}_w\right\},\mathbf{F}\doteq \left\{f_s,{\mathbf{F}}_w\right\}$$

(25)

to represent the relevant macroforces.

Remark 12. 

It follows from the above Remark that we can either consider the case $\left\{\mathbf{W},{\mathbf{F}}_{wk}\right\}$ or $\left\{{\mathbf{W}}_k,{\mathbf{F}}_w\right\}$. In  both cases, we obtain the same thermodynamics. A NFl parameter can be treated as a deterministic parameter for whichq${}_k=$q$,\forall k$, as the probability of q  is unity (certainty).

Remark 13. 

The work parameter may be a function of time t. By taking one of the components of the work parameter $\mathbf{w}$ to be simply t, it is also possible to include t as a separate parameter in $\mathcal{H}\left(\mathbf{x}\left|t,\mathbf{W}\right.\right)$ as is common in mechanics [164].

Definition 11. 

In general, $p_k$ are functions of the microquantity ${\mathbf{X}}_k$ or ${\mathbf{Z}}_k$ in ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{S}}_{\mathbf{Z}}$, respectively, and are implicit functions of t through the latter; they may also depend explicitly on time t if not unique in the state space. For an EQ or an IEQ macrostate, $p_k$ have no explicit dependence on t; see Section 12 for details. As $p_k$ always satisfies the sum rule (see Equation (13)) over any $\mathfrak{M}$, it is also an ensemble quantity because of this, and should be treated as a mixed microquantity; it is not determined by ${\mathfrak{m}}_k$ alone so it is not a true microquantity.

Definition 12. 

The collection $\left\{{\mathfrak{m}}_k,p_k\right\}$ provides a complete microscopic or statistical mechanical description of thermodynamics of any arbitrary macrostate $\mathfrak{M}$ in some state space $\mathfrak{S}$ in which one deals with macroscopic or ensemble averages using $\left\{p_k\right\}$ (see Definition 7) over $\left\{{\mathfrak{m}}_k\right\}$ of microstate variables.

2.7. Entropy and Stochastic Description

Definition 13. 

A state function entropy S for ${\mathfrak{M}}_{eq}$ or ${\mathfrak{M}}_{ieq}$ is defined thermodynamically by the Gibbs fundamental relation up to a constant.

Definition 14. 

Statistical entropy S, often called the Gibbs entropy, for $\mathfrak{M}$ is defined by its microstates by the Gibbs formulation (see Equation (116)),

$$S\equiv 〈S〉={\sum }_k{p}_k{S}_k=-{\sum }_k{p}_{k}ln{p}_k,$$

(26a)

with its differential given by

$$dS=d〈S〉=-{\sum }_k({\eta }_k+1)d{p}_k\equiv -{\sum }_k{\widehat{\eta }}_{k}d{p}_k$$

(26b)

where  $S_k$ is defined by

$$S_k\equiv -{\eta }_k\doteq -ln{p}_k;$$

(27a)

in terms of Gibbs’ index of probability ([48], p. 16)

$${\eta }_k\doteq ln{p}_k,$$

(27b)

and where we have also introduced

$${\widehat{\eta }}_k\doteq {\eta }_k+1.$$

(27c)

Remark 14. 

The quantity $S_k$ and any deterministic function of it are mixed microquantities for the simple reason that $p_k$ satisfies the sum rule in Equation (13), which requires considering all the microstates; see also Definition 11. However, S is a macroquantity that is also a state variable.

This property of $S_k$ should not be forgotten.

Remark 15. 

Being additive, S is extensive. As a consequence, $S_k$ must be extensive.

As $\tilde{\Sigma }$ is taken to be in EQ, its Hamiltonian is defined by its observable $\tilde{\mathbf{X}}$; the internal variable $\tilde{\xi }$ plays no role. Thus, we will express its Hamiltonian as

$$\tilde{\mathcal{H}}\left(\tilde{\mathbf{x}}\left|\tilde{\mathbf{w}}\right.\right).$$

(28a)

We will also assume that $\tilde{\Sigma }$ is weakly interacting with $\Sigma$, a point discussed carefully in Section 5.6. By neglecting their mutual interaction, we have quasi-additivity of their Hamiltonians to determine the Hamiltonian of ${\Sigma }_0$:

$${\mathcal{H}}_0\left({\mathbf{x}}_0\left|\mathbf{W},\tilde{\mathbf{w}}\right.\right)\approx \mathcal{H}\left(\mathbf{x}\left|\mathbf{W}\right.\right)+\tilde{\mathcal{H}}\left(\tilde{\mathbf{x}}\left|\tilde{\mathbf{w}}\right.\right),$$

(28b)

and states the quasi-additivity of the microstate energies; see Equation (119). We also assume the following additivity in this case:

$${\mathbf{W}}_0\equiv \mathbf{W}+\tilde{\mathbf{w}};$$

(28c)

see also Equation (118a).

2.8. Reduction

Very often, we need to define an ensemble average over a composite system such as ${\Sigma }_0$ formed by two or more systems. We focus on ${\Sigma }_0=\Sigma \cup \tilde{\Sigma }$. A microquantity q${}_{0k_0}$ associated with ${\Sigma }_0$ may also refer to a microquantity q${}_k$ associated with $\Sigma$, or a microquantity ${\tilde{\mathsf{q}}}_{\tilde{k}}$ associated with $\tilde{\Sigma }$.

Definition 15. 

The ensemble average over ${\mathfrak{m}}_{0k_0}$ of a composite microquantity  q${}_{0k_0}$ of ${\Sigma }_0$ is given by the joint probability

$$p_{0k_0}\equiv p\left(\left.k\right|\tilde{k}\right)p_{\tilde{k}}=p_{k}p\left(\left.\tilde{k}\right|k\right)$$

(29)

to be used in the following two equivalent ways:

$$\begin{array}{cc}\hfill q_0& ={\sum }_k{p}_k{\sum }_{\tilde{k}}\frac{p\left(\left.k\right|\tilde{k}\right)}{p_k}{p}_{\tilde{k}}{q}_{0k_0}\hfill \end{array}$$

(30a)

$$\begin{array}{cc}\hfill & ={\sum }_k{p}_k{\sum }_{\tilde{k}}p\left(\left.\tilde{k}\right|k\right)q_{0k_0}.\hfill \end{array}$$

(30b)

This averaging is properly discussed in Section 7. The  conditional probabilities $p\left(\left.k\right|\tilde{k}\right)$ and $p\left(\left.\tilde{k}\right|k\right)$ contain all the information about the correlation between $\Sigma$ and $\tilde{\Sigma }$ due to their mutual interaction, which will be considered in detail in Section 5.6 and Section 7. Here, we use the above definition to define the conditional microquantity q${}_{0k}$ given that $\Sigma$ is in the microstate ${\mathfrak{m}}_k$.

Definition 16. 

The reduction of the composite microquantity $q_{0k_0}$ to a conditional ${\Sigma }_0$-microquantity q${}_{0k}$ is defined by

$$q_{0k}\doteq {\sum }_{\tilde{k}}\frac{p\left(\left.k\right|\tilde{k}\right)}{p_k}{p}_{\tilde{k}}{q}_{0k_0}={\sum }_{\tilde{k}}p\left(\left.\tilde{k}\right|k\right)q_{0k_0}.$$

(31)

Here, the conditional microquantity q${}_{0k}$ associated with ${\Sigma }_0$ carries the suffix k and not $k_0$, and is obtained under the condition that Σ is in the microstate ${\mathfrak{m}}_k$, and requires conditionally averaging over all the microstates ${\tilde{\mathfrak{m}}}_{\tilde{k}}$ of $\tilde{\Sigma }$ using the reduced or conditional probability $p\left(\left.k\right|\tilde{k}\right)/p_k$; see Section 7 for details.

It is evident, but also easily verified, that the conditional microquantity associated with $\Sigma$ is the same as q${}_k$. For ${\tilde{\mathrm{q}}}_{\tilde{k}}$, we find that

$${\tilde{\mathrm{q}}}_k\doteq {\sum }_{\tilde{k}}p\left(\left.\tilde{k}\right|k\right){\tilde{\mathrm{q}}}_{\tilde{k}},$$

(32)

and can be very different from ${\tilde{\mathrm{q}}}_{\tilde{k}}$.

Claim 7. 

When the two bodies in the above definition are quasi-independent (see Definition 28 and Section 7.3 for full details), then

$$p\left(\left.k\right|\tilde{k}\right)\approx p_k,p\left(\left.\tilde{k}\right|k\right)\approx p_{\tilde{k}}.$$

(33)

Remark 16. 

A composite microquantity ${\chi }_{0k_0}^j$ and a medium microquantity ${\tilde{\chi }}_{\tilde{k}}^j$ are easily reduced to the conditional microquantities ${\chi }_{0k}^j$ and ${\tilde{\chi }}_k^j$ ascribed to ${\mathfrak{m}}_k$, respectively, by  using quasi-independence condition as

$${\chi }_{0k}^j\approx {\sum }_{\tilde{k}}{p}_{\tilde{k}}{\chi }_{0k_0}^j,{\tilde{\chi }}_k^j\approx {\sum }_{\tilde{k}}{p}_{\tilde{k}}{\tilde{\chi }}_{\tilde{k}}^j={\tilde{\chi }}^j,$$

(34)

so that their averages following Equation (12) finally give ${\overline{\chi }}_0^j$ and ${\tilde{\chi }}^j$ approximately compared to its exact formulation in Equation (31). Here, the  conditional quantities ${\chi }_{0k}^j$ and ${\tilde{\chi }}_k^j$ require conditional averaging over all the microstates ${\tilde{\mathfrak{m}}}_{\tilde{k}}$ with their probabilities $p_{\tilde{k}}$, given that Σ is in the microstate ${\mathfrak{m}}_k$; the reduced or conditional probability approximately becomes unity due to Equation (33). The last equation in Equation (34) follows from Theorem 1.

Remark 17. 

The above reduction plays a very important role in the formulation of the NEQ statistical mechanics (μNEQT) of the system Σ by reducing all microquantities in ${\Sigma }_0$ to conditional microquantities under the condition that Σ is in microstate ${\mathfrak{m}}_k$.

For a medium microquantity ${\tilde{\mathrm{q}}}_{\tilde{k}}$, we obtain a very important result, which we quote as a Theorem because of its extreme importance.

Theorem 1. 

Under quasi-independence approximation, the conditional ${\tilde{q}}_k$ is simply given by the macroquantity $\tilde{q}$:

$${\tilde{q}}_k\approx \tilde{q},\forall k.$$

(35)

Proof. 

By replacing ${\tilde{\chi }}_{\tilde{k}}^j$ by ${\tilde{\mathrm{q}}}_{\tilde{k}}$ and ${\tilde{\chi }}_k^j$ by ${\tilde{\mathrm{q}}}_k$ in Equation (34), we obtain the ensemble average on the right side, which proves the theorem. ☐

The application and general proof of this important theorem is deferred to Section 7.5, where it is restated slightly differently as Theorem 7, where we justify Remark 16, which is used in the simple proof given above.

2.9. Process Quantities

Remark 18. 

For a state variable  q$\in \left\{S,\mathbf{Z}\right\}$ for ${\Sigma }_b$, its microstate analog  q${}_k$ is trivially identified as the microstate value  q  takes on ${\mathfrak{m}}_k$, and appears as the coefficient of $p_k$ in the right-hand side of Equation (12), the ensemble average. We now consider the process quantity$d_{qk}$ and consider its ensemble average

$$〈d\mathrm{q}〉\equiv \widehat{A}d{\mathrm{q}}_k\doteq {\sum }_k{p}_{k}d{\mathrm{q}}_k$$

(36a)

if we follow the convention adopted in Equation (12). However, $〈dq〉$ above is not the same as

$$d\widehat{A}q\equiv dq\mathrm{or}d\overline{q}\left(t\right)\mathrm{or}d〈q〉\doteq {\sum }_k{p}_{k}d{\overline{q}}_k\equiv d\left({\sum }_k{p}_k{q}_k\right);$$

(36b)

we have also introduced the microstate analog $d{\overline{q}}_k$ for dq  or $d〈q〉$ to make sure that we distinguish dq${}_k$ and

$$d{\overline{q}}_k={\left(d\widehat{A}\mathsf{q}\right)}_k\doteq d{\mathrm{q}}_k+{\mathrm{q}}_{k}d{\eta }_k,$$

(36c)

so that $d〈q〉=\widehat{A}d\overline{q}$. This distinction becomes very important for  q$=E$ and S, as we will see in Section 10.1; see also Definition 23.

Definition 17. 

In mechanics, the generalized or BI-microwork by ${\Sigma }_b$ with parameter $\mathbf{W}$ is defined as the microwork done by the fluctuating microforce ${\mathbf{F}}_{wk}$

$$d{W}_k\doteq {\mathbf{F}}_{wk}·d\mathbf{W}=-(\partial E_k/\partial \mathbf{W})·d\mathbf{W}=-d{E}_k,$$

(37a)

with ${\mathbf{F}}_{wk}$ in its component form is given by

$${\mathbf{F}}_{wk}=(P_k,...,{\mathbf{A}}_k)=({\mathbf{f}}_{wk},{\mathbf{A}}_k);$$

(37b)

here, ⋯ denotes microfields corresponding to the rest of the state variables in $\mathbf{w}$ besides V, and

$${\mathbf{f}}_{wk}\doteq -\partial E_k/\partial \mathbf{w},{\mathbf{A}}_k\doteq -\partial E_k/\partial \xi ,$$

(37c)

with ${\mathbf{A}}_k$ representing the microaffinity.

With fluctuating $\left\{{\mathbf{W}}_k\right\}$, it is defined as the microwork done over the fluctuating generalized displacement $d{\mathbf{W}}_k$

$$d{W}_k\doteq {\mathbf{F}}_{\mathrm{w}}·d{\mathbf{W}}_k=-(\partial E_k/\partial {\mathbf{W}}_k)·d{\mathbf{W}}_k;$$

(38)

see Claim 6. The generalized or BI-macrowork done by ${\Sigma }_{\mathrm{b}}$ after ensemble averaging (see Equation (19)) in both approaches are the same:

$$dW=〈dW〉={\mathbf{F}}_{\mathrm{w}}·d\mathbf{W}$$

(39)

Explicitly, we express ${\mathbf{F}}_{\mathrm{w}}$ in its component form as

$${\mathbf{F}}_{\mathrm{w}}=(P\left(t\right),...,\mathbf{A}\left(t\right))=({\mathbf{f}}_{\mathrm{w}}\left(t\right),\mathbf{A}\left(t\right));$$

(40)

see Figure 1. Here, ⋯ denotes the macrofields corresponding to the rest of the state variables in $\mathbf{w}$ besides V, and

$${\mathbf{f}}_{\mathrm{w}}\doteq -\partial E/\partial \mathbf{w}.$$

(41)

The SI-affinity

$$\mathbf{A}\doteq -\partial E/\partial \xi$$

(42)

corresponding to $\xi$ [12,51] is nonzero, except in EQ, when it vanishes: ${\mathbf{A}}_{\mathrm{eq}}\equiv {\mathbf{A}}_0=0=0$ [13,51].

The SI-macrowork $d{W}_{\xi }$ done by $\Sigma$ as the internal variable $\xi$ varies is

$$d{W}_{\xi }\equiv d_{\mathrm{i}}{W}_{\xi }\doteq \mathbf{A}·d\xi \ge 0.$$

(43)

Even for an isolated NEQ system, $d{W}_{\xi }$ will not vanish; it vanishes only in EQ, since $\xi$ does no work when ${\mathbf{A}}_0=0$. Because of this, $d_{\mathrm{e}}{W}_{\xi }\equiv 0$ so that $d{W}_{\xi }\equiv d_{\mathrm{i}}{W}_{\xi }$. However, ${\mathbf{f}}_{\mathrm{w}},d\tilde{W}$ and $d_{\mathrm{e}}W$ are unaffected by the presence of $\xi$.

Definition 18. 

In statistical mechanics, generalized or SI-microheat for ${\Sigma }_b$ is defined as

$$d{Q}_k\doteq -T({\eta }_k+1)d{\eta }_k\equiv -T{\widehat{\eta }}_{k}d{\eta }_k;$$

(44a)

see Equation (255). The average of $d{Q}_k$ is the generalized or SI-macroheat

$$dQ\doteq {\sum }_k{E}_{k}d{p}_k\equiv TdS.$$

(44b)

Remark 19. 

We will use “generalized” or “SI” interchageably in this review.

Conclusion 1. 

The SI-macroheat or the generalized macroheat $dQ$

$$dQ\doteq TdS,$$

(45)

is identified as the Clausius equality; see Remark 42.

This interesting equality should be distinguished from the well-known Clausius inequality

$$d_{\mathrm{e}}Q\doteq T_0{d}_{\mathrm{e}}S\le T_{0}dS.$$

(46)

Thus, Equation (93a) allows us to uniquely identify generalized heat and work as independent of each other.

Remark 20. 

The d in $dW,dQ,d{W}_k$, and $d{Q}_k$ does not denote any differential operator on some quantity $W,Q,W_k$, and $Q_k$, respectively. Conventionally, one uses a symbol đ or some other symbols in thermodynamics to emphasize this distinction. However, we follow the standard notation of mechanics for $dW$ and $d{W}_k$ to emphasize these mechanical concept of work. We also use the same symbol for $dQ$ and $d{Q}_k$. If we extend Equation (36a) to also include $d{W}_k$ and $d{Q}_k$, then we could also use $\overline{dW}$ and $\overline{dQ}$ for $dW$ and $dQ$, respectively, but we will use the simpler notation $dW$ and $dQ$. This should not cause any confusion.

It follows from Equations (45) and (46) that the irreversible macroheat is

$$d_{\mathrm{i}}Q=\left\{\begin{array}{c}(T-T_0)dS+T_0{d}_{\mathrm{i}}S\\ (T-T_0)d_{\mathrm{e}}S+T{d}_{\mathrm{i}}S\end{array}\right.;$$

(47)

see also Equation (142).

Definition 19. 

Changes in quantities such as $S,E,V,\cdots$ in an infinitesimal process $\delta \mathcal{P}$ are denoted by $d_{\alpha }S,d_{\alpha }E,d_{\alpha }V,\cdots$; changes during a finite process $\mathcal{P}$ are denoted by ${\Delta }_{\alpha }S,{\Delta }_{\alpha }E,{\Delta }_{\alpha }V,\cdots$. All of these are process quantities, which also include $d_{\alpha }W,d_{\alpha }Q,{\Delta }_{\alpha }W$, and ${\Delta }_{\alpha }Q$.

Definition 20. 

The path ${\gamma }_{\mathcal{P}}$ of a macrostate $\mathfrak{M}$ is the path it takes in ${\mathfrak{S}}_{\mathbf{Z}}$ during a process $\mathcal{P}$. The trajectory ${\gamma }_k$ is the trajectory a microstate ${\mathfrak{m}}_k$ takes in time in $\mathfrak{S}$ during the process $\mathcal{P}$. The path and the trajectories are uniquely specified in ${\mathfrak{S}}_{\mathbf{Z}}$ if $\left\{{\mathfrak{m}}_k\right\}$ are uniquely specified in it.      

2.10. ${\Sigma }_0$ (Isolated Body) and $\tilde{\Sigma }$(Medium)

Remark 21. 

As an isolated body cannot exchange anything with its surroundings, we must always have

$$d_e{\theta }_0\left(t\right)\equiv 0,d_e{\theta }_{0k_0}\left(t\right)\equiv 0,d_{\mathrm{e}}{\theta }_{0k}\left(t\right)\equiv 0,\forall k_0,k;$$

(48a)

see Definition 3. The last equality emerges from reduction; see Remark 16.

Remark 22. 

For a medium $\tilde{\Sigma }$, which is assumed to be in EQ and weakly interacting with and quasi-independent of the system Σ in microstate ${\mathfrak{m}}_k$, we must have

$$d_{\mathrm{i}}\tilde{\theta }\left(t\right)\equiv 0,d_{\mathrm{i}}{\tilde{\theta }}_{\tilde{k}}\left(t\right)\ne 0,d_{\mathrm{i}}{\tilde{\theta }}_k\left(t\right)=0$$

(49)

after reduction of $d_i{\tilde{\theta }}_{\tilde{k}}\left(t\right)$ from $\tilde{k}$ to k. The last equality follows from Theorem 1 by replacing ${\tilde{q}}_k$ by $d_{\mathrm{i}}{\tilde{\theta }}_k=d_i\tilde{\theta }$, a NFl macroquantity, even though $d_i{\tilde{\theta }}_{\tilde{k}}$ is a Fl microquantity over ${\tilde{\mathfrak{m}}}_{\tilde{k}}$.

Remark 23. 

As we always use microstates ${\mathfrak{m}}_k$’s with fluctuating energies $E_k$, we find it useful and simple to use the notation in which $\mathbf{W}$ is fluctuating with ${\mathbf{W}}_k$ over microstates. This means that we will consider the state variable $\mathbf{Z}$ fluctuating with ${\mathbf{Z}}_k$ over microstates as if we are dealing with the fixed field approach with $\mathbf{Z}$ given by the extension of Equation (19)

$$\mathbf{Z}={\sum }_k{p}_k{\mathbf{Z}}_k.$$

(50)

The approach also covers the fluctuating workfield approach if we simply replace each ${\mathbf{W}}_k$ by a fixed $\mathbf{W}$.

3. Mathematical Digression on $\left\{{\mathit{d}}_{\mathit{\alpha }}\right\}$

In NEQT, there are various forms of work and heat $\left[dW\right]$ and $\left[dQ\right]$. Therefore, it is necessary to distinguish between them. Let us consider the Clausius equality in Equation (45) relating the SI-macroheat $dQ$ and the entropy change $dS$. It would be naïve to take this equality to conjecture that

$$d_{\alpha }Q=T{d}_{\alpha }S,$$

for the simple reason that the exchange macroheat $d_{\mathrm{e}}Q$ is a MI-quantity so it must be determined by the medium alone. The presence of T in the above conjecture $d_{\mathrm{e}}Q=T{d}_{\mathrm{e}}S$ raises doubts about the conjecture as T has nothing to do with the medium. Therefore, it is important to understand the role of the operators $d_{\alpha }$, which is explained in this section. This makes this section extremely important in the review.

3.1. Generalizing $d\equiv d_e+d_i$

The linear operators $d_{\alpha }$ satisfy not only the identities in Equations (14a) and (14b), but also the following identities:

$$\begin{array}{cc}\hfill d_{\alpha }(a{\mathrm{q}}_1+b{\mathrm{q}}_2)& =a{d}_{\alpha }{\mathrm{q}}_1+b{d}_{\alpha }{\mathrm{q}}_2,\hfill \\ \hfill d_{\alpha }\left({\mathrm{q}}_1{\mathrm{q}}_2\right)& =\mathrm{q}{}_1{d}_{\alpha }{\mathrm{q}}_2+\left(d_{\alpha }{\mathrm{q}}_2\right){\mathrm{q}}_2;\hfill \end{array}$$

(51)

here q${}_1$ and q${}_2$ are two extensive random variables, and a and b are two pure numbers.

The generalization of de Groot–Prigogine notation in Notation 3 provides a very compact description of NEQ processes in the $\mu$NEQT. The original notation [13,51] is restricted to the entropy, particle number, energy, and volume changes $dS,dN,dE$, and $dV$, respectively, for ${\Sigma }_{\mathrm{b}}$; see Figure 1 for $dZ\to dX=dS,dN,dE$ and $dV$:

$$\begin{array}{cc}\hfill d\left[S\right]& \equiv d_e\left[S\right]+d_{\mathrm{i}}\left[S\right],\hfill \end{array}$$

(52a)

$$\begin{array}{cc}\hfill d\left[N\right]& \equiv d_e\left[N\right]+d_{\mathrm{i}}\left[N\right],\hfill \end{array}$$

(52b)

$$\begin{array}{cc}\hfill d\left[E\right]& \equiv d_e\left[E\right]+d_{\mathrm{i}}\left[E\right],\hfill \end{array}$$

(52c)

$$\begin{array}{cc}\hfill d\left[V\right]& \equiv d_e\left[V\right]+d_{\mathrm{i}}\left[V\right],\hfill \end{array}$$

(52d)

As no internal process can change the energy [12], we have

$$d_{\mathrm{i}}E\equiv 0.$$

(53a)

The surprising fact is that $d_{\mathrm{i}}{E}_k\ne 0$, as we will establish below; see Theorem (6). Similarly,

$$d_{\mathrm{i}}V=0.$$

(53b)

We have also assumed that $d_{\mathrm{i}}N=0$, but this is no consequence as we are assuming no chemical reaction in the review. We should emphasize that the partitions above have nothing to do with the partitions in Equations (238) and (247a), respectively. The original partition in Equation (52b) is not relevant in the review as we do not consider any chemical reaction, so $dN\equiv d_{\mathrm{e}}N$. Observe that the above partitions are defined only for macroscopic extensive observables for a body. We have extended the notation to not only all extensive state variables in $\left[\chi \right]$ but for $\left[d_{\alpha }W\right],\left[d_{\alpha }Q\right]$ for any body ${\Sigma }_{\mathrm{b}}$. We thus have

$$\begin{array}{cc}\hfill d{W}_k& =d_e{W}_k+d_{\mathrm{i}}{W}_k,d{Q}_k=d_e{Q}_k+d_{\mathrm{i}}{Q}_k,\hfill \end{array}$$

(54a)

$$\begin{array}{cc}\hfill dW& =d_{e}W+d_{\mathrm{i}}W,dQ=d_{e}Q+d_{\mathrm{i}}Q;\hfill \end{array}$$

(54b)

For ${\Sigma }_{\mathrm{b}}$ an isolated system ${\Sigma }_0$, it follows from Equation (48a) that

$$d_{\mathrm{e}}{W}_0\equiv 0,d_{\mathrm{e}}{Q}_0\equiv 0.$$

(55a)

For ${\Sigma }_{\mathrm{b}}$ a medium $\tilde{\Sigma }$, it follows from Equation (49) that

$$d_{\mathrm{i}}\tilde{W}\equiv 0,d_{\mathrm{i}}\tilde{Q}\equiv 0.$$

(55b)

Note that $dW,dQ$, etc., do not represent changes in any SI-macrovariable; see Remark 20.

Remark 24. 

We mostly focus on $\left\{q_k\right\}$ or $\left\{d_{\alpha }{q}_k\right\}$ in the μNEQT, from which we obtain the information about the corresponding macroquantity  q  or $d_{\alpha }$q,  respectively, by ensemble averaging. The approach in this sense is to effectively discuss $\left[q\right]$ or $\left[d_{\alpha }q\right]$, without explicitly showing the suffix k, unless clarity is needed. We will, however, use ${\left[q\right]}_k$ or ${\left[d_{\alpha }q\right]}_k$ when we consider specific cases.

We now consider the three systems separately for clarity below so we need $\left[\mathrm{q}\right],\left[d_{\alpha }\mathrm{q}\right],\left[\tilde{\mathrm{q}}\right],$$\left[d_{\alpha }\tilde{\mathrm{q}}\right]$, and $\left[{\mathrm{q}}_0\right],\left[d_{\alpha }{\mathsf{q}}_0\right]$ for $\Sigma ,\tilde{\Sigma }$ (not necessarily in EQ) and ${\Sigma }_0$, respectively, which satisfy additivity for ${\Sigma }_0$ so that

$${\left[{\mathrm{q}}_0\right]}_{k_0}={\left[\mathrm{q}\right]}_k+{\left[\tilde{\mathrm{q}}\right]}_{\tilde{k}},{\left[d_{\alpha }{\mathrm{q}}_0\right]}_{k_0}={\left[d_{\alpha }\mathrm{q}\right]}_k+{\left[d_{\alpha }\tilde{\mathrm{q}}\right]}_{\tilde{k}},$$

(56)

where we have explicitly shown microstate indices for $\Sigma ,\tilde{\Sigma }$, and ${\Sigma }_0$; here and in the following, q$\in \chi$, and $d_{\alpha }$q$\in d_{\alpha }\theta$. For these equations to hold, we need to assume that $\Sigma$ and $\tilde{\Sigma }$ interact so weakly that their interactions can be neglected (recall that $\left[E\right]$ is one of the possible $\left[\mathrm{q}\right]$) and that $\Sigma$ and $\tilde{\Sigma }$ are quasi-independent [148]; see Section 7.3. We also consider their partitions as shown in Equation (14a).

Remark 25. 

The medium $\tilde{\Sigma }$ in Equation (56) need not be in EQ, so Equation (56) also applies to a system Σ consisting of two subsystems ${\Sigma }_1$ and ${\Sigma }_2$ interacting with each other satisfying quasi-additivity and quasi-independence. All we need to do is to take ${\Sigma }_0\to \Sigma ,\Sigma \to {\Sigma }_1$, and $\tilde{\Sigma }\to {\Sigma }_2$. We can also have Σ embedded in a medium $\tilde{\Sigma }$, distinct from the previous $\tilde{\Sigma }$. It follows from Equation (56) that

$$q_k=q_{k_1}+q_{k_2},d_{\alpha }{q}_{k_0}=d_{\alpha }{q}_{k_1}+d_{\alpha }{q}_{k_2},$$

(57a)

and

$$q=q_1+q_2,d_{\alpha }q=d_{\alpha }{q}_1+d_{\alpha }{q}_2.$$

(57b)

Explicitly, we have

$$\begin{array}{cc}\hfill dq& =d{q}_1+d{q}_2,d{q}_k=d{q}_{1k_1}+d{q}_{2k_2},\hfill \\ \hfill d_{e}q& =d_e{q}_1+d_e{q}_2,d_e{q}_k=d_e{q}_{1k_1}+d_e{q}_{2k_2},\hfill \\ \hfill d_{i}q& =d_i{q}_1+d_i{q}_2,d_i{q}_k=d_i{q}_{1k_1}+d_i{q}_{2k_2}.\hfill \end{array}$$

(58)

in which we must treat $d_e$q${}_j,d_e{\mathsf{q}}_{j{k}_j},j=1,2$, carefully. As usual, $\left[d_e\mathsf{q}\right]$ is the exchange with $\tilde{\Sigma }$, but $\left[d_e{\mathsf{q}}_1\right],\left[d_e{\mathsf{q}}_2\right]$ each have two exchanges; one exchange involving the suffix m is with $\tilde{\Sigma }$, and the other exchange is with the other subsystem. Thus, we have

$$\left[d_e{\mathsf{q}}_1\right]=\left[d_e{\mathsf{q}}_{1m}\right]+\left[d_e{\mathsf{q}}_{12}\right],\left[d_e{\mathsf{q}}_2\right]=\left[d_e{\mathsf{q}}_{2m}\right]+\left[d_e{\mathsf{q}}_{21}\right],$$

(59)

in which $\left[d_e{\mathsf{q}}_{12}\right],\left[d_e{\mathsf{q}}_{21}\right]$ stand for mutual exchanges between the subsystems.

Remark 26. 

For an isolated Σ in Equation (58), we must have ${\left[d_e\mathsf{q}\right]}_k=$${\left[d_e{\mathsf{q}}_{1\mathrm{m}}\right]}_{k_1}={\left[d_e{\mathsf{q}}_{1\mathrm{m}}\right]}_{k_2}=0$ (see Remark 21), so

$${\left[d_e{\mathrm{q}}_1\right]}_{k_1}=-{\left[d_e{\mathrm{q}}_2\right]}_{k_2}.$$

(60)

Remark 27. 

It follows from Remark 26 that

$$d_{\mathrm{e}}{W}_k=0,d_{\mathrm{e}}{W}_{1k_1}=-d_{\mathrm{e}}{W}_{2k_2}.$$

(61)

We now turn back to discussing a system embedded in a medium as above, and prove the following important theorem.

Theorem 2. 

We consider the system Σ and the medium $\tilde{\Sigma }$ (not necessarily in EQ) forming the isolated system ${\Sigma }_0$. We prove two important identities that are extremely useful in the μNEQT:

$$\begin{array}{cc}\hfill {\left[d_{\mathrm{e}}\mathrm{q}\right]}_k& \doteq -{\left[d_{\mathrm{e}}\tilde{\mathrm{q}}\right]}_{\tilde{k}}=-{\left[d\tilde{\mathrm{q}}\right]}_{\tilde{k}}+{\left[d_{\mathrm{i}}\tilde{\mathrm{q}}\right]}_{\tilde{k}},\hfill \end{array}$$

(62a)

$$\begin{array}{cc}\hfill {\left[d{\mathrm{q}}_0\right]}_{k_0}& \equiv {\left[d\mathrm{q}\right]}_k+{\left[d\tilde{\mathrm{q}}\right]}_{\tilde{k}}={\left[d_{\mathrm{i}}{\mathrm{q}}_0\right]}_{k_0}={\left[d_{\mathrm{i}}\mathrm{q}\right]}_k+{\left[d_{\mathrm{i}}\tilde{\mathrm{q}}\right]}_{\tilde{k}}.\hfill \end{array}$$

(62b)

Proof. 

As ${\Sigma }_0$ is isolated, there cannot be any exchange quantity, so $\left[d_e{\mathrm{q}}_0\right]\equiv 0$. It follows from Equation (60) that

$${\left[d_e{\mathrm{q}}_0\right]}_{k_0}={\left[d_e\mathrm{q}\right]}_k+{\left[d_e\tilde{\mathrm{q}}\right]}_{\tilde{k}}\equiv 0.$$

The identity in Equation (62a) immediately follows. Again using the second equation in Equation (56) for $d_{\alpha }=d$, and using ${\left[d_e{\mathsf{q}}_0\right]}_{k_0}=0$ proves the second identity, after using Equations (14a) and (14b) in ${\left[d{\mathrm{q}}_0\right]}_{k_0}$. This case is appropriate for treating $\tilde{\Sigma }$ as another system. ☐

For $\tilde{\Sigma }$ in EQ, $d_{\mathrm{i}}\tilde{\mathrm{q}}=0$ but not $d_{\mathrm{i}}{\tilde{\mathrm{q}}}_{\tilde{k}}$, as it is an outcome of a random variable $d_{\mathrm{i}}\tilde{\mathrm{q}}$; see Remark 22. Thus,

$$d{\tilde{\mathrm{q}}}_{\tilde{k}}=d_{\mathrm{e}}{\tilde{\mathrm{q}}}_{\tilde{k}}+d_{\mathrm{i}}{\tilde{\mathrm{q}}}_{\tilde{k}};d_{\alpha }{\mathrm{q}}_{0k_0}=d_{\alpha }{\mathrm{q}}_k+d_{\alpha }{\tilde{\mathrm{q}}}_{\tilde{k}},$$

(63a)

which should undergo reduction as our interest is to investigate $\Sigma$ in ${\mathfrak{m}}_k$. This is done in Section 7.5, where we find that

$$d_{\mathrm{e}}{\mathrm{q}}_k=-d_{\mathrm{e}}{\tilde{\mathrm{q}}}_k=-d_{\mathrm{e}}\tilde{\mathrm{q}}=d_{\mathrm{e}}\mathrm{q},\forall k,$$

(64a)

showing that exchange microquantities are not random variables; see Theorem 7. For the macrostate, we have

$$d\tilde{\mathrm{q}}=d_{\mathrm{e}}\tilde{\mathrm{q}}=-d_{\mathrm{e}}\mathrm{q},d{\mathrm{q}}_0=d_{\mathrm{i}}\mathrm{q}.$$

(64b)

For $\left[\mathrm{q}\right]=\left[\mathbf{Z}\right]$ and $\tilde{\Sigma }$ in EQ, we have from Equation (62b) and the general additivity

$${\left[{\mathbf{Z}}_0\right]}_{k_0}\equiv {\left[\mathbf{Z}\right]}_k+{\left[\tilde{\mathbf{Z}}\right]}_{\tilde{k}},$$

obtained by extending Equation (28c), the identity

$$d\left[{\mathbf{Z}}_0\right]=d_{\mathrm{e}}\left[\mathbf{Z}\right]+d_{\mathrm{i}}\left[\mathbf{Z}\right]+d\left[\tilde{\mathbf{Z}}\right]=d_{\mathrm{i}}\left[\mathbf{Z}\right],$$

(65a)

which shows that

$$d_{\mathrm{e}}\left[\mathbf{Z}\right]=-d\left[\tilde{\mathbf{Z}}\right]=-d_{\mathrm{e}}\left[\tilde{\mathbf{Z}}\right]$$

(65b)

in accordance with Equation (62a). We thus have

$$d{\mathbf{Z}}_{0k}=d_{\mathrm{i}}{\mathbf{Z}}_k,d{\mathbf{Z}}_0=d_{\mathrm{i}}\mathbf{Z},$$

(65c)

where all quantities pertaining to ${\mathfrak{m}}_{0k_0}$ have been reduced (see Definition 4) and we have used the fact that after reduction (see Remark 22),

$$d_{\mathrm{i}}{\tilde{\mathbf{Z}}}_k=d_{\mathrm{i}}\tilde{\mathbf{Z}}=0,\forall k.$$

For q$=\left[E\right]$ for a macrostate $\mathfrak{M}$, we have

$$d{E}_{0k}=d_{\mathrm{i}}{E}_k,d{E}_0=d_{\mathrm{i}}{E}_0=d_{\mathrm{i}}E=0;$$

(66)

the last equation follows from Equation (53a).

For q$=\left[S\right]$ for a macrostate $\mathfrak{M}$, we have the standard result

$$d{\left[S\right]}_0\equiv d_{\mathrm{i}}\left[S\right],$$

(67a)

from which we obtain

$$d{S}_{0k}\equiv d_{\mathrm{i}}{S}_k$$

(67b)

giving the internal entropy generation, which has no particular sign, and

$$d{S}_0=d_{\mathrm{i}}S\ge 0$$

(67c)

for the irreversible entropy generation. We similarly have

$$d{W}_{0k}=d_{\mathrm{i}}{W}_k,d{Q}_{0k}=d_{\mathrm{i}}{Q}_k,$$

(68a)

after reducing all quantities pertaining to ${\mathfrak{m}}_{0k_0}$. For a macrostate $\mathfrak{M}$,

$$d{W}_0=d_{\mathrm{i}}W\ge 0;d{Q}_0=d_{\mathrm{i}}Q\ge 0;$$

(69)

see Equation (145). Here, $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}Q$ are the irreversible macrowork done by and macroheat generation due to internal processes in $\Sigma$; see Theorem 4.

Claim 8. 

The nonnegative inequalities for macroquantities $d_{\mathrm{i}}$q in the above equations are in accordance with the second law, where ensemble averaging at each instant plays a central role. Because of this relationship with the second law, we call these quantities irreversible. There is no sign requirement for corresponding microquantities $d_{\mathrm{i}}$q${}_k$ that do not require such averaging. To make this clear distinction, we call these microquantities simply internal.

The discussion above finally justifies Conclusion 2 of several micro- and macroworks that are distinct in nature. Intuitively, the generalized microwork $d{W}_k$ denotes the mechanical work done by the system, a part

$$d_{\mathrm{e}}{W}_k=d_{\mathrm{e}}W=-d_{\mathrm{e}}\tilde{W}=-d\tilde{W}$$

(70a)

of which is transferred to ${\tilde{\Sigma }}_{\mathrm{w}}$ through exchange and $d_{\mathrm{i}}{W}_k$ is internally spent to overcome internal processes due to the microforce force imbalance ($\mu$FI) within $\Sigma$. Of the three, only $d{W}_k$ and $d_{\mathrm{i}}{W}_k$ are the outcomes of random variables $d\mathbb{W}$ and $d_{\mathrm{i}}\mathbb{W}$, respectively.

Similarly, there are several micro- and macroheats that are distinct in nature. Of $d{Q}_k$, a part

$$d_{\mathrm{e}}{Q}_k=d_{\mathrm{e}}Q=-d_{\mathrm{e}}\tilde{Q}=-d\tilde{Q}$$

(70b)

is transferred from ${\tilde{\Sigma }}_{\mathrm{h}}$ through exchange and $d_{\mathrm{i}}{Q}_k$ is internally generated by internal processes within $\Sigma$. Of the three, only $d{Q}_k$ and $d_{\mathrm{i}}{Q}_k$ are the outcomes of random variables $d\mathtt{Q}$ and $d_{\mathrm{i}}\mathtt{Q}$, respectively. Similar comments also apply to $d{S}_k,d_{\mathrm{e}}{S}_k$, and $d_{\mathrm{i}}{S}_k$.

What has been said above can be summarized as follows (also see Claim 15):

Summary 1. 

dq${}_k=(d{S}_k,d{E}_k,d{W}_k,d{Q}_k)$ and $d_i$q${}_k=(d_i{S}_k,d_{i,}{E}_k,d_i{W}_k,d_i{Q}_k)$are random variables and fluctuate around their respective averagesdq and$d_i$q, so they have values on both sides of their averages.

This justifies Remark 22.

3.2. Consequences of Theorem 60

We show the importance of the above theorem about exchange microquantities, which is why they have been extensively exploited in modern NEQ statistical mechanics ($\mathring{\mu }$NEQT). We will only consider the case of fixed work parameter so we have fluctuating microforces associated with the random variable ${\mathbb{F}}_{\mathrm{w}}$. The discussion is easily extended to fluctuating work parameter. We first consider an NEQ $\Sigma$ in the microstate ${\mathfrak{m}}_k$. The microwork $d{W}_k$ is given in Equation (37a); see also Equation (40). The same equation, applied to $\tilde{\Sigma }$ in the microstate ${\tilde{\mathfrak{m}}}_{\tilde{k}}$, gives

$$d{\tilde{W}}_{\tilde{k}}\doteq {\tilde{\mathbf{f}}}_{\mathrm{w}\tilde{k}}·d\tilde{\mathbf{w}}={\tilde{\mathbf{f}}}_{\mathrm{w}\tilde{k}}·d_{\mathrm{e}}\tilde{\mathbf{w}}=-{\tilde{\mathbf{f}}}_{\mathrm{w}\tilde{k}}·d_{\mathrm{e}}\mathbf{w},$$

(71a)

where we have used the fact that $\tilde{\Sigma }$ is in EQ so ${\tilde{E}}_{\tilde{k}}$ does not depend on the internal variable $\tilde{\xi }$ (see Equation (28a)), so that ${\tilde{\mathbf{F}}}_{\mathrm{w}\tilde{k}}\to {\tilde{\mathbf{f}}}_{\mathrm{w}\tilde{k}},$$d\tilde{\mathbf{W}}\to d\tilde{\mathbf{w}}=d_{\mathrm{e}}\tilde{\mathbf{w}}$, and $d_{\mathrm{i}}\tilde{\mathbf{w}}=0$. We have also used Equation (65b) to set $d\tilde{\mathbf{w}}=-d_{\mathrm{e}}\mathbf{w}$ in the last equation. Thus,

$$d{\tilde{W}}_{\tilde{k}}=d_{\mathrm{e}}{\tilde{W}}_{\tilde{k}}=-d_{\mathrm{e}}{W}_k,$$

(71b)

where we have also used Equation (70a) to derive the last equation.

Remark 28. 

A careful reader will notice that we have an equality between two quantities having different and independent suffixes $\tilde{k}$ and k. This implies that we can change one index, say k, and not change $\tilde{k}$. As the equality again remains valid, both sides must be independent of the suffixes. It will be justified later in Section 7.5 in a different way.

Thus,

$$d{\tilde{W}}_{\tilde{k}}=d\tilde{W}=d_{\mathrm{e}}\tilde{W},d_{\mathrm{e}}{W}_k=d_{\mathrm{e}}W,$$

(72a)

and

$$d_{\mathrm{e}}W=-d\tilde{W}=-d_{\mathrm{e}}\tilde{W}.$$

(72b)

This is consistent with Equation (64b) as expected. Explicitly, we have

$$d_{\mathrm{e}}{\tilde{W}}_k=-{\tilde{\mathbf{f}}}_{\mathrm{w}}·d_{\mathrm{e}}\mathbf{w}=d\tilde{W},{\tilde{\mathbf{f}}}_{\mathrm{w}}\equiv {\sum }_{\tilde{k}}{p}_{\tilde{k}}{\tilde{\mathbf{f}}}_{\mathrm{w}\tilde{k}}={\mathbf{f}}_{0\mathrm{w}},\forall k,$$

(73a)

where ${\mathbf{f}}_{0\mathrm{w}}$ refers to ${\Sigma }_0$. Thus, the exchange microwork is

$$d_{\mathrm{e}}{W}_k\doteq {\mathbf{f}}_{0\mathrm{w}}·d_{\mathrm{e}}\mathbf{w}=d_{\mathrm{e}}W,\forall k.$$

(73b)

We now identify the internal microwork $d_{\mathrm{i}}{W}_k$:

$$d{W}_{0k}=d{W}_k+d{\tilde{W}}_k=d{W}_k-d_{\mathrm{e}}W=d_{\mathrm{i}}{W}_k,$$

(74)

and is explicitly given by

$$d_{\mathrm{i}}{W}_k=({\mathbf{f}}_{wk}-{\mathbf{f}}_{0\mathrm{w}})·d_{\mathrm{e}}\mathbf{w}+{\mathbf{f}}_{\mathrm{w}k}·d_{\mathrm{i}}\mathbf{w}+{\mathbf{A}}_k·d\xi ,$$

(75)

where we have allowed the possibility of an internal change $d_{\mathrm{i}}\mathbf{w}=d\mathbf{w}-d_{\mathrm{e}}\mathbf{w}$, similar to $d_{\mathrm{i}}\xi =d\xi$. Such a situation arises if $\mathbf{w}$ refers to polarization or magnetization, which can change due to internal processes.

We now turn to the physical significance of the three different terms in the internal microwork $d_{\mathrm{i}}{W}_k$:

• The first term is the internal microwork due to force imbalance ${\mathbf{f}}_{\mathrm{w}k}-{\mathbf{f}}_{0\mathrm{w}}$ between the SI-microforce of $\Sigma$, and the MI-macroforce of $\tilde{\Sigma }$.
• The second term is the internal microwork due to the internal displacement $d_{\mathrm{i}}\mathbf{w}$ by the SI-microforce ${\mathbf{f}}_{\mathrm{w}k}$ of $\Sigma$.
• The last term is due to the internal variable displacement by the SI-microaffinity ${\mathbf{A}}_k$.

We introduce the internal microforce imbalance ($\mu$FI, $\mu$ for micro) between $\Sigma$ and $\tilde{\Sigma }$, and the internal SI-microforce

$$\Delta {\mathbf{F}}_{\mathrm{w}k}\doteq ({\mathbf{f}}_{wk}-{\mathbf{f}}_{0\mathrm{w}},{\mathbf{A}}_k),{\mathbf{f}}_{\mathrm{w}k},$$

(76a)

respectively, and the corresponding displacements

$$(d_{\mathrm{e}}\mathbf{w},d\xi ),d_{\mathrm{i}}\mathbf{w}$$

(76b)

to reproduce Equation (75).

The corresponding macroforce imbalance and the internal macroforce are given by

$$\Delta {\mathbf{F}}_{\mathrm{w}}=({\mathbf{f}}_{\mathrm{w}}-{\mathbf{f}}_{0\mathrm{w}},\mathbf{A}),{\mathbf{f}}_{\mathrm{w}},$$

(76c)

with the same displacements as above. Here, we will take a more general view of $\mathbf{A}$, and also extend its definition to $\mathbf{X}$. For $\mathbf{w}$, this means that we can treat ${\mathbf{f}}_{\mathrm{w}}-{\mathbf{f}}_{0\mathrm{w}}$ also as an affinity. By including $\Delta F^{\mathrm{h}}\doteq T_0-T$ also as an affinity [134], we can include it with $\Delta {\mathbf{F}}_{\mathrm{w}}$ to form an extended set of thermodynamic forces or macroforce imbalances [51]:

$$\Delta \mathbf{F}\doteq (T_0-T,{\mathbf{f}}_{\mathrm{w}}-{\mathbf{f}}_{0\mathrm{w}},\mathbf{A}).$$

(76d)

Claim 9. 

The extended set $\Delta \mathbf{F}$ of thermodynamic forces in Equation (76d) must vanish in EQ. However, $\Delta {\mathbf{F}}_{\mathrm{w}k}$ need not vanish even in EQ.      

3.3. Some Simple Examples

As an example, we focus on the case with $\mathbf{W}=(V,\xi ),\tilde{w}=\left(\tilde{V}\right)$. The corresponding ${\mathbf{f}}_{0\mathrm{w}}$ is replaced by $P_0$ of $\tilde{\Sigma }$ so that (setting $d_{\mathrm{i}}V=0$)

$$\begin{array}{cc}\hfill d{W}_k& =P_{k}dV+A_{k}d\xi ,\hfill \end{array}$$

(77a)

$$\begin{array}{cc}\hfill d_e{W}_k& =P_{0}dV,\forall k,d_{e}W=P_{0}dV;\hfill \end{array}$$

(77b)

$$\begin{array}{cc}\hfill d_{\mathrm{i}}{W}_k& =(P_k-P_0)dV+A_{k}d\xi ,\forall k;\hfill \end{array}$$

(77c)

we can identify the two internal parts

$$d_{\mathrm{i}}{W}_{kV}=(P_k-P_0)dV,d_{\mathrm{i}}{W}_{k\xi }=A_{k}d\xi$$

(78)

that make up $d_{\mathrm{i}}{W}_k$. The corresponding macroworks are given by

$$\begin{array}{cc}\hfill dW& =PdV+Ad\xi ,d_{e}W=P_{0}dV,\hfill \end{array}$$

(79a)

$$\begin{array}{cc}\hfill d_{\mathrm{i}}W& =(P-P_0)dV+Ad\xi .\hfill \end{array}$$

(79b)

The results $d_{\mathrm{e}}W=P_{0}dV,d_{\mathrm{i}}W=(P-P_0)dV$ in the absence of $\xi$ are well-known in classical thermodynamics [51]. We identify the irreversible macrowork $d{W}_V$ and $d{W}_{\xi }$ due to V and $\xi$, respectively, from Equation (78):

$$d_{\mathrm{i}}{W}_V=(P-P_0)dV\ge 0,d_{\mathrm{i}}{W}_{\xi }=Ad\xi \ge 0.$$

(80)

The above example describes a possible NEQ situation in Figure 3a of a gas of volume V in a cylinder with a movable piston forming the system $\Sigma$ described by $\mathbf{W}=(V,\xi )$ by considering its microstate $\mathfrak{M}$ very close to ${\mathfrak{M}}_{\mathrm{eq}}$ so that only one internal variable is sufficient to describe it uniquely by treating $\mathfrak{M}={\mathfrak{M}}_{\mathrm{ieq}}$. A possible choice of $\xi$ can be rationalized as follows. We imagine the gas to be divided into two parts of volumes $V_1,V_2$ and uniform number densities $n_1,n_2$, respectively, by an imaginary wall, with the region next to the piston designated as $V_1$. The entire volume is not uniform if $n_1\ne n_2$, which we assume. We now define

$$\xi \doteq V_1/n_1-V_2/n_2,$$

recalling that $V=V_1+V_2$; see Section 4 for a generalization to describe $\mathfrak{M}={\mathfrak{M}}_{\mathrm{ieq}}$ far away from ${\mathfrak{M}}_{\mathrm{eq}}$ that will require many internal variables. In a given microstate ${\mathfrak{m}}_k$, the SI-pressure $P_k$ and the affinity $A_k$ form the corresponding microforce ${\mathbf{F}}_{\mathrm{w}k}=(P_k,A_k)$ (see Equation (17a)) with

$$P_k=-\partial E_k/\partial V,A_k=-\partial E_k/\partial \xi .$$

(81)

The corresponding generalized microwork $d{W}_k$ is given in Equation (77a). For the medium, the generalized microforce ${\tilde{f}}_{\mathrm{w}\tilde{k}}=P_{\tilde{k}}$ determines the generalized microwork $d{\tilde{W}}_{\tilde{k}}=P_{\tilde{k}}d\tilde{V}=-P_{\tilde{k}}dV$ in Equation (71a). The conditional microforce ${\tilde{\mathbf{f}}}_{\mathrm{w}k}$ from Theorem 1 is equal to ${\tilde{\mathbf{f}}}_{\mathrm{w}}=P_0,\forall k$; here, $P_0$ is the external pressure on the piston. Thus, the conditional microwork is $d{\tilde{W}}_k=d\tilde{W}=-P_{0}dV$ so that $d_{\mathrm{e}}{W}_k=-d{\tilde{W}}_k=P_{0}dV=d_{\mathrm{e}}W$ given in Equation (77b). The internal microwork in the gas is $d_{\mathrm{i}}{W}_k$ in Equation (77a).

The irreversible macrowork $d_{\mathrm{i}}W=(P-P_0)dV+Ad\xi$ must be nonnegative as we prove in Theorem 4. For  this to be true, each term must be nonnegative; see Equation (80). Indeed, it is easy to verify that $d_{\mathrm{i}}{W}_V\doteq (P-P_0)dV\ge 0$. For $P>P_0$, the gas must expand so $dV>0$. For $P<P_0$, the gas must contract so $dV<0$. In both cases, the product satisfies the inequality.

This will become more clear by the following example of a spring discussed below.

The pressure difference $\Delta P=P-P_0$ (see Figure 3a) plays an important role as a macroforce imbalance in capturing dissipation. Only under mechanical equilibrium do we have the imbalance vanish ($\Delta P=0$). This imbalance is a general feature but its importance at the microstate level in NEQ statistical mechanics has not been recognized. The following examples will make it abundantly clear that a nonzero microforce imbalance like $\Delta P_k=P_k-P_0$ is just as common even in classical mechanics whenever there is absence of mechanical equilibrium as in thermodynamics, EQ or otherwise. This is because the determination of various microforces and microworks are oblivious to any stochasticity; see Remark 30 for $d_{\alpha }W=-d_{\alpha }{E}_{\mathrm{w}}$. As a consequence, there are no restrictions on the sign of $d_{\mathrm{i}}{W}_k$ as it is purely a mechanical quantity. Therefore, our second example below covers classical mechanics as well as thermodynamics; see also Conclusion 3.

Consider a general but purely classical mechanical one-dimensional massless spring of arbitrary Hamiltonian $\mathcal{H}\left(x\right)$ with one end fixed at an immobile wall on the left and the other end with a mass m free to move; see Figure 3b with vacuum and no fluid filling the cylinder. We consider a particular microstate ${\mathfrak{m}}_k$ of energy $E_k$ given by $\mathcal{H}$. The center of mass of m is located at x from the left wall. The free end is pulled mechanically by an external force (not necessarily a constant) $F_0$ applied at time $t=0$ and changes x; thus, x acts as a work parameter. We do not show the center-of-mass momentum p, as it plays no role in determining work.

Initially the spring is undisturbed and has zero SI restoring spring microforce $F_{\mathrm{w}k}=-\partial E_k/\partial x$; see Equation (17a). The microwork done by $F_{\mathrm{w}k}$ is the SI-work $d{W}_k=F_{\mathrm{w}k}dx$ as given in Equation (37a). The total microforce

$$F_{\mathrm{t}k}=F_0+F_{\mathrm{w}k}$$

(82)

represents the microforce imbalance ($\mu$FI) $F_{\mathrm{t}k}\lessgtr 0$ as discussed later in Section 6.4; recall that $F_0$ and $F_{\mathrm{w}k}$ point in opposite directions so $F_{\mathrm{t}k}$ is a difference $\Delta F_{\mathrm{w}k}=F_{\mathrm{w}k}-\left|F_0\right|$. There is no mechanical equilibrium unless $\Delta F_{\mathrm{w}k}=0$ and the spring continues to stretch or contract, thereby giving rise to an oscillatory motion that will go on forever. During each oscillation, $\Delta F_{\mathrm{w}k}$ is almost always nonzero, except when the mass is momentarily at the equilibrium (mechanical) position of the spring where $\Delta F_{\mathrm{w}k}=0$. The SI-microwork done by $F_{\mathrm{w}k}$ is the spring work (see Equation (37a))

$$d{W}_k=-d{E}_k,$$

(83)

while the microwork performed by the external source is $d_{\mathrm{e}}W=-F_{0}dx$. Being a purely mechanical example, there is no dissipation. Despite this, we can introduce using our notation

$$d_{\mathrm{i}}{W}_k\doteq d{W}_k-d_{\mathrm{e}}W\equiv \Delta F_{\mathrm{w}k}dx;$$

(84)

this microwork can be of either sign (no second law here) and represents the work done by the $\mu$FI $F_{\mathrm{t}k}=\Delta F_{\mathrm{w}k}$. Thus,

Conclusion 2. 

$d{W}_k,d_{\mathrm{e}}{W}_k$ and $d_{\mathrm{i}}{W}_k$ represent different kinds of mechanical work, a result that has nothing to do with dissipation but only with the microforce imbalance; among these, only the generalized work $d{W}_k$ is a SI microwork.

3.4. Manipulations with $d_{\alpha }$

As introduced in Equation (9), $d_{\alpha }$ can be applied to micro- and macroquantities in the collection such as $d_{\alpha }{E}_k,d_{\alpha }{W}_k,d_{\alpha }{Q}_k$, etc., and $d_{\alpha }E,d_{\alpha }W,d_{\alpha }Q$, etc.

Definition 21. 

Micropartition: The micropartition of the BI-$d{\theta }_k\left(t\right)$ for ${\Sigma }_b$ is given in Equation (14b), in which $d_{\mathrm{e}}{\theta }_k\left(t\right)$ is the change due to exchange with its surroundings, and $d_{\mathrm{i}}{\theta }_k\left(t\right)\left(t\right)$ is the internal change within ${\Sigma }_{\mathrm{b}}$.

The corresponding partitions for $d{E}_k$ and $d{S}_k$ are given in Equation (15), and those for $d{W}_k$ and $d{Q}_k$ in Equation (54a). For a Fl-$\mathbf{W}$, we have

$$d{\mathbf{W}}_k\doteq d_{\mathrm{e}}{\mathbf{W}}_k+d_{\mathrm{i}}{\mathbf{W}}_k.$$

(85)

The micropartition also applies to $d{p}_k$:

$$d{p}_k\doteq d_{\mathrm{e}}{p}_k+d_{\mathrm{i}}{p}_k,$$

(86a)

We define

$$d_{\alpha }{\eta }_k\doteq \frac{d_{\alpha }{p}_k}{p_k}.$$

(86b)

Definition 22. 

Macropartition: The macropartition of $d\theta \left(t\right)$ for ${\Sigma }_{\mathrm{b}}$ is given in Equation (14a). It consists of two parts; the exchange $d_{\mathrm{e}}\theta$ is the change due to exchange with its surroundings, and $d_{\mathrm{i}}\theta$ is the irreversible change occurring within ${\Sigma }_{\mathrm{b}}$.

For the average in Equation (19) or for a NFl-$\mathbf{W}$, we have

$$d\mathbf{W}\doteq d_{\mathrm{e}}\mathbf{W}+d_{\mathrm{i}}\mathbf{W}.$$

(87)

In a process, $\chi$ undergoes infinitesimal changes $d_{\alpha }{\chi }_k$ at fixed $p_k$, or infinitesimal changes $d_{\alpha }{p}_k$ at fixed ${\chi }_k$. The changes result in two distinct ensemble averages or process quantities.

Definition 23. 

Infinitesimal macroquantities $〈d_{\alpha }\mathrm{q}〉,$q$\in \chi =\left\{S,\mathbf{Z}\right\}$ are ensemble averages

$$d_{\alpha }{\mathrm{q}}_{\mathrm{m}}=〈d_{\alpha }\mathrm{q}〉=\widehat{A}{d}_{\alpha }{\mathrm{q}}_k\doteq {\sum }_k{p}_k{d}_{\alpha }{\mathrm{q}}_k,$$

(88a)

at fixed $\left\{p_k\right\}$ so they are isentropic. They generalize the earlier definition in Equation (36a). We identify them as mechanical macroquantity and write them as $d_{\alpha }$q${}_{\mathrm{m}}$ for brevity. Infinitesimal macroquantities

$$d_{\alpha }{\mathrm{q}}_{\mathrm{s}}\doteq 〈\mathrm{q}{d}_{\alpha }\eta 〉\doteq {\sum }_k{\mathrm{q}}_k{d}_{\alpha }{p}_k,$$

(88b)

which are ensemble averages involving $\left\{d_{\alpha }{p}_k\right\}$ with a concomitant change $dS$ in the entropy. We identify them as stochastic macroquantities and write them as $d_{\alpha }$q${}_{\mathrm{s}}$ for brevity. Together, they determine the change $d_{\alpha }$q:

$$d_{\alpha }\mathrm{q}\equiv d_{\alpha }\overline{\mathrm{q}}\doteq d_{\alpha }{\mathrm{q}}_{\mathrm{w}}+d_{\alpha }{\mathrm{q}}_{\mathrm{s}},\mathrm{q}\in \left\{S,\mathbf{Z}\right\}.$$

(89)

Remark 29. 

The above equation shows that we must carefully distinguish $d_{\alpha }$ q $=d_{\alpha }\overline{\mathrm{q}}$ and $d_{\alpha }$ q ${}_{\mathrm{w}}=\overline{d_{\alpha }\mathrm{q}}$; their difference, the commutator ${\widehat{C}}_{\alpha }$q, is the stochastic quantity $d_{\alpha }$q${}_{\mathrm{s}}$, discussed in Section 10:

$${\widehat{C}}_{\alpha }\mathrm{q}=d_{\alpha }\mathrm{q}-d_{\alpha }{\mathrm{q}}_{\mathrm{m}};$$

(90)

see Equation (229).

Remark 30. 

For E, the above distinction is the content of the extension of the first law or the law of the conservation of energy

$$d_{\alpha }E=d_{\alpha }Q-d_{\alpha }W.$$

(91)

We immediately identify that

$$d_{\alpha }Q=d_{\alpha }{E}_{\mathrm{s}},d_{\alpha }W=-d_{\alpha }{E}_{\mathrm{w}}.$$

(92)

For $d_{\alpha }=d,d_{\mathrm{e}}$, we have the SI- and MI-formulation of the first law given by (recall that $dE\equiv d_{\mathrm{e}}E$ as $d_{\mathrm{i}}E\equiv 0$)

$$\begin{array}{cc}\hfill dE& =dQ-dW,\hfill \end{array}$$

(93a)

$$\begin{array}{cc}\hfill d_{e}E& =d_{e}Q-d_{e}W.\hfill \end{array}$$

(93b)

Remark 31. 

The SI-formulation of the first law in Equation (93a) shows that $dE$ can be uniquely partitioned into a stochastic component $dQ$ determined by $dS$ and a mechanical component $dW$ determined by $d\mathbf{W}$, which have independent origins.      

Traditionally, the first law is expressed in terms of the change in the energy caused by exchange quantities and is written as

$$dE=d_{\mathrm{e}}Q-d_{\mathrm{e}}W.$$

(94)

As the exchange form of $dE$ is written as $d_{\mathrm{e}}E$ (see Equation (52c)), this is equivalent to the first law in Equation (93b).

We now prove the following important thermodynamic identity as a theorem [75,76,134,148,149].

Theorem 3. 

For any NEQ process $\mathcal{P}$,

$$d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W\ge 0.$$

(95)

Proof. 

For $d_{\alpha }=d_{\mathrm{i}}$ in Equation (91), and using Equation (53a), we have

$$d_{\mathrm{i}}E=d_{\mathrm{i}}Q-d_{\mathrm{i}}W=0,$$

(96)

from which follows the following important thermodynamic identity $d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W$. We defer the proof of the inequality to a later part of the review. ☐

The above equality emphasizes the well-known fact (first discovered in 1798 by Count Rumford of Bavaria [165]) that the irreversible macrowork is always equal in its value but not in its cause (see later) to the irreversible macroheat. The inequality is governed by the second law. The analysis also demonstrates the important fact that the first law in Equation (93a) can be applied either to an exchange process in Equation (93b) or to an interior process in Equation (96). Indeed, in the last formulation, the law is also applicable to an isolated system for which it is replaced by

$$d{E}_0=d{Q}_0-d{W}_0=0.$$

(97)

Definition 24. 

For any body ${\Sigma }_{\mathrm{b}}$, we simply refer to $d{W}_k$ and $d{Q}_k$ as generalized or BI-microwork and generalized or BI-microheat or simply microwork and microheat, respectively. Similarly, we refer to $dW$ and $dQ$ as generalized or BI-macrowork and generalized or BI-macroheat or simply macrowork and macroheat, respectively. We will always refer to $d_{\mathrm{e}}{W}_k$ and $d_{\mathrm{e}}{Q}_k$ as exchange microwork and exchange microheat, respectively. We use exchange macrowork and exchange macroheat for $d_{\mathrm{e}}W$ and $d_{\mathrm{e}}Q$, respectively. As there is no irreversibility in mechanics, we use internal microheat for $d_{\mathrm{i}}{Q}_k$ and internal microwork for $d_{\mathrm{i}}{W}_k$, respectively; see Claim 8. We never the use the prefix irreversible for these or other internal microquantities. We use irreversible macroheat for $d_{\mathrm{i}}Q$ and irreversible macrowork for $d_{\mathrm{i}}W$, respectively.

As the system $\Sigma$ is of primary interest in the $\mu$NEQT, we will always reduce any microquantity associated with $\tilde{\Sigma }$ and ${\Sigma }_0$ to refer to the microstate ${\mathfrak{m}}_k$. Thus, all microquantities for any ${\Sigma }_{\mathrm{b}}$ will carry the suffix k of ${\mathfrak{m}}_k$. We will usually refer to $d{\tilde{W}}_k$ as the external microwork to distinguish it from the microwork $d{\tilde{W}}_{\tilde{k}}$ done by $\tilde{\Sigma }$ in its microstate ${\tilde{\mathfrak{m}}}_{\tilde{k}}$. We will use microenergy change, exchange microenergy change, and internal microenergy change for $d{E}_k,d_{\mathrm{e}}{E}_k,$ and $d_{\mathrm{i}}{E}_k$. We will refer to $d_{\alpha }{S}_k$ as microentropy change, even though both $d_{\alpha }{S}_k$ and $d_{\alpha }{Q}_k$ are mixed microquantities; see Remark 14.

4. Internal Variables

Let us consider two noninteracting mechanical systems ${\Sigma }_1$ and ${\Sigma }_2$ that form a composite system $\Sigma$, which we take to be isolated. We assume that both ${\Sigma }_1$ and ${\Sigma }_2$ are physically “similar” in that each requires the same set of NFl-state variable $\mathbf{W}$ having r components, so separately they are described by Hamiltonians $E_{1k_1}={\mathcal{H}}_{1k_1}\left({\mathbf{W}}_1\right)$ and $E_{2k_2}={\mathcal{H}}_{2k_2}\left({\mathbf{W}}_2\right)$ for ${\mathfrak{m}}_{1k_1}$ and ${\mathfrak{m}}_{2k_2}$ of ${\Sigma }_1$ and ${\Sigma }_2$, respectively. We assume that the number of particles $N_1\in {\mathbf{W}}_1\doteq ({\mathbf{w}}_1,{\xi }_1)$ and $N_2\in {\mathbf{W}}_2\doteq ({\mathbf{w}}_2,{\xi }_2)$ are kept fixed in the two microstates so their total N is also fixed for each microstate ${\mathfrak{m}}_k$ of $\Sigma$ given by

$${\mathfrak{m}}_k={\mathfrak{m}}_{1k_1}\otimes {\mathfrak{m}}_{2k_2}.$$

(98)

As the particle numbers are fixed, we do not consider them to be part of the work sets anymore. We choose to express the combined Hamiltonian as

$$E_k\doteq {\mathcal{H}}_k({\mathbf{Z}}_1,{\mathbf{W}}_2)={\mathcal{H}}_{1k_1}\left({\mathbf{W}}_1\right)+{\mathcal{H}}_{2k_2}\left({\mathbf{W}}_2\right)$$

(99)

of ${\mathfrak{m}}_k$, which is a function of $2r+2$ state variables (which includes the microenergies $E_{1k_1}$ and $E_{2k_2}$ of ${\Sigma }_1$ and ${\Sigma }_2$, respectively), from which we construct the following independent combinations:

$$\mathbf{Z}\doteq {\mathbf{Z}}_1+{\mathbf{Z}}_2,\widehat{\xi }\doteq {\mathbf{Z}}_1/n_1-{\mathbf{Z}}_2/n_2,$$

(100)

so that we can equivalently express ${\mathcal{H}}_k({\mathbf{Z}}_1,{\mathbf{W}}_2)$ as ${\mathcal{H}}_k(\widehat{\mathbf{W}},\xi )$ of $2(r+1)$ variables, which excludes $E_k$ as explained below; here, $n_1=N_1/N$ and $n_2=N_2/N$,

$$\widehat{\mathbf{W}}\doteq {\mathbf{W}}_1+{\mathbf{W}}_2=({\mathbf{w}}_1+{\mathbf{w}}_2,{\xi }_1+{\xi }_2)$$

(101)

is the total initial work variable set, and $\xi$ is the new set of internal variables beyond those included in ${\mathbf{Z}}_1$ and ${\mathbf{Z}}_2$. In addition, the excluded $E_k\doteq E_{1k_1}+E_{2k_2}$ is the microenergy of ${\mathfrak{m}}_k$, and carries the suffix k. The choice of new arguments for ${\mathcal{H}}_k({\mathbf{W}}_1,{\mathbf{W}}_2)$ is convenient as it allows it to be expressed as ${\mathcal{H}}_k\left(\mathbf{W}\right)$ in terms of the set formed by $2r+1$ variables

$$\mathbf{W}\doteq (\widehat{\mathbf{W}},\widehat{\xi })$$

(102a)

of the composite system $\Sigma$, as is also done for ${\Sigma }_1$ and ${\Sigma }_2$. The set of internal variables

$$\xi \doteq ({\xi }_1+{\xi }_2,\widehat{\xi })$$

(102b)

denotes the set of internal variables for $\Sigma$.

Manipulating $\mathbf{W}$ will change the energy $E_k$ of $\Sigma$. Thus,

$$d{E}_k=\frac{\partial E_{1k_1}}{\partial {\mathbf{W}}_1}·d{\mathbf{W}}_1+\frac{\partial E_{2k_2}}{\partial {\mathbf{W}}_2}·d{\mathbf{W}}_2.$$

(103a)

It is easy to check that $d{E}_k$ is also given by

$$d{E}_k=\frac{\partial E_k}{\partial \mathbf{W}}·d\mathbf{W}=\frac{\partial E_k}{\partial \widehat{\mathbf{W}}}·d\widehat{\mathbf{W}}+\frac{\partial E_k}{\partial \xi }·d\widehat{\xi },$$

(103b)

so both representations of ${\mathcal{H}}_k$ are equivalent in all ways.

The choice of $\widehat{\xi }$ in terms of $n_1$ and $n_2$ ensures that it vanishes if the two systems form a uniform system $\Sigma$ for which we must have ${\mathbf{Z}}_1/N_1={\mathbf{Z}}_2/N_2$. However, other choices for $\widehat{\xi }$ can also be made as long as $\widehat{\xi }$ remains independent of $\widehat{\mathbf{W}}$.

Let us consider a simple example in which we only allow the energy E and volume V for each each system ($r=1$). We have ${\xi }_{\mathrm{V}},{\xi }_{\mathrm{E}k}$ as work variables in forming $\mathbf{W}$. In this case, we have $E_k=E_{1k_1}+E_{2k_2}$ for the microstate energy and $V=V_1+V_2$ for the total volume. By definition,

$${\xi }_{\mathrm{E}k}=E_{1k_1}/n_1-E_{2k_2}/n_2,{\xi }_{\mathrm{V}}=V_1/n_1-V_2/n_2.$$

(104a)

The microstate energy

$$E_k(V,{\xi }_{\mathrm{V}},{\xi }_{\mathrm{E}k})=E_{1k_1}\left(V_1\right)+E_{2k_2}\left(V_2\right)$$

(104b)

is a function of three ($2r+1$) variables. We first consider ${\xi }_{\mathrm{V}}$. We have for $P_k$, using Equation (81),

$$P_k=n_1{P}_{1k_1}+n_2{P}_{2k_2},A_{\mathrm{V}k}=n_1{n}_2(P_{1k_1}-P_{2k_2}),$$

(104c)

where we have used $V_1=n_{1}V+n_1{n}_2{\xi }_{\mathrm{V}}$ and $V_2=n_{2}V-n_1{n}_2{\xi }_{\mathrm{V}}$. As V is NFl, $P_k$ is Fl over ${\mathfrak{m}}_k$, as we have learned.

We now use ${\xi }_{\mathrm{E}k}$ to express $E_{k_1}=n_1{E}_k+n_1{n}_2{\xi }_{\mathrm{E}k}$ and $E_{k_2}=n_1{E}_k-n_1{n}_2{\xi }_{\mathrm{E}k}$. Differentiating Equation (104b) with respect to $E_k$ and ${\xi }_{\mathrm{E}k}$, respectively, and using Equation (42), we obtain

$$1=n_1+n_2,A_{\mathrm{E}}=0,$$

(104d)

where $A_{\mathrm{E}}$ (see Equation (18)) is NFl, so it has no suffix k.

As $\Sigma$ is an isolated system, it is deterministic. So the observables ($E_{k_1},E_{k_2},V_1,V_2$) remain constant, which means that $E_k$ and ${\xi }_{\mathrm{E}k},\forall k$, will remain constant in time. If we allow a mutual interaction so that there is a possible energy (or volume) transfer between ${\Sigma }_1$ and ${\Sigma }_2$, then this will be characterized by oscillating ${\xi }_{\mathrm{E}k}$ and ${\xi }_{\mathrm{V}}$ due to energy and volume transfers, respectively, back and forth between the two systems. On the other hand, if the interacting ${\Sigma }_0$ become stochastic, as discussed in Section 7, it will obey the second law and ${\xi }_{\mathrm{E}k}$ and ${\xi }_{\mathrm{V}}$ will eventually vanish. This case is studied later, where it is shown that macroheat flows from hot to cold.

The above discussion can be easily extended to a composite system composed of $m>2$ subsystems by the trick proposed by Gujrati in ([77], Section 3). The trick is very simple. We use the collection $\mathbf{W}=(\widehat{\mathbf{W}},\widehat{\xi })$ introduced above for the composite system. We consider two such composite systems, and introduce their work parameters ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$, which are used in Equation (103b) for each one of them. We now treat each as a system so that we have two new systems ${\Sigma }_1$ and ${\Sigma }_2$ that form a new composite system $\Sigma$. We use ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$ to obtain the new collection of $(\widehat{\mathbf{W}},\widehat{\xi })$ as introduced above. This set defines a new $\mathbf{W}=(\widehat{\mathbf{W}},\widehat{\xi })$ for the new composite system, which now has $m=4$ subsystems. We then treat two such composite systems and treat each as a system to form another new composite system with $m=8$, and so on to finally consider a composite system formed of m subsystems. We thus claim the following:

Claim 10. 

The internal energy $E_k$ of the microstate ${\mathfrak{m}}_k$ of a composite system of m subsystems is a function of the work set$\left\{{\mathbf{W}}_1,{\mathbf{W}}_2,\cdots ,{\mathbf{W}}_m\right\}$ composed of their work parameters, and can be expressed as a function of

$$\widehat{\mathbf{W}}\doteq {\mathbf{W}}_1+{\mathbf{W}}_2+\cdots +{\mathbf{W}}_m$$

and a set $\widehat{\xi }$ of internal variables [77]; together, they form the set $\mathbf{W}$ for the composite system, as shown in Equation (102a).

Claim 11. 

We see that the new combination $\mathbf{\xi }$ is the set of internal variables, which also plays an important role in the unique description of the composite system. As the uniqueness is just as important in a thermodynamic consideration, which will be taken up in the following sections, internal variables will play just as important a role there as here.

The above discussion is for a mechanical system with no interaction, but is easily extended to the case in which the two systems are interacting, as will be done in the following sections. The internal variables discussed above relate to a particular microstate ${\mathfrak{m}}_k$ so some of them may carry the suffix k, and should be denoted as a internal microvariable ${\xi }_k$. To see this, we recall that the microenergy $E_k$ carries the suffix k so any internal variable formed from microenergies of ${\Sigma }_1$ and ${\Sigma }_2$ will carry it as was the case for ${\xi }_{\mathrm{E}k}$ constructed above. The discussion is also easily extended to include thermodynamics, where the internal macrovariable $\xi$ obeys the restrictions imposed by the second law; see Equation (43) and Corollary 1. In this case, ${\mathbf{W}}_{k_l}$ of the lth subsystem will also include the internal variable ${\xi }_{k_l}$, not to be confused with ${\xi }_k$ for the system. It is clear that the complications due to ${\xi }_{k_l}$ are avoided if each subsystem is in EQ so that ${\xi }_{k_k}$’s do not exist, as was the simple example considered above. Then there is a maximum number $n^{*}$ of internal macrovariables in $\xi$ that is determined by m. This has been discussed in recent publications [77,78], to which we refer the reader. By the addition of the suffix, it should be obvious that the above discussion is easily extended to Fl work parameter, such as Fl volume $V_k$ for ${\mathfrak{m}}_k$, so that all microstates experience the same pressure P; see Equation (18). Thus, the above concept of internal variables is quite general. However, for the notational simplicity, we will not add the suffix to $\mathbf{W}$ and $\xi$ unless needed for clarity by clearly specifying the situation.

5. Fundamentals of the $\mathbf{\mu }$NEQT

In this section, we will usually talk about a system, but the discussion is valid for any body ${\Sigma }_{\mathrm{b}}$. The most convenient and most common framework of describing a thermodynamic system $\Sigma$ is in terms of the SI-set $\mathbf{X}=(E,V,N,\cdots )$ of its extensive macroscopic observables, which results in the SI-set $\mathbf{f}$ of the generalized macroforces (see Equation (25)) and the state space ${\mathfrak{S}}_{\mathbf{X}}$ that is sufficient to uniquely describe the EQ system and its macrostate ${\mathfrak{M}}_{\mathrm{eq}}$. A very important SI quantity in thermodynamics is the entropy S that in EQ is uniquely determined by $\mathbf{X}$ so that $S_{\mathrm{eq}}\doteq S\left(\mathbf{X}\right)$ is a state function of ${\mathfrak{M}}_{\mathrm{eq}}$. For an NEQ macrostate $\mathfrak{M}$, S will not be a state function in ${\mathfrak{S}}_{\mathbf{X}}$, so it will depend explicitly on time. In this case, $\mathbf{X}$ no longer forms the set of state variables to uniquely describe $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{X}}$, and both $\mathfrak{M}$ and S have an explicit t-dependence; see Equation (141) for the latter. This is true whether the system is noninteracting (i.e., isolated) or interacting (i.e., interacts with a medium $\tilde{\Sigma }$, which is external to the system $\Sigma$); see Figure 1.

With respect to microstates ${\mathfrak{m}}_k$, the interaction between $\Sigma$ and $\tilde{\Sigma }$ causes MI-exchange $\left[d_{\mathrm{e}}\mathbf{X}\right]$, which is then used to identify $\left[d_{\mathrm{i}}\mathbf{X}\right]\doteq \left[d\mathbf{X}\right]-\left[d_{\mathrm{e}}\mathbf{X}\right]$; see Notation (11a). In general, the  SI-change $\left[d\mathsf{q}\right]$ can be partitioned into $\left[d_{\mathrm{e}}\mathsf{q}\right]$ and $\left[d_{\mathrm{i}}\mathsf{q}\right]$ in accordance with Equations (14b) and (14a), respectively, in which the MI-exchange between $\Sigma$ and $\tilde{\Sigma }$ is caused by their interaction and $\left[d_{\mathrm{i}}\mathsf{q}\right]$ is the change brought about by internal processes within $\Sigma$. In particular, $d_{\mathrm{i}}{\mathsf{q}}_k$ represents the internal microchange, while $d_{\mathrm{i}}\mathsf{q}$ the irreversible macrochange. The  SI-force corresponding to $\mathbf{w}$ is $\left[{\mathbf{f}}_{\mathrm{w}}\right]$; see Equation (37c). There is no microanalog of $f_{\mathrm{s}}$ introduced in Equation (24).

The above discussion is restricted to any ${\mathfrak{M}}_{\mathrm{eq}}$ that is uniquely specified in ${\mathfrak{S}}_{\mathbf{X}},\mathbf{X}=(E,\mathbf{w})$. In an NEQ macrostate ${\mathfrak{M}}_{\mathrm{neq}}$, ${\mathfrak{S}}_{\mathbf{X}}$ is no longer a convenient state space as it cannot specify ${\mathfrak{M}}_{\mathrm{neq}}$ NEQ macrostate uniquely. This loss of uniqueness for ${\mathfrak{M}}_{\mathrm{neq}}$ has been a major obstacle in formulating an NEQ thermodynamics that can be as robust and complete as the classical EQ thermodynamics. All competing NEQT approaches belong to $\mathring{\mathrm{M}}\mathrm{NEQT}$ as discussed in Section 1 and deal only with exchange quantities that can be uniquely described in ${\mathfrak{S}}_{\mathbf{X}}$, as the medium $\tilde{\Sigma }$ is always taken to be in EQ. Thus, they cannot offer any help to overcome the nonuniqueness of ${\mathfrak{M}}_{\mathrm{neq}}$.

We consider this loss of uniqueness to be the main issue in improving our current incomplete understanding of NEQ processes. Our approach to overcome this loss is to describe ${\mathfrak{M}}_{\mathrm{neq}}$ in an appropriately enlarged state space to ${\mathfrak{S}}_{\mathbf{Z}}$ by including internal variable set [12,13,18,42,51,108,134,148,166,167,168] $\xi$ and identifying $\mathbf{Z}\doteq \mathbf{X}\cup \xi$ as the set of state variables to uniquely specify ${\mathfrak{M}}_{\mathrm{neq}}$. The internal variables also play a very dominant role in glassy and granular materials [169,170,171,172,173]. In all previous theories involving internal variables, they are introduced almost in an ad hoc manner without providing any physical insight into their origin. In contrast, our approach to introduce them differs from other approaches by providing a very clear and physical prescription, as discussed in Section 4. As ${\mathfrak{M}}_{\mathrm{eq}}$ describes a uniform system [33], ${\mathfrak{M}}_{\mathrm{neq}}$ invariably requires some sort of nonuniformity, as in a composite system $\Sigma ={\cup }_i{\Sigma }_i$ composed of various subsystems ${\Sigma }_i$. At the mechanical level, this nonuniformity is captured by the parameters of the SI-Hamiltonians of ${\Sigma }_i$, as was the case with two subsystems in Equation (99). The internal variables as they appear in Equation (100) are mathematically required to ensure that the number of independent variables on both sides in Equation (99) are exactly the same. While their forms may not be unique, they must be independent. In terms of $\mathbf{Z}$, we now have a complete SI-specification of ${\mathfrak{m}}_k$ of $\Sigma$, assuming a certain choice of $\xi$. This is the uniqueness we are looking for to develop the NEQ statistical mechanics. As  discussed in Section 4, $\xi$ cannot be controlled from the outside of $\Sigma$. Therefore, its variation is due to internal processes only and may be controlled by the second law. It should be obvious from the discussion in Section 4 that $\xi$ for a purely mechanical system such as ${\mathfrak{m}}_k$ cannot have any connection with the second law. Only in the presence of stochasticity required for a thermodynamic system will its average behavior be governed by the second law, so it also plays an important role in our approach. However, the requirement of including internal variables for a complete specification is a mechanical necessity due to nonuniformity, but becomes critical in the NEQ statistical mechanics. We direct readers to Section 5.7 for a simple example that clarifies its importance.

In the following, we will be considering the state space ${\mathfrak{S}}_{\mathbf{Z}}$ in which the entropy is a state function $S\left(\mathbf{Z}\right)$ so that we will be dealing with ${\mathfrak{M}}_{\mathrm{ieq}}$; see Definition 13. This means that $\left\{p_k\right\}$ are uniquely defined to specify ${\mathfrak{M}}_{\mathrm{ieq}}$. However, $\left\{{\mathfrak{m}}_k\right\}$ themselves are independent of this particular choice of $\left\{p_k\right\}$, simply because $\left\{{\mathfrak{m}}_k\right\}$ are determined by the deterministic Hamiltonian of $\Sigma$ as discussed in Section 1, so they remain oblivious to their probabilities. It is this independence of $\left\{{\mathfrak{m}}_k\right\}$ and $\left\{p_k\right\}$ that allows us to develop the $\mu$NEQT as a mechanical theory that is modified by stochasticity by extending the conventional similar approach in the $\mu$EQT [33,54].

Let us consider an infinitesimal change $d\mathbf{Z}$ in $\mathbf{Z}$ that takes ${\mathfrak{M}}_{\mathrm{ieq}}^{\prime }={\mathfrak{M}}_{\mathrm{ieq}}\left(\mathbf{Z}\right)$ to ${\mathfrak{M}}_{\mathrm{ieq}}^{″}={\mathfrak{M}}_{\mathrm{ieq}}(\mathbf{Z}+d\mathbf{Z})$ both belonging to ${\mathfrak{S}}_{\mathbf{Z}}$. If the system always stays within ${\mathfrak{S}}_{\mathbf{Z}}$ during this change, then the change is carried out along an IEQ process in ${\mathfrak{S}}_{\mathbf{Z}}$. It is ${\mathfrak{M}}_{\mathrm{ieq}}$ during this change so that $d{W}_k=-d{E}_k$. If intermediate macrostates leave ${\mathfrak{S}}_{\mathbf{Z}}$ during this change, then the change is not carried out along an IEQ process in ${\mathfrak{S}}_{\mathbf{Z}}$. Nevertheless, the microenergy change $d{E}_k=-d{W}_k$ between ${\mathfrak{M}}_{\mathrm{ieq}}^{\prime }$ and ${\mathfrak{M}}_{\mathrm{ieq}}^{″}$ is the same in both situations. In other words, $d{E}_k=-d{W}_k$ is the same between ${\mathfrak{M}}_{\mathrm{ieq}}^{\prime }$ and ${\mathfrak{M}}_{\mathrm{ieq}}^{″}$, regardless of the nature of the process.

We will focus on an isolated composite system $\Sigma$ in microstate ${\mathfrak{m}}_k$ made of two subsystems ${\Sigma }_1$ in microstate ${\mathfrak{m}}_{k_1}$ and ${\Sigma }_2$ in microstate ${\mathfrak{m}}_{k_2}$; recall Remark 25. Following from Remarks 21 and 26, we now conclude that

$$d{\mathsf{q}}_k\equiv d_{\mathrm{i}}{\mathsf{q}}_k=d{\mathsf{q}}_{k1}+d{\mathsf{q}}_{k2}.$$

In particular, we have

$$d_{\mathrm{i}}{W}_k=d{W}_{k_1}+d{W}_{k_2};$$

(105)

we can use Equation (37a) for NFl ${\mathbf{W}}_l$ and Equation (38) for Fl ${\mathbf{W}}_{k_l},l=1,2$, to determine $d{W}_{k_l},l=1,2$.

Let us consider one of the above three bodies and focus on its $\mathbf{W}$. For NFl $\mathbf{W}$, the corresponding generalized microforce ${\mathbf{F}}_{\mathrm{w}k}$ is Fl as shown in Equation (17a). For Fl ${\mathbf{W}}_k$, the corresponding generalized microforce ${\mathbf{F}}_{\mathrm{w}}$ is NFl, as shown in Equation (18). Including $E_k$, which is always FL, we see that $\mathbf{Z}$ for the body is Fl in the latter case.

As shown in Equation (20), the BI-microwork $d{W}_k={\mathbf{F}}_{\mathrm{w}k}·d\mathbf{W}$ and $d{W}_k={\mathbf{F}}_{\mathrm{w}}·d{\mathbf{W}}_k$ defined mechanically as force × displacement in the two cases are the same, and are fluctuating over $\left\{{\mathfrak{m}}_k\right\}$ as expected due to the ubiquitous Fl microforce and Fl work parameter, respectively. The mechanically defined macrowork $dW$ in each case will result in the irreversible macrowork $d_{\mathrm{i}}W\ge 0$ in accordance with the second law. It follows from Equation (105) that each side represents a mechanical microwork, showing that even $d_{\mathrm{i}}{W}_k$ is a mechanical quantity. It follows from Theorem 6 that $d_{\mathrm{i}}{E}_k=-d_{\mathrm{i}}{W}_k$, again emphasizing that $d_{\mathrm{i}}{W}_k$ has a mechanical origin. However, the second law puts no restriction on the Fl mechanical microanalog $d_{\mathrm{i}}W$. For the example of the spring with the force imbalance given in Equation (82) with NFl x, the internal microwork is given in Equation (84) and can be of any sign according to the signature of the internal microforce imbalance $\Delta F_{\mathrm{w}k}$. In the presence of any microforce imbalance (see Conclusion 2) in an NEQ system, $d_{\mathrm{i}}{W}_k$ will not vanish, even if its average does. The following Remark emphasizes these points.

Remark 32. 

The internal microwork $d_{\mathrm{i}}{W}_k$ within an isolated Σ due to Fl internal microforces or Fl work parameter is ubiquitous. Its presence has a purely mechanical origin, as seen in Equation (84) or in Equation (78) for NFl $\mathbf{W}$. For Fl ${\mathbf{W}}_k$, because of their mechanical nature, different additive parts of $d_{\mathrm{i}}{W}_k$ given in (78) are independent of $p_k$ in that they remain the same between ${\mathfrak{M}}_{\mathrm{ieq}}^{\prime }$ and ${\mathfrak{M}}_{\mathrm{ieq}}^{″}$, both belonging to ${\mathfrak{S}}_{\mathbf{Z}}$, regardless of the processes between them. Despite this, the macroscopic analogs of each of these parts and $d_{\mathrm{i}}W$ are controlled by the second law; see Corollary 1. It follows that in general, determining $d_{\mathrm{i}}{W}_k$ from SI-$d{W}_k$ will be a convenient way to discuss the statistical mechanics of NEQ systems; see Section 2.

We now put down the set of axioms for the formulation of the $\mu$NEQT that are in addition to the axioms put forward by Callen [3]. Callen only discusses a system in equilibrium, so his two most important axioms are about the existence of the entropy function and of the stable equilibrium for EQ macrostates. We extend these axioms to NEQ macrostates below.

Axiom 1. 

Fundamental Axiom The thermodynamic behavior of a system is not the behavior of a single sample, but the average behavior of a large number of independent samples, prepared identically under the same macroscopic conditions at time $t=0$.

Axiom 2. 

Axiom of Entropy Function Existence There exists an entropy function $S\left(\mathfrak{M}\right)$ for $\mathfrak{M}$ in any state space, which may be a function of the state variables in that state space and time t.

Axiom 3. 

Axiom of IEQ Any ${\mathfrak{M}}_{\mathrm{neq}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ can always be turned into a unique ${\mathfrak{M}}_{\mathrm{ieq}}$ in a suitably enlarged state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}\supset {\mathfrak{S}}_{\mathbf{Z}},{\mathbf{Z}}^{\prime }=\mathbf{Z}\cup {\xi }^{\prime }$ so the thermodynamic and statistical entropies are identical; see Proposition 1 and Section 12.6 for details.

Axiom 4. 

Axiom of Stability The unique macrostate ${\mathfrak{M}}_{\mathrm{ieq}}$ for a given $\mathbf{Z}$ is stable in ${\mathfrak{S}}_{\mathbf{Z}}$ in that the system does not leave it if already there or returns to it if disturbed. A stable macrostate satisfies the stability conditions

$$d^{2}S<0,d^{2}E>0.$$

(106)

If we consider the matrix $\mathbf{J}$ formed by ${\partial }^{2}S/\partial Z_j{Z}_{j^{\prime }}$, or the matrix $\mathbf{K}$ formed by ${\partial }^{2}E/\partial {\zeta }_j{\zeta }_{j^{\prime }}$, then all the principle minors of the determinant of $\mathbf{J}$ must be strictly negative, or the determinant of $\mathbf{K}$ must be strictly positive. By allowing $\mathbf{Z}$ to vary, ${\mathfrak{M}}_{\mathrm{ieq}}$ moves to the most stable macrostate ${\mathfrak{M}}_{\mathrm{eq}}$, in which all thermodynamic forces (see Equation (76d)) vanish.

We do not consider the stability border $d^{2}S\to 0,d^{2}E\to 0$ in the review.

It is an observed fact that nature, in her inorganic as well as organic forms, is driven towards greater stability. This tendency is just as ubiquitous in physics as it is in biology. Anything in nature that is capable of changing always changes eventually into an unchanging stable form, even in an explosion. This is also true of the Belusov reaction [51], undergoing oscillations initially but eventually ending into a stable macrostate.

Axiom 5. 

Axiom of quasi-additivity Any quantity $\left[\mathrm{q}\right]$ satisfies the principle of quasi-additivity

$$\left[\mathrm{q}\right]\approx {\sum }_j{\left[\mathrm{q}\right]}_j.$$

(107a)

The above axiom also applies to $\left[S\right]$, the entropy, but requires the following additional axiom of quasi-independence, to be discussed later in Section 7.3.

Axiom 6. 

Axiom of Quasi-independence For entropy to be quasi-additive, as

$$\left[S\right]\approx {\sum }_j{\left[S\right]}_j,$$

(108)

requires the property of quasi-independence (see Claim 7) between different parts of the system.

Axiom 7. 

Axiom of Reduction All microquantities carrying the suffix $\tilde{k}$ and $k_0$, and associated with $\tilde{\Sigma }$ and ${\Sigma }_0$, respectively, must be reduced to microquantities carrying the suffix k under the condition that $\Sigma$ is in the microstate ${\mathfrak{m}}_k$ in order to assess their influence on ${\mathfrak{m}}_k$.

The discussion of the rules for reduction is postponed to Section 7.4.

5.1. Fundamental Axiom

To avoid any influence of the possible changes in the system brought about by measurements, we instead prepare a large number $N_{\mathrm{S}}$ of samples or replicas under identical macroscopic conditions. The replicas are otherwise independent of each other in that they evolve independently in time. This is consistent with the requirement that different measurements should not influence each other. In the rest of this review, we will use the same term ensemble to collectively represent the samples. The average over these samples of some thermodynamic quantity then determines the thermodynamic property of the system. As the replica approach plays a central role in our formalism, we state its importance as Axiom 1, which was first proposed in [79].

Such an approach is standard in equilibrium statistical mechanics [11,33,34,36,54], but it must also apply to systems not in equilibrium. For the latter, this averaging must be carried out by ensuring that all samples have identical history, i.e., prepared at the same time $t=0$. This is obviously not an issue for systems in equilibrium. We refer the reader to a great discussion about the status of statistical mechanics and its statistical nature by Tolman ([54], Section 25), where he clearly puts down this viewpoint of statistical mechanics as follows. We quote from p. 65:

“The methods are essentially statistical in character and only purport to give results that may be expected on the average rather than precisely expected for any particular system.....The methods being statistical in character have to be based on some hypothesis as to a priori probabilities, and the hypothesis chosen is the only postulate that can be introduced without proceeding in an arbitrary manner....”

Tolman [54] then goes on to argue on p. 67 that what statistical mechanics should strive for is to ensure

“...that the averages obtained on successive trials of the same experiment will agree with the ensemble average, thus permitting any particular individual system to exhibit a behavior in time very different from the average;”

see also the last paragraph on p. 106 in Jaynes [174].

5.2. Parameter Description

As said earlier, E is always treated as a random variable E taking the values $\left\{E_k\right\}$ that fluctuate over $\left\{{\mathfrak{m}}_k\right\}$, regardless of how $\mathbf{W}$ is treated. The most convenient description of a system is to use the NFl-$\mathbf{W}$ so it is the same for all $\left\{{\mathfrak{m}}_k\right\}$. Per Claim 3, this results in a random SI-variable ${\mathbb{F}}_{\mathrm{w}}$ with its outcome ${\mathbf{F}}_{\mathrm{w}k}$ (see Equation (17a)) fluctuating over $\left\{{\mathfrak{m}}_k\right\}$ so its ensemble average is the generalized (mechanical) macroforce ${\mathbf{F}}_{\mathrm{w}}$; see Equation (17b). In contrast, the conjugate field $\beta =1/T$ for Fl-E is fixed.

It is possible to use a mixed parameter approach. We consider $\mathbf{W}$ having two nonoverlapping subsets ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$, with ${\mathbf{W}}_1$ a NFl-parameter ${\mathbf{W}}^{\mathrm{NF}}$. The remaining subset ${\mathbf{W}}_2$ is Fl-parameter set ${\mathbf{W}}^{\mathrm{F}}$ taking the values $\left\{{\mathbf{W}}_k^{\mathrm{F}}\right\}$ over $\left\{{\mathfrak{m}}_k\right\}$. We impose the consistency condition on ${\mathbf{W}}_k^{\mathrm{F}}$ (see Claims 3 and 5) so that the corresponding field ${\mathbf{F}}_{\mathrm{w}k}^{\mathrm{F}}=-\partial E_k/\partial {\mathbf{W}}_k^{\mathrm{F}}={\mathbf{F}}_{\mathrm{w}}^{\mathrm{F}},\forall k$; see Equation (18). For a null set ${\mathbf{W}}^{\mathrm{NF}}$, we retrieve the field-parameter description in Claim 3. As before, the consistency requires obtaining the same MNEQT, so we must have

$$〈{\mathbf{W}}^{\mathrm{F}}〉={\mathbf{W}}_2,〈{\mathbf{F}}_{\mathrm{w}}^{\mathrm{NF}}〉={\mathbf{F}}_{\mathrm{w}1};$$

(109)

see Condition 1.

To clarify the above distinction, we consider the simpler case of NFl-$\mathbf{W}=(V,\xi )$ for a system. The energy E is a random variable E taking Fl-values $\left\{E_k\right\}$; their average value is determined by a fixed $f_{\mathrm{s}}=-T$; see Equation (24) and Claim 5. In this ensemble, $T,V$ and $\xi$ are fixed so we can also call it a $(T$-V-$\xi )$-ensemble. In this case, $E_k,P_k$, and $A_k$ are fluctuating over $\left\{{\mathfrak{m}}_k\right\}$. If we take ${\mathbf{W}}^{\mathrm{NF}}=\left(\xi \right)$ and ${\mathbf{W}}^{\mathrm{F}}=\left(V\right)$, then $E_k,V_k$, and $A_k$ are fluctuating over $\left\{{\mathfrak{m}}_k\right\}$ with $T,P$ and $\xi$ kept fixed in this ensemble, which we can call a $(T$-P-$\xi )$-ensemble. We can also consider an ensemble with ${\mathbf{W}}^{\mathrm{NF}}=\left(V\right)$ and ${\mathbf{W}}^{\mathrm{F}}=\left(\xi \right)$. In this ensemble, $E_k,P_k$, and ${\xi }_k$ are fluctuating over $\left\{{\mathfrak{m}}_k\right\}$; $T,V$ and A are kept fixed so we can call it a $(T$-V-$A)$-ensemble. For these ensembles to represent the same physical system thermodynamically, we must have $V=〈V〉,P=〈P〉,\xi =〈\xi 〉$, and  $A=〈A〉$ in accordance with Equation (109).

Remark 33. 

An NEQ ensemble is specified by the set of its NFl quantities ${\mathbf{W}}^{\mathrm{NF}}$ and ${\mathbf{F}}_{\mathrm{w}}^{\mathrm{F}}$.

5.3. Ensemble of Replicas

The discussion here provides an extension of the ideas valid for thermodynamic equilibrium macrostates ${\mathfrak{M}}_{\mathrm{eq}}$ to not only nonequilibrium macrostates ${\mathfrak{M}}_{\mathrm{neq}}$ but also to macrostates ${\mathfrak{M}}_{\det }$. The latter are governed by deterministic dynamics in which microstate probabilities remain constant, as will be justified below; see Claim 12. The premise of the extension to ${\mathfrak{M}}_{\mathrm{neq}}$ is that these ideas must be just as valid for them, as they are based on thermodynamics being an experimental science [79]. Thermodynamics (equilibrium and nonequilibrium) requires verification by performing the experiment many times over. The same premise also applies to ${\mathfrak{M}}_{\det }$. Therefore, we consider all these macrostates in the following, and simply use $\overline{\mathfrak{M}}$ to stand for all these states. We must prepare many copies or replicas $\mathcal{N}>>1$ of the system at the same time t under identical conditions specified by the set of extensive variables $\mathbf{Z}\left(t\right)$ that can be used to also study how the system evolves in time. We identify a replica as simply representing an “instantaneous state” of the system, i.e., one of the microstates ${\mathfrak{m}}_k$. The collection of all replicas at each instant t is the ensemble, which is specified by the set $\mathbf{Z}\left(t\right)$ and $\mathcal{N}$. The ensemble then becomes the representation of the macrostate $\overline{\mathfrak{M}}$. Any quantity q$(\mathbf{Z},t)$ of interest associated with $\overline{\mathfrak{M}}$ is then identified as an instantaneous average over these replicas or samples, and is an explicit function of the set $\mathbf{Z}$ and possibly t. For simplicity, we will usually suppress $\mathbf{Z}$ and only exhibit the explicit dependence on t in q. By definition, the ensemble average is given by

$$\mathsf{q}\left(t\right)\mathrm{or}\overline{\mathsf{q}}\left(t\right)\mathrm{or}<\mathsf{q}>\left(t\right)\doteq \frac{1}{\mathcal{N}}{\displaystyle \sum _{k=1}^W}{\mathcal{N}}_k\left(t\right){\mathsf{q}}_k,$$

(110)

where q${}_k$ is the value of q in the kth microstate ${\mathfrak{m}}_k$, ${\mathcal{N}}_k\left(t\right)$ denotes the number of samples in the kth microstate ${\mathfrak{m}}_k$ at time t, and W is the total number of distinct microstates, which we assume is finite right now. We also assume $\left\{{\mathcal{N}}_k\left(t\right)\right\}$ to be a countable set. It should be obvious that $\mathcal{N}>>W$ for the above definition to make sense. The overbar on or the angular bracket around q in Equation (110) are used to indicate the average q, which is also represented simply as q, following the acceptable tradition in thermodynamics. We will use all three notations to indicate the average in this review as need be.

5.4. Concept of Probability

We now introduce the concept of ensemble probability

$$p_k\left(t\right)\equiv \underset{\mathcal{N}\to \infty }{lim}{\mathcal{N}}_k\left(t\right)/\mathcal{N},{\displaystyle \sum _{k=1}^W}{p}_k\left(t\right)\equiv 1,$$

(111)

which is valid even if $W\to \infty$. As is well-known [114], the probabilities require the formal limit $\mathcal{N}\to \infty$, which is going to be implicit in the following. This justifies Equation (12).

It should be stressed that the concept of probability introduced in Equation (111) is also valid for a Hamiltonian system with deterministic dynamics. All one needs to do is to prepare an ensemble with a given number ${\mathcal{N}}_k$ of replicas. As these numbers will not change because the dynamics is deterministic, $p_k$ will not change.

It should be noted that ${\mathfrak{m}}_k$, and hence the value q${}_k$ on it, depend on $\mathbf{Z}\left(t\right)$ explicitly, but may also depend on t explicitly. In general, $p_k\left(t\right)$ will be time-dependent as determined by the history of the process. They become history-independent and constant in time t for ${\mathfrak{M}}_{\mathrm{eq}}$. As we will soon see, they remain constant in a mechanical evolution of ${\mathfrak{M}}_{\det }$. In this sense, there is a close parallel between ${\mathfrak{M}}_{\mathrm{eq}}$ and ${\mathfrak{M}}_{\det }$, as discussed below.

The average of the state variable $\mathbf{Z}$, using the tradition in thermodynamics, is simply written as $\mathbf{Z}$ (see Equation (110)):

$$\mathbf{Z}\equiv \sum _{k=1}^W{p}_k\left(t\right){\mathbf{Z}}_k;$$

(112)

here ${\mathbf{Z}}_k$ is the value of $\mathbf{Z}$ in ${\mathfrak{m}}_k$. We will also extend this tradition to ${\mathbf{F}}_{\mathrm{w}}$ in Equation (40) so that

$${\mathbf{F}}_{\mathrm{w}}\equiv \sum _{k=1}^W{p}_k\left(t\right){\mathbf{F}}_{\mathrm{w}k},$$

(113)

where, as usual, ${\mathbf{F}}_{\mathrm{w}k}$ is the value of ${\mathbf{F}}_{\mathrm{w}}$ in ${\mathfrak{m}}_k$.

Claim 12. 

The $p_k$ defined above in Equation (111) remains a constant of motion for a deterministic system.

This is easy to rationalize as follows. Consider a collection of microstates $\left\{{\mathfrak{m}}_k\right\}$ of a system with $\left\{{\mathcal{N}}_k\right\}$ copies at some initial time $t=0$. In a deterministic evolution, ${\mathcal{N}}_k$’s do not change, which justifies the above claim.

Definition 25. 

To distinguish the usage of constant probabilities for deterministic systems with the usage of probabilities for thermodynamic systems, where they may change spontaneously without any external intervention, we will use the term stochastic for this aspect of probabilistic behavior in $\mathfrak{M}$, but not in ${\mathfrak{M}}_{\det }$.

We clarify this point further. Consider an isolate system that is not in EQ. This means that, according to the Boltzmann principle, not all microstates are equally probable. In time, the system will come to equilibrium by ensuring that all microstates become equally probable. This shows how a thermodynamic system behaves in a way that allows $p_k$ to change in time even without any external intervention. For a deterministic system such as a loaded die, this will never happen even if it is disturbed by the performance of mechanical work, like throwing, an external intervention.

For a thermodynamic system in EQ, $\left\{p_k\right\}$ remains invariant (constant) in time. In this regard, such a system is identical to a deterministic system that obeys Liouville theorem [164], since it is well-known that an EQ system also obeys the theorem [33]. The reason is very simple. The various members of the above ensemble in EQ occupy various microstates with equal probability with the maximum entropy as shown in Section 5.5. This entropy remains a constant of motion for the EQ system.

Remark 34. 

An EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ under fixed conditions of the surroundings so $p_k$’s do not change is no different than a deterministic macrostate ${\mathfrak{M}}_{\det }$, except that the former has a well-defined notion of temperature but the latter has no such notion.

5.5. Statistical Entropy for $\mathfrak{M}\left(t\right)$

We provide a very general statistical formulation of S for a general system $\Sigma$ that is applicable to mechanical as well as thermodynamic systems. It will be shown to be identical to the thermodynamic entropy S by appealing to the third law. Our derivation demonstrates that the concept of entropy in general is of a statistical nature. We consider a state $\mathfrak{M}\left(t\right)\equiv \mathfrak{M}\left(\mathbf{Z}\right(t),t)$ of $\Sigma$ at a given instant t. We focus on a macrostate $\mathfrak{M}\left(t\right)$ of $\Sigma$ at a given instant t, which refers to the sets $\mathbf{m}=\left\{{\mathfrak{m}}_k\right\}$ and $\mathbf{p}=\left\{p_k\right\}$ of microstates and their probabilities, respectively. We consider Fl-$\mathbf{W}$ but the discussion is also valid for NFl-$\mathbf{W}$ by simply setting ${\mathbf{W}}_k=\mathbf{W},\forall k$. The microstates are specified by $(E_k\left(t\right),{\mathbf{W}}_k\left(t\right))$, which along with $\mathbf{p}$ need not uniquely specify the macrostate $\mathfrak{M}\left(t\right)$. In the following, we will use the set $\mathbf{Z}\left(t\right)$ for $\mathbf{m}$ for simplicity. We will also denote $\mathbf{Z}\left(t\right)$ by $\overline{\mathbf{Z}}$ so that we can separate out the explicit variation due to t in addition to the implicit variation in t due to $\overline{\mathbf{Z}}$, if any. For simplicity, we suppress t in $\mathfrak{M}$ in the following. For the computation of combinatorics, the  probabilities are handled as described in Section 5.4. We follow the notation used there, choosing $\mathcal{N}=\mathcal{C}W\left(\overline{\mathbf{Z}}\right)$ with $\mathcal{C}$ some large integer constant, and $W\left(\overline{\mathbf{Z}}\right)$ the number of distinct microstates ${\mathfrak{m}}_k$ in the ensemble or the sample space $\Gamma \left(\overline{\mathbf{Z}}\right)$ spanned by $\left\{{\mathfrak{m}}_k\right\}$. We will see that $W\left(\overline{\mathbf{Z}}\right)$ is determined by ${\mathfrak{m}}_k$’s having nonzero probabilities [79]. We will call them available microstates.

The ensemble $\Gamma \left(\overline{\mathbf{Z}}\right)$ above is a generalization of the ensemble introduced by Gibbs, except that the latter is restricted to an equilibrium system, whereas $\Gamma \left(\overline{\mathbf{Z}}\right)$ refers to the system in any arbitrary macrostate so that $p_k$ in Equation (111) may be time-dependent, and may not be unique. The samples are, by definition, independent of each other so that there are no correlations among them. Because of this, we can treat the samples in $\Gamma \left(\overline{\mathbf{Z}}\right)$ to be the outcomes of some random variable, the macrostate $\mathfrak{M}\left(t\right)$. This independence property of the outcomes is crucial in the following. Each sample of $\mathfrak{M}\left(t\right)$ is one of a microstate in $\Gamma \left(\overline{\mathbf{Z}}\right)$. They may be equiprobable but not necessarily. The number of ways $\mathcal{W}$ to arrange the $\mathcal{N}$ samples into $W\left(\overline{\mathbf{Z}}\right)$ distinct microstates is

$$\mathcal{W}\equiv \mathcal{N}!/{\displaystyle \prod _k}{\mathcal{N}}_k\left(t\right)!.$$

(114)

Taking its natural log, as proposed by Boltzmann, to obtain an additive quantity per sample as described in Section 5.6 (see also Axiom 6), we obtain

$$S\equiv (1/\mathcal{N})ln\mathcal{W},$$

(115)

and using Stirling’s approximation, we see easily that it can be written as the ensemble average (see Equations (12) and (26a)),

$$S(\overline{\mathbf{Z}},t)\equiv -〈\eta \left(t\right)〉\equiv -{\displaystyle \sum _{k=1}^{W\left(\overline{\mathbf{Z}}\right)}}{p}_k\left(t\right)ln{p}_k\left(t\right),$$

(116)

of the negative of Gibbs’ index of probability ([48], p. 16)

$${\eta }_k\left(t\right)\doteq ln{p}_k\left(t\right).$$

(117)

We have shown an explicit time dependence in S, which is distinct from the implicit time dependence in $\overline{\mathbf{Z}}$, to merely express the fact that it may not be a state function in ${\mathfrak{S}}_{\overline{\mathbf{Z}}}$, i.e., that $\mathfrak{M}$ may not be uniquely specified in ${\mathfrak{S}}_{\overline{\mathbf{Z}}}$. The above derivation clearly shows that Equation (116), which is identical in form to Equation (26a), justifies the latter for an arbitrary $\mathfrak{M}$.

The identification of entropy in Equation (116) with the Gibbs formulation of entropy is a time-honored practice for nonequilibrium states since the days of Gibbs ([48] see, in particular, chapters 11 and 12, where time dependence is discussed), and has been discussed by Tolman ([54], Ch. 13, and in particular pp. 538–539), Jaynes [174], Rice and Gray [55], and Rice [57], to name a few. There is no restriction on $p_j\left(t\right)$. In particular, they do not have to be given by probabilities valid for equilibrium states; see also Sethna ([36], Section 5.3.1). The definition merely follows from the observation that the index of probability is an additive quantity for independent replicas (see Fundamental Axiom) and that the entropy is merely its average value (with a negative sign). Tolman takes great care in establishing that this formulation of the entropy satisfies the second law ([54], Section 130). Tolman also shows that the Boltzmann definition of entropy is a special case of the general formulation due to Gibbs ([54], see the derivation of Equation (131.2)), just as we have argued; see Equation (208).

The identification of the entropy with the negative of the Boltzmann H-function ([54], see p. 561), the latter describing a nonequilibrium state, should leave no doubt in anyone’s mind that the Gibbs formulation of the entropy can be applied equally well to an equilibrium or a nonequilibrium system. Nevertheless, we should point out that not all subscribe to this viewpoint of ours about the Gibbs formulation of entropy, because they insist that the Gibbs entropy is a constant of motion [135]. This constancy follows immediately from the application of Liouville’s theorem in classical mechanics [32,33,34,36,54], valid for a system described by a Hamiltonian, as discussed above and as we have already discussed in Section 5.4. We thus see that our formulation of the entropy in EQ is consistent with this theorem.

The above derivation is based on fundamental principles of combinatorics and additivity, and does not require the notion of equilibrium or nonequilibrium in the system; therefore, it is always applicable for any arbitrary macrostate $\mathfrak{M}\left(t\right)$ including that of a determining system; see Claim 12. To the best of our knowledge, even though such an expression has been extensively used in the literature for NEQ entropy, it has been used by simply appealing to the information entropy [72,175]. Thus, Equation (116) is a generalization of Equation (26a) to the general case, and thus justifies it for $\mathfrak{M}\left(t\right)$. We now generalize Claim 12 as follows:

Claim 13. 

The probability $p_k$ and the Gibbs entropy (see Equation (26a)) is easy to define for a $\mathfrak{M}\left(t\right)$ including that of a deterministic Hamiltonian system. As the probability and the entropy for ${\mathfrak{M}}_{\det }$ do not change as a function of time, we show in Section 10.1 that the concepts of microheat and macroheat cannot be associated with a Hamiltonian system, although the concepts of microwork and macrowork are defined.

The distinction between the Gibbs’ statistical entropy and the thermodynamic entropy should be emphasized. The latter appears in the Gibbs fundamental relation that relates the energy change $dE$ with the entropy change $dS$, as is well-known in classical thermodynamics, and as we will also demonstrate below; see also Equation (93a). The concept of microstates is irrelevant for this, as it is a purely thermodynamic relation. On the other hand, the Gibbs’ statistical entropy is solely determined by $\left\{{\mathfrak{m}}_k\right\}$, so it is a statistical quantity. It then becomes imperative to show their equivalence, mainly because the statistical entropy is based on the Boltzmann idea. This equivalence has been justified elsewhere [75,76], and is summarized in the following Remark.

Remark 35. 

Because of this equivalence, we will no longer make any distinction between the statistical Gibbs entropy and the thermodynamic entropy and will use the standard notation S for both of them for a macrostate ${\mathfrak{M}}_{\mathrm{ieq}}$, of which ${\mathfrak{M}}_{\mathrm{eq}}$ is a special case.

Remark 36. 

The Gibbs entropy appears as an instantaneous ensemble average; see Definition 7. This average should be contrasted with a temporal average in which a macroquantity q is considered as the average over a long period ${\tau }_0$ of time

$$\mathrm{q}=\frac{1}{{\tau }_0}{\int }_0^{{\tau }_0}\mathrm{q}\left(t\right)dt,$$

where q$\left(t\right)$ is the value of q  at time t [33]. For an EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$, both definitions give the same result provided ergodicity holds. The physics of this average is that q$\left(t\right)$ at t represents a microstate of ${\mathfrak{M}}_{\mathrm{eq}}$. As ${\mathfrak{M}}_{\mathrm{eq}}$ is invariant in time, these microstates belong to ${\mathfrak{M}}_{\mathrm{eq}}$, and the time average is the same as the ensemble average if ergodicity holds. However, for an NEQ macrostate ${\mathfrak{M}}_{\mathrm{neq}}\left(t\right)$, which continuously changes with time, the  temporal average is not physically meaningful as the microstate at time t corresponds to ${\mathfrak{M}}_{\mathrm{neq}}\left(t\right)$ and not to ${\mathfrak{M}}_{\mathrm{neq}}(t=0)$ in that the probabilities and $\mathbf{Z}$ are different in the two macrostates. Only the ensemble average makes any sense at any time t, as discussed in [176]. Because of this, we only consider ensemble averages in this review.

A word of caution must be offered. If S is not a state function, it cannot be measured or computed. Thus, while the statistical entropy can be computed in principle in all cases if $\left\{p_k\right\}$ is known, there is no way to compare its value with thermodynamic entropy in all cases. Thus, no comment can be made about their relationship in general for an arbitrary $\mathfrak{M}\left(t\right)$. We have only established their equivalence for ${\mathfrak{M}}_{\mathrm{ieq}}$ for which the two entropies are the same.

Remark 37. 

We have summarized our approach for an arbitrary macrostate in Axiom 3, which allows us to identify the two entropies in all cases. Thus, we only need to investigate the μNEQT for ${\mathfrak{M}}_{\mathrm{ieq}}$ to also cover $\mathfrak{M}$; see Section 5.9.

5.6. Principle of Additivity

5.6.1. Additivity

We consider a system $\Sigma$ consisting of two nonpenetrating sub-bodies ${\Sigma }_1$ and ${\Sigma }_2$ at present, each specified by ${\mathbf{W}}_1$ and ${\mathbf{W}}_2$. Later, we will generalize to any number of sub-bodies ${\Sigma }_j$. The principle of additivity states that $\Sigma$ is specified by $\mathbf{W}$ given by

$$\mathbf{W}\doteq {\sum }_j{\mathbf{W}}_j.$$

(118a)

This principle is self-evident for nonpenetrating systems. For example, the  number of particles

$$N\equiv {\sum }_j{N}_j$$

(118b)

remains an identity. (This remains true even if the bodies are interpenetrating, for which the volumes may not be additive). For  nonpenetrating bodies, however, their volumes become additive:

$$V={\sum }_j{V}_j,$$

(118c)

which we will assume in this review. We will call the case of nonpenetrating bodies the discrete approach. It is evident that in this approach, the principle of additivity is valid for any number of sub-bodies ${\Sigma }_j,j=1,2,\cdots$. In this case, the sum in the above equations is over all sub-bodies.

We now show that the above sample average in Equation (110) also follows immediately from the principle of additivity of quantities that are additive; see Claim 2. One considers a very large macroscopic system ${\Sigma }_0$ of $N_0\equiv$$N\mathcal{N}$ particles and imagines dividing it into a large number $\mathcal{N}$ of macroscopically large and nonoverlapping parts of equal size N, each representing a microstate of the system $\Sigma$. As the parts are macroscopically large, they will act almost independently; see Section 7.3 for details. How well this condition is satisfied depends on how large the parts are. In principle, they can be made arbitrary large to ensure their complete independence. At the same time t, these parts will be in microstates ${\mathfrak{m}}_k$ of $\Sigma$ with probabilities $p_k\left(t\right)$. The additivity principle states that any extensive thermodynamic quantity $X\left(t\right)$ of the system ${\Sigma }_0$ is the sum of this quantity over its various macroscopically large parts. This principle is consistent with the definition of the average in (110). One can also think of the $\mathcal{N}$ parts as representing the same measurement that has been repeated $\mathcal{N}$ times on samples prepared under identical macroscopic conditions at the same instant t.

5.6.2. Quasi-Additivity

We have enunciated the principle of additivity for $\mathbf{W}$ above. The  energy E plays a very different role because of mutual interactions between various sub-bodies. We again restrict to only two sub-bodies for simplicity, which can be later generalized to any number of sub-bodies. We assume that they are weakly interacting so that their energies are quasi-additive, which we express in a form using $j=1,2$:

$$E={\sum }_j{E}_j+U_{\mathrm{int}}\approx {\sum }_j{E}_j,$$

(119)

where $U_{\mathrm{int}}$ is the weak interaction energy between ${\Sigma }_1$ and ${\Sigma }_2$, and can be neglected to a good approximation provided

$$U_{\mathrm{int}}<<E_{\mathrm{sm}}\doteq min\left\{E_j\right\}.$$

(120)

We can extend the discussion to many sub-bodies $\left\{{\Sigma }_j\right\},j=1,2,\cdots$, by defining $U_{\mathrm{int}}$ as the net interaction energies between all of them:

$$U_{\mathrm{int}}\doteq {\sum }_{j>l}{E}_{jl},$$

where $E_{jl}$ is the interaction energy between ${\Sigma }_j$ and ${\Sigma }_l$. The inequality in Equation (120) can be made as precise as we wish by making N extremely large compared to various sub-bodies.

Remark 38. 

With quasi-additivity for energies, we can extend the principle of additivity from $\mathbf{W}$ to

$$\mathbf{Z}\approx {\sum }_j{\mathbf{Z}}_j,$$

(121)

by including quasi-additivity for the energies; see Claim 2.

However, the quasi-additivity of the entropy is altogether a different issue. The entropy additivity is strictly valid if $\Sigma$ and $\tilde{\Sigma }$ are (statistically) independent [3], i.e., noninteracting. However, this independence is not of any physical interest as $\Sigma$ and $\tilde{\Sigma }$ must be interacting with each other for any interesting thermodynamics; otherwise, there is no need to consider $\tilde{\Sigma }$, and the issue of additivity does not arise. Thus, we are inclined to consider them to be quasi-independent. To the best of our knowledge, the discussion of quasi-independence and its distinction from interactions between $\Sigma$ and $\tilde{\Sigma }$ that are weak has been carefully presented elsewhere ([148], $S_{\mathrm{corr}}$ was called $S_{\mathrm{int}}$ there; however, $S_{\mathrm{corr}}$ seems to be more appropriate) for the first time, which we summarize below. The  presence of interparticle interactions that determine E and $\tilde{E}$ for $\Sigma$ and $\tilde{\Sigma }$, respectively, results in the thermodynamic concept of correlation lengths in them. The correlation length ${\lambda }_{\mathrm{corr}}>a$ is a property of macrostates, and can be much larger than the interparticle interaction length a between particles depending on the macrostate. In general, ${\lambda }_{\mathrm{corr}}>>a$. A simple well-known example is of the correlation length ${\lambda }_{\mathrm{corr}}$ of a nearest neighbor Ising model, which increases very rapidly as the critical point is reached, and where it can be much larger than the nearest neighbor distance a between the spins. Two interacting Ising systems at the same temperature cannot be “independent”, so the additivity of entropy for ${\Sigma }_0$ is replaced by the following:

$$S_0({\mathbf{X}}_0,t)=S(\mathbf{X}\left(t\right),t)+\tilde{S}\left(\tilde{\mathbf{X}}\left(t\right)\right)+S_{\mathrm{corr}}\left(t\right),$$

(122a)

where $S_{\mathrm{corr}}\left(t\right)$ is a correction term to the entropy due to correlation that is present between $\Sigma$ and $\tilde{\Sigma }$ due to their mutual interaction. If the linear sizes l and $\tilde{l}$ of the two bodies are much larger compared to ${\lambda }_{\mathrm{corr}}$, then this correlation becomes almost nonexistent. In this case, $S_{\mathrm{corr}}\left(t\right)$ can be neglected to a good approximation so that

$$S_0({\mathbf{X}}_0,t)\approx S(\mathbf{X}\left(t\right),t)+\tilde{S}\left(\tilde{\mathbf{X}}\left(t\right)\right),$$

(122b)

provided $\tilde{l},l>>{\lambda }_{\mathrm{corr}}$. Under this condition, $\Sigma$ and $\tilde{\Sigma }$ are said to be quasi-independent [148], which ensures that their entropies become quasi-additive. This distinction is usually not made explicit in the literature. Usually, $\tilde{l}>>l$, but this condition was not used above so the above additivity is valid for any two bodies for which $\tilde{l},l>>{\lambda }_{\mathrm{corr}}$. For $\tilde{\Sigma }$ representing a medium, $\tilde{S}$ has no explicit time dependence as it is assumed to be in equilibrium, and ${\mathbf{X}}_0$ remains constant for the isolated system ${\Sigma }_0$.

The above quasi-additivity principle is applicable to microstates of $\Sigma$ as well. We now focus on classical microstates represented by the sub-bodies, and apply the discussion to only two sub-bodies representing $\Sigma ={\Sigma }_1$ and $\tilde{\Sigma }={\Sigma }_2$ forming the isolated system ${\Sigma }_0$  as they are central to our statistical mechanics. We consider the energies of the microstates ${\mathfrak{m}}_{0k_0},{\mathfrak{m}}_k$, and  ${\tilde{\mathfrak{m}}}_{\tilde{k}}$. They are related as follows:

$$E_{0k_0}=E_k+{\tilde{E}}_{\tilde{k}}+E_{k,\tilde{k}}$$

(123a)

where we have also included the interaction energy $E_{k,\tilde{k}}$ due to $U_{\mathrm{int}}$, which is usually negligible relative to $E_k,$ ${\tilde{E}}_{\tilde{k}}$. These energies are independent of the macrostates and, therefore, independent of quantities such as the temperatures and probabilities that specify macrostates of various bodies forming the system. The energies corresponding to their macrostates are related by

$$E_0=E+\tilde{E}+U_{\mathrm{int}};$$

(123b)

see Equation (119). Again, the smallness of $E_{k,\tilde{k}}$ results in its average $U_{\mathrm{int}}$ obtained by using $p_{0k_0}$ and $E_{k,\tilde{k}}$ in Equation (112), being negligible relative to E and $\tilde{E}$.

Remark 39. 

The assumption to neglect $E_{k,\tilde{k}}$ or $U_{\mathrm{int}}$ merely makes Σ and $\tilde{\Sigma }$ satisfy the principle of additivity. We will make this assumption in this review extensively.

Remark 40. 

From now on, we will usually replace the sign “≈” by “=” unless clarity is needed.

Remark 41. 

Throughout this review, we will think of the above approximate equalities as equalities to make the energies additive by neglecting the interaction energy between Σ and $\tilde{\Sigma }$, which is a standard practice in the field, but also assuming quasi-independence between them to make the entropies to be additive, which is not usually mentioned as a requirement in the literature.

5.7. $\Sigma$ in Internal EQ (IEQ)

The central concept of the $\mu$NEQT is that of the internal equilibrium (IEQ) according to which the entropy S of an NEQ macrostate is a state function of the state variables in the enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$ [134,148,149]. The enlargement of the space relative to the EQ state space ${\mathfrak{S}}_{\mathbf{X}}$ is due to independent internal variables [13,18,51,108], which is sufficient to uniquely specify $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{Z}}$. We denote such a state by ${\mathfrak{M}}_{\mathrm{ieq}}$. The same state cannot be uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$ or any other extended state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$ that does not have the same set of internal variables as in $\mathbf{Z}$.

We give a simple example to clarify why and how internal variables are useful for describing an NEQ state. Consider the case of two identical bodies ${\Sigma }_1$ and ${\Sigma }_2$ in thermal contact at different temperatures $T_1\left(t\right)$ and $T_2\left(t\right)$ and energies $E_1\left(t\right)$ and $E_2\left(t\right)$, respectively; we ignore other observables $N,V$, etc. Thus, $\mathbf{X}=(E\dot{)}$ for each system. We assume that each one is in an EQ state of its own at each instant. Together, they form an isolated composite system $\Sigma$, whose entropy $S(E_1,E_2)=S_1\left(E_1\right)+S_2\left(E_2\right)$ is a function of two variables at each instant t, and can be written as a state function in the enlarged state space formed by

$$E=E_1+E_2=const,\xi \left(t\right)=E_1-E_2.$$

(We have neglected the interaction energy $E_{12}$ between ${\Sigma }_1$ and ${\Sigma }_2$ here per Remark 39.) This situation should be compared with its mechanical analog in Section 4, and in particular with Equation (104a) for ${\xi }_{\mathrm{E}k}$; here, $n_1=n_2=1/2$. The discussion there was purely mechanical so there was no dissipation.

We are in a position now to understand how dissipation emerges in thermodynamics. As the system approaches EQ, $E_1\to E_2$ so that $\xi \to 0$. This also means that $T_1\left(t\right)\to T_2\left(t\right)=T_{\mathrm{eq}}$, the EQ temperature. The first thing we learn from this simple example is that it clearly shows how the t-dependence in $S(E,t)\equiv S(E_1,E_2)$ can be replaced by invoking an extensive internal variable $\xi \left(t\right)$ so that the entropy can be treated as a state function $S(E,\xi )$ in the enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$ spanned by E and $\xi$. In other words, the system is in an IEQ state. In general, we will need to enlarge ${\mathfrak{S}}_{\mathbf{X}}$ by introducing an appropriate number of internal variables to form ${\mathfrak{S}}_{\mathbf{Z}}$ in which the system is in IEQ. Thus, we can always express S in an IEQ state as a state function

$$S=S_{\mathrm{ieq}}=S\left(\mathbf{Z}\right)$$

(124)

in the appropriately enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$. This is carefully discussed in Section 12, where we take a different approach. As $S_1\left(E_1\right)$ and $S_2\left(E_2\right)$, being in EQ, have their maximum value for given $E_1\left(t\right)$ and $E_2\left(t\right)$, $S(E,\xi )$ also has its maximum value for given $E\left(t\right)$ and $\xi \left(t\right)$, but this value increases as $\xi \to 0$, and EQ is achieved. In general, ${\mathfrak{M}}_{\mathrm{ieq}}$ has the maximum possible entropy for the given $\mathbf{Z}$, and continues to increase as $\mathbf{Z}$ changes and EQ is reached. For this IEQ state, it is trivial to show that the temperature ($1/T=\partial S/\partial E$; see Equation (129)) of $\Sigma$ is

$$T\left(t\right)=2T_1{T}_2/(T_1+T_2)$$

(125a)

and its affinity $T\partial S/\partial \xi$ (see Equation (133)) is given by

$$A\left(t\right)=(T_2-T_1)/(T_1+T_2).$$

(125b)

At equilibrium, $T_1=T_2=T_{\mathrm{eq}}$ and $\xi =0,A=0$. Thus, $T_1$ and $T_2$ may be very different, yet the system as a whole can be treated as being in IEQ with a unique temperature $T\left(t\right)$, any temperature difference $T_2\left(t\right)-T_1\left(t\right)$ between its parts not withstanding. The discussion can be extended easily to the case when the two bodies are in IEQs and also when they are of different sizes. In all cases, a unique temperature in accordance with Equation (129) can be defined for the composite system [77,78]. Once it is determined, we do not have to worry about the internal temperature difference between ${\Sigma }_1$ and ${\Sigma }_2$. Any internal heat transfer between them is captured by

$$\beta A\left(t\right)d\xi =d_{\mathrm{i}}S=d{E}_1({\beta }_1-{\beta }_2),$$

(126)

as can be easily verified; here $d_{\mathrm{i}}S$ is the irreversible entropy generation due to macroheat exchange [51]. We thus see the affinity for $\xi$ is given by

$$A\left(t\right)=\frac{T{d}_{\mathrm{i}}S}{d\xi }=\frac{d{E}_1}{d\xi }\frac{({\beta }_1-{\beta }_2)}{\beta },$$

(127)

which vanishes as EQ is reached, a well-known feature [51] of classical thermodynamics. The analysis clearly shows how thermodynamics brings in dissipation in a mechanical system, showing the consistency of our approach using internal variables.

5.8. Gibbs Fundamental Relations for ${\mathfrak{M}}_{\mathrm{ieq}}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ and ${\mathfrak{S}}_{\zeta }$

We first consider the state space ${\mathfrak{S}}_{\mathbf{Z}}$ in which ${\mathfrak{M}}_{\mathrm{ieq}}\left(\mathbf{Z}\right)$ is uniquely specified. In this space, the state function $S\left(\mathbf{Z}\right)$ results in the general form of the Gibbs fundamental relation

$$dS\left(\mathbf{Z}\right)=\frac{\partial S}{\partial E}dE+\frac{\partial S}{\partial \mathbf{W}}·d\mathbf{W}$$

(128a)

for the entropy, from which follows the Gibbs fundamental relation for $E\left(\zeta \right)$ in ${\mathfrak{S}}_{\zeta }$ spanned by $\zeta \doteq (S,\mathbf{W})$,

$$dE\left(\zeta \right)=\frac{\partial E}{\partial S}dS+\frac{\partial E}{\partial \mathbf{W}}·d\mathbf{W}.$$

(128b)

Introducing the SI-temperature $T=1/\beta$ as

$$T\doteq \partial E/\partial S,\beta =\partial S/\partial E,$$

(129)

and re-expressing the generalized macroforce in Equation (18) as

$${\mathbf{F}}_{\mathrm{w}}=-\partial E/\partial \mathbf{W}=T\partial S/\partial \mathbf{W},$$

(130)

we rewrite Equations (128a) and (128b) as

$$\begin{array}{cc}\hfill dS& =\beta dE+\beta dW\hfill \end{array}$$

(131a)

$$\begin{array}{cc}\hfill dE& =TdS-dW\hfill \end{array}$$

(131b)

in terms of SI macroquantities; here, we have introduced SI-macrowork $dW$ as the generalized macrowork

$$dW\doteq {\mathbf{F}}_{\mathrm{w}}·d\mathbf{W}\equiv T\frac{\partial S}{\partial \mathbf{W}}·d\mathbf{W}$$

(132)

done by the system. The derivative with respect to $\xi$ determines the affinity

$$\mathbf{A}\doteq T\left(\partial S/\partial \xi \right)=-\left(\partial E/\partial \xi \right),$$

(133)

which vanishes in equilibrium so that ${\mathbf{A}}_{\mathrm{eq}}={\mathbf{A}}_0=0$. Thus, in general, ${\mathbf{F}}_{\mathrm{w}}=({\mathbf{f}}_{\mathrm{w}},\mathbf{A})$, where

$${\mathbf{f}}_{\mathrm{w}}=-\partial E/\partial \mathbf{w}=T\partial S/\partial \mathbf{w}$$

(134)

is the generalized macrowork due to $\mathbf{w}$.

Remark 42. 

Comparing Equation (131) with Equation (93a) allows us to verify Conclusion 1 for the Clausius equality.

This equality must be distinguished from $d_{\mathrm{e}}Q$ in Equation (46). Thus, Equation (93a) allows us to uniquely identify the generalized macroheat $dQ=TdS$ determined by $dS$ and the generalized macrowork determined by $d\mathbf{W}$ to be independent of each other as they belong to orthogonal subspaces in the subspace ${\mathfrak{S}}_{\zeta }$; see also Section 10.2. Both are SI-macroquantities. The  resulting thermodynamics has been identified as the MNEQT. In terms of various components of ${\mathbf{F}}_{\mathrm{w}}$, the generalized macrowork is

$$dW=PdV-\mu dN+\cdots +\mathbf{A}·d\xi .$$

(135a)

We can identify various components of the macrowork as $d{W}_V=PdV,d{W}_N=\mu dN,\cdots ,$$d{W}_{{\xi }_1}=A_{1}d{\xi }_1,\cdots ,$ using an obvious notation. The  missing terms denote the contribution from the rest of the variables not shown, and 

$$P\doteq -\partial E/\partial V,\mu \doteq \partial E/\partial N,\cdots ,\mathbf{A}\doteq -\partial E/\partial \xi ,$$

(135b)

are the SI-fields associated with $\mathbf{W}$, with changes $d\mathbf{W}=$ $dV,dN,\cdots ,d\xi$ being the changes in it.

In the $\mathring{\mathrm{M}}\mathrm{NEQT}$, the first law in Equation (94) refers to exchange macroheat $d_{\mathrm{e}}Q=T_0{d}_{\mathrm{e}}S$ (see Equation (46)) and macrowork

$$d_{\mathrm{e}}W=P_0{d}_{\mathrm{e}}V-{\mu }_0{d}_{\mathrm{e}}N+\cdots ;$$

(135c)

in terms of the fields (the temperature $T_0$, pressure $P_0$, chemical potential ${\mu }_0$,⋯) of the medium and the corresponding macroscopic exchange quantities in all cases, regardless of the irreversibility. As the medium is in EQ, there is no contribution due to $\xi$ in $d_{\mathrm{e}}W$ as the corresponding contribution ${\mathbf{A}}_0·d\xi$ vanishes due to the fact that the affinity ${\mathbf{A}}_0\equiv 0$ for the medium. Our sign convention is that $d_{\mathrm{e}}Q$ is positive when it is added to $\Sigma$, and $d_{\mathrm{e}}W$ is positive when it is transferred to $\tilde{\Sigma }$.

It follows from Equations (135a) and (135c) that the irreversible macrowork, also known as dissipative work, is

$$d_{\mathrm{i}}W=(P-P_0)dV-(\mu -{\mu }_0)dN+\cdots +\mathbf{A}·d\xi \ge 0.$$

(136)

The coefficients $P-P_0,\mu -{\mu }_0,\cdots ,\mathbf{A}$ are commonly known as thermodynamic forces or macroforce imbalances [51], which vanish in EQ; see Section 6.4.

Remark 43. 

We have included the term associated with N for completeness in Equations (135a), (135c) and (136). We will no longer consider this term anymore.

We should compare the above equations with Equation (79). Once $d_{\mathrm{e}}W$ or $dW$ has been identified, the use of the first law allows us to uniquely determine $d_{\mathrm{e}}Q$ or $dQ$, respectively.

It is clear that the root cause of dissipation is the macroforce imbalance. It drives the system towards equilibrium [41,42,75,76,134,148,149,150,152,153]. It arises due to the imbalance between the external and the average internal forces performing work; the microforce imbalance is introduced in the following section. The average force imbalances give rise to an internal work $d_{\mathrm{i}}W$ due to all kinds of force imbalances. The irreversible or dissipated work is given in Equation (136), which is generated within $\Sigma$.

If we include the relative velocity between a Brownian particle ${\Sigma }_{\mathrm{BP}}$ and the medium to account for the Brownian motion [148,157], we must account for [148] an additional term $-\mathbf{V}·d{\mathbf{P}}_{\mathrm{BP}}$ in $d_{\mathrm{i}}W$ due to the relative velocity $\mathbf{V}$:

$$d_{\mathrm{i}}W=(P-P_0)dV-\mathbf{V}·d{\mathbf{P}}_{\mathrm{BP}}+Ad\xi ;$$

(137)

here, $d{\mathbf{P}}_{\mathrm{BP}}={\mathbf{F}}_{\mathrm{wBP}}dt$ is the change in the linear momentum of the Brownian particle experiencing a macroforce ${\mathbf{F}}_{\mathrm{wBP}}$. To see it, we recognize that $\mathbf{V}·d{\mathbf{P}}_{\mathrm{BP}}$ must be nonpositive to comply with the second law. Thus, ${\mathbf{F}}_{\mathrm{wBP}}$ must be antiparallel to $\mathbf{V}$ and describes the frictional drag. This is discussed in detail in Ref. [157]. Thus, the force is reviewed in Section 14 as the role of friction in the Langevin equation turns out to be different in the two NEQ thermodynamics. We will come back to this term later when we consider the motion of a particle attached to a spring; see Figure 3b, a system also studied by Jarzynski, so that a comparison can be made.

The irreversible macroheat $d_{\mathrm{i}}Q$ in all cases is given by Equation (47), and shows that it does not vanish when $T=T_0$, provided $d_{\mathrm{i}}S>0$. This means that the irreversible macrowork is present even if there is no temperature difference, such as in an isothermal process, as long as there exists some nonzero thermodynamic force or irreversibility. The resulting irreversible entropy generation is then given by $d_{\mathrm{i}}S$. We summarize this [51] as

Conclusion 3. 

To have dissipation, it is necessary and sufficient to have a nonzero thermodynamic force. In its absence, there can be no dissipation regardless of the time dependence of the work process; see also Remark 32. This understanding of dissipation becomes clear from the microscopic source of dissipation in Proposition 2.

5.9. Time-Dependent Gibbs Fundamental Relations for ${\mathfrak{M}}_{\mathrm{nieq}}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$

We now consider the generalization of the Gibbs fundamental relation for ${\mathfrak{M}}_{\mathrm{nieq}}$, which is not uniquely specified in ${\mathfrak{S}}_{\mathbf{Z}}$ or ${\mathfrak{S}}_{\zeta }$, by starting from Equation (295a) having an explicit time dependence that comes from “hidden” internal variables ${\xi }^{\prime }$ in ${\mathfrak{S}}_{\mathbf{Z}}$. From the state function entropy $S\left({\mathbf{Z}}^{\prime }\left(t\right)\right)$ for ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ in ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$, we have

$$dS\left({\mathbf{Z}}^{\prime }\left(t\right)\right)=\frac{\partial S}{\partial E}dE+\frac{\partial S}{\partial \mathbf{W}}·d\mathbf{W}+\frac{\partial S}{\partial {\xi }^{\prime }}·d{\xi }^{\prime },$$

where $\mathbf{W}$ is the work variable in ${\mathfrak{S}}_{\mathbf{Z}}$. Expressing the last term as

$$\frac{\partial S}{\partial {\xi }^{\prime }}·\frac{d{\xi }^{\prime }}{dt}dt,$$

we obtain the following generalization of the Gibbs fundamental relation for ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$:

$$dS(\mathbf{Z}\left(t\right),t)=\frac{\partial S}{\partial E}dE+\frac{\partial S}{\partial \mathbf{W}}·d\mathbf{W}+\frac{\partial S}{\partial t}dt,$$

(138a)

where

$$\frac{\partial S}{\partial t}\doteq \frac{\partial S}{\partial {\xi }^{\prime }}·\frac{d{\xi }^{\prime }}{dt}\ge 0.$$

(138b)

Definition 26. 

As the presence of $\partial S/\partial t$ above in ${\mathfrak{S}}_{\mathbf{Z}}$ is due to “hidden” internal variables in ${\xi }^{\prime }$, we will call it the hidden entropy generation rate, and 

$$d_{\mathrm{i}}{S}^{\mathrm{hid}}\left(t\right)=\frac{\partial S}{\partial t}dt=\frac{\partial S}{\partial {\xi }^{\prime }}·d{\xi }^{\prime }\ge 0,$$

(139a)

the hidden entropy generation. It results in a hidden irreversible macrowork

$$d_{\mathrm{i}}{W}^{\mathrm{hid}}\doteq T{d}_{\mathrm{i}}{S}^{\mathrm{hid}}={\mathbf{A}}^{\prime }·d{\xi }^{\prime },$$

(139b)

in ${\mathfrak{S}}_{\mathbf{Z}}$ due to the hidden internal variable with affinity ${\mathbf{A}}^{\prime }$.

In ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$, we can identify the temperature T as the thermodynamic temperature in ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$ by the standard definition. It is clear from the above discussion that $\partial S\left({\mathbf{Z}}^{\prime }\left(t\right)\right)/\partial E$ in ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$ has the same value as $\partial S\left(\mathbf{Z}\right(t),t)/\partial E$ in ${\mathfrak{S}}_{\mathbf{Z}}$. However, there is an alternative definition of a temperature for $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ as

$$dQ(\mathbf{Z}\left(t\right),t)/dS(\mathbf{Z}\left(t\right),t)=T_{\mathrm{arb}}^{\mathrm{alt}}(\mathbf{Z}\left(t\right),t),$$

while $T\left({\mathbf{Z}}^{\prime }\left(t\right)\right)=dQ\left({\mathbf{Z}}^{\prime }\left(t\right)\right)/dS\left({\mathbf{Z}}^{\prime }\left(t\right)\right)$ for ${\mathfrak{M}}_{\mathrm{ieq}}$ in ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$. It is easy to see that they are not the same as macroheats $dQ\left({\mathbf{Z}}^{\prime }\left(t\right)\right)=dE\left(t\right)+dW\left({\mathbf{Z}}^{\prime }\left(t\right)\right)$  and $dQ\left(\mathbf{Z}\right(t),t)=dE\left(t\right)+dW\left(\mathbf{Z}\right(t),t)$ are not the same as macroworks. Thus, this definition is not a thermodynamic temperature for $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{Z}}$. Therefore, we are now set to identify $T_{\mathrm{arb}}$ (see also Equation (257)) as a thermodynamic temperature of ${\mathcal{M}}_{\mathrm{arb}}$ by this T.

Remark 44. 

$1/T_{\mathrm{arb}}\doteq \partial S(\mathbf{Z}\left(t\right),t)/\partial E$ in ${\mathfrak{S}}_{\mathbf{Z}}$ is identified by the same derivative in the Gibbs fundamental relation in ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$ as follows:

$$\frac{1}{T_{\mathrm{arb}}}=\frac{\partial S\left({\mathbf{Z}}^{\prime }\left(t\right)\right)}{\partial E}\equiv \frac{1}{T\left(\mathbf{Z}\right(t\left)\right)},$$

(140a)

while the alternative nonthermodynamic temperature satisfies

$$T_{\mathrm{arb}}^{\mathrm{alt}}(\mathbf{Z}\left(t\right),t)=T\left(\mathbf{Z}\left(t\right)\right)[1+d_{\mathrm{i}}{S}^{\mathrm{hid}}/dS(\mathbf{Z}\left(t\right),t)],$$

(140b)

as is easily verified.

Remark 45. 

As discussed above and as will be discussed in detail in Section 12.1, a macrostate ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ with $S\left(\mathbf{Z}\right(t),t)$ can be converted to ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ with a state function $S\left({\mathbf{Z}}^{\prime }\left(t\right)\right)$ in an appropriately chosen state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}\supset {\mathfrak{S}}_{\mathbf{Z}}$ by finding the appropriate window in which ${\tau }_{\mathrm{obs}}$ lies as well. The needed additional internal variable ${\xi }^{\prime }$ determines the hidden entropy generation rate $\partial S/\partial t$ in Equation (138b) due to the non-IEQ nature of ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, and ensures validity of the Gibbs relation in Equation (138a) for it, thereby not only providing a new interpretation of the temporal variation of the entropy due to hidden variables but also extending the MNEQT to ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$.

The above discussion strongly points towards the following possible proposition.

Proposition 1. 

The MNEQT provides a very general framework to study any ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, since it can be converted into a ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ in an appropriately chosen state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}$, with $d_{\mathrm{i}}{S}^{\mathrm{hid}}\left(t\right)$ originating from hidden internal variable ${\xi }^{\prime }$.

We now consider a process $\mathcal{P}$ to be studied in ${\mathfrak{S}}_{\mathbf{Z}}$. It is natural to think of at least the initial macrostate ${\mathfrak{M}}^{\mathrm{in}}$ of $\mathcal{P}$ as being uniquely identified as ${\mathfrak{M}}_{\mathrm{ieq}}^{\mathrm{in}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$. During the process, $\mathfrak{M}\left(t\right)$ along $\mathcal{P}$ may turn into ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ or remain ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$. The former has been studied above. The latter can happen under the following two cases:

(i) all internal variables in $\xi$ remain out of equilibrium;

(ii) internal variables in a subset ${\xi }^{\prime }\subset \xi$ have equilibrated so that the affinity ${\mathbf{A}}^{\prime }=T\partial S/$$\partial {\xi }^{\prime }$ vanishes.

In both cases, $\mathfrak{M}\left(t\right)$ remains ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, except that in (ii), $\mathfrak{M}\left(t\right)$ can also be treated as ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ in the proper subspaces between ${\mathfrak{S}}_{{\mathbf{Z}}^{″}}\subset {\mathfrak{S}}_{\mathbf{Z}}$ and ${\mathfrak{S}}_{\mathbf{Z}}$, with ${\mathbf{Z}}^{″}$ defined by ${\mathbf{Z}}^{″}\cup {\xi }^{\prime }=\mathbf{Z}$. Even though ${\mathbf{A}}^{\prime }=0$ in these subspaces so that $d_{\mathrm{i}}{S}^{\mathrm{hid}}\left(t\right)=0$ and $d_{\mathrm{i}}{W}^{\mathrm{hid}}\left(t\right)=0$, the Fl microaffinity ${\mathbf{A}}_k^{\prime }\ne 0$ in these subspaces, and will still play an important role in the $\mu$NEQT. Therefore,

Remark 46. 

We will use the state space ${\mathfrak{S}}_{\mathbf{Z}}$ to construct the NEQ statistical mechanics in (i) and (ii) without affecting the hidden entropy generation and hidden irreversible macrowork. This allows us to use ${\mathfrak{S}}_{\mathbf{Z}}$ over the entire process.

Remark 47. 

In a process $\mathcal{P}$ resulting in ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, it is natural to assume that the terminal macrostates in $\mathcal{P}$ are ${\mathfrak{M}}_{\mathrm{ieq}}$ so the affinity corresponding to ${\xi }^{\prime }$ must vanish in them.

The above discussion can be easily applied to consider the case ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime }}\subset {\mathfrak{S}}_{\mathbf{Z}}$, in which internal variables in a subset ${\xi }^{\prime }$ of $\xi$ have equilibrated. The result is summarized in the following:

Remark 48. 

By replacing $\mathbf{Z}$ by $\mathbf{X}$, and  ${\mathbf{Z}}^{\prime }$ by $\mathbf{Z}$, we can also express the Gibbs fundamental relation for any NEQ macrostate in ${\mathfrak{S}}_{\mathbf{X}}$ as

$$dS(\mathbf{X}\left(t\right),t)=\frac{\partial S}{\partial E}dE+\frac{\partial S}{\partial \mathbf{w}}·d\mathbf{w}+\frac{\partial S}{\partial t}dt,$$

(141)

by treating ${\mathfrak{M}}_{\mathrm{neq}}$ as ${\mathfrak{M}}_{\mathrm{ieq}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$. In an NEQ process $\overline{\mathcal{P}}$ between two EQ macrostates but resulting in ${\mathfrak{M}}_{\mathrm{ieq}}\left(t\right)$ between them in ${\mathfrak{S}}_{\mathbf{Z}}$, the affinity corresponding to ξ must vanish in the terminal EQ macrostates of $\overline{\mathcal{P}}$.

Equation (141) proves extremely useful to describe ${\mathfrak{M}}_{\mathrm{neq}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ as it may not be easy to identify $\xi$ in all cases.

Remark 49. 

The explicit time dependence in the entropy for ${\mathfrak{M}}_{\mathrm{neq}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{M}}_{\mathrm{nieq}}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ is solely due to the internal variables, which do not affect the validity of the Clausius equality $dQ=TdS$ (Equation (45)), with T defined as the inverse of $\partial S/\partial E$ at fixed $\mathbf{w},t$ or $\mathbf{W},t$ in the two state spaces, respectively; see Equation (129). As a consequence, Equation (47) remains valid for any $\mathfrak{M}$.

5.10. Consequences of the Second Law

Theorem 4. 

As a consequence of the second law, the irreversible macrowork $d_{\mathrm{i}}W$ (see Equation (136)) which is equal in magnitude to the macroheat $d_{\mathrm{i}}Q$ (see Equation (95)) for any $\mathfrak{M}$ is nonnegative in any real process.

Proof. 

Using Equation (47), we find

$$\begin{array}{c}{T}_0{d}_{\mathrm{i}}S=(T_0-T)dS+d_{\mathrm{i}}W\\ T{d}_{\mathrm{i}}S=(T_0-T)d_{\mathrm{e}}S+d_{\mathrm{i}}W\end{array}\ge 0,$$

(142)

where the inequality follows from the second law $d_{\mathrm{i}}S\ge 0$ in Equation (67c); we assume T and $T_0$ to be nonnegative. Therefore, each of the two independent contributions in each equation must be nonnegative. This thus proves that

$$d_{\mathrm{i}}W=d_{\mathrm{i}}Q\ge 0.$$

(143)

☐

Corollary 1. 

Different components of $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}Q$ for any $\mathfrak{M}$ must be individually nonnegative.

Proof. 

Consider the independent components such as $d_{\mathrm{i}}{W}_{\mathrm{V}},d_{\mathrm{i}}{W}_{\xi }$, etc., of $d_{\mathrm{i}}W$. As $d_{\mathrm{i}}W$ is nonnegative, each component must be nonnegative.   □

This proves the inequalities in Equations (43) and (80). In addition, it shows that each term on the right in Equation (75) is nonnegative. We thus have a proof of a part of Remark 32 that deals with the consequences of the second law.

Corollary 2. 

In any real process,

$$(T_0-T)d_{\mathrm{e}}S\ge 0;(T_0-T)dS\ge 0.$$

Proof. 

The corollary follows from the preceding theorem.    □

The first inequality merely states the well-known fact of thermodynamics that macroheat $d_{\mathrm{e}}Q=T_0{d}_{\mathrm{e}}S$ flows from “hot” to “cold”. The second inequality also states a well-known fact about the stability in thermodynamics, which requires the entropy to increase with temperature. As EQ is reached, $T\to T_0$ either from above ($T>T_0$) or from below ($T<T_0$). In the former case, S decreases, while it increases in the latter case.

Corollary 3. 

For an isolated system $(dS\equiv d_{\mathrm{i}}S)$ or for $T=T_0$,

$$T{d}_{\mathrm{i}}S=d_{\mathrm{i}}W\ge 0.$$

(144)

Proof. 

Setting $d_{\mathrm{e}}S=0$ for an isolated system or $T=T_0$ in Equation (142) proves the theorem immediately.    □

The inequalities in Equation (142) follow from the second law $d_{\mathrm{i}}S\ge 0$ in Equation (67c). Each term on the right side, being independent of each other, must be nonnegative separately, which yields

$$(T_0-T)dS\ge 0,(1-T/T_0)d_{\mathrm{e}}Q\ge 0,d_{\mathrm{i}}W\ge 0$$

(145)

as consequences of the second law. In view of Equation (95), the last inequality above proves the last two inequalities in Equation (69).

5.11. Assumptions

We list the two important assumptions of our approach. They can be relaxed but we will not do that in this review.

5.11.1. N Fixed for $\Sigma$

In order to fix the size of $\Sigma$, we need to specify one of its extensive state variables. Usually, N is kept fixed to ensure a fixed size. Therefore, N is not considered part of $\mathbf{X}=(E,V,\cdots )$ and $\mathbf{Z}$ from now on [177]. This also means that (i) there is no chemical reaction, and (ii) there is EQ with respect to the chemical potential. Most of the time, we will simplify the discussion by using a single internal variable; the extension to many internal variables is trivial.

Our primary interest is in studying an irreversible process $\mathcal{P}$, which in MNEQT requires the existence of thermodynamic forces [51]. Their absence signifies that $\mathcal{P}$ represents a reversible process. It should be stressed that our notation is designed in such a way that the investigation can also apply directly to the (isolated) NEQ system ${\Sigma }_0$, if need be, for which no exchange with the outside is possible. In that case, the external driving must be replaced by spontaneous processes going on within ${\Sigma }_0$ that drive it towards equilibrium. During this drive, there is dissipation within ${\Sigma }_0$ that is found to contribute to work fluctuations in the $\mu$NEQT. As is well-known, such spontaneous fluctuations are not directly captured in the $\mathring{\mu }$NEQT, the microstate extension of the $\mathring{\mathrm{M}}\mathrm{NEQT}$. This makes our approach superior.

5.11.2. $\tilde{\Sigma }$ Always in EQ

We will assume $\tilde{\Sigma }$ to be always in equilibrium (which requires it to be extremely large compared to $\Sigma$, as noted above). Any irreversibility going on within ${\Sigma }_0$ due to internal dissipation, internal motion, internal nonuniformities, etc., is ascribed to $\Sigma$ alone. Moreover, we assume additivity of volume, a weak interaction between, and  quasi-independence of, $\Sigma$ and $\tilde{\Sigma }$; the last two conditions, respectively, ensure that the energies and entropies are additive [75,76,134,148,149] but also impose some restriction on the size of $\Sigma$ in that it cannot be too small. In particular, the size should be at least as big as the correlation length for quasi-independence as discussed there. In this study, we will assume that all required conditions necessary for the above-mentioned additivity are met.

6. Mechanical Aspects

We will consider a system in this section, but the arguments are valid for any system $\Sigma$.

6.1. Microstate Evolution in ${\mathfrak{S}}_{\mathbf{Z}}$

The traditional formulation of statistical thermodynamics [33,48,79] is built on a mechanical approach in which ${\mathfrak{m}}_k$ follows its classical or quantum mechanical Hamiltonian evolution dictated by its SI-Hamiltonian ${\mathcal{H}}_k=\mathcal{H}\left(\left.{\mathbf{x}}_k\left(t\right)\right|\mathbf{W}\left(t\right)\right)$, which suffices to provide the deterministic mechanical description with NFl-$\mathbf{W}$. We will see below that k does not change as $\mathbf{W}$ changes in a process $\mathcal{P}$. We will only consider a classical case system $\Sigma$, for which the change in ${\mathcal{H}}_k$ in $\mathcal{P}$ is

$$d{\mathcal{H}}_k=\frac{\partial {\mathcal{H}}_k}{\partial {\mathbf{x}}_k\left(t\right)}·d{\mathbf{x}}_k\left(t\right)+\frac{\partial {\mathcal{H}}_k}{\partial \mathbf{W}\left(t\right)}·d\mathbf{W}\left(t\right).$$

(146)

The first term on the right, due to the dynamical variations of ${\mathbf{x}}_k$ in the system, vanishes identically due to Hamilton’s equations of motion for any ${\mathfrak{m}}_k$. Thus, for fixed $\mathbf{W}$, the energy $E_k\left(\mathbf{W}\right)=\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{W}\right)$ of ${\mathfrak{m}}_k$ remains constant in time due to deterministic Hamiltonian dynamics. Only the variation $d\mathbf{W}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ generates any change in $E_k$. Consequently, we can write

$$d{\mathcal{H}}_k=d{\mathcal{H}}_k^{\left(\mathrm{w}\right)}=\frac{d{\mathcal{H}}_k}{\partial \mathbf{W}\left(t\right)}·d\mathbf{W}\left(t\right)$$

(147)

for all ${\mathfrak{m}}_k$, which clearly shows that only the variation $d{\mathcal{H}}_k^{\left(\mathrm{w}\right)}$ due to $d\mathbf{W}$ is relevant. This is indicated by the superscript w on $d{\mathcal{H}}_k^{\left(\mathrm{w}\right)}$. We do not worry about how ${\mathbf{x}}_k$ changes dynamically in $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{W}\right)$ from now on, and focus, instead, on the state space ${\mathfrak{S}}_{\mathbf{Z}}$, in which we can simply express the Hamiltonian as ${\mathcal{H}}_k\left(\mathbf{W}\right)$ for any microstate, remembering that its value $E_k\left(\mathbf{W}\right)$ is a point in ${\mathfrak{S}}_{\mathbf{Z}}$.

6.2. SI-Microwork in ${\mathfrak{S}}_{\mathbf{Z}}$

The point $E_k\left(\mathbf{W}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ undergoes a change due to $d\mathbf{W}$ given by

$$d{E}_k=\frac{\partial E_k}{\partial \mathbf{W}}·d\mathbf{W}=-d{W}_k,$$

(148)

where

$$d{W}_k={\mathbf{F}}_{\mathrm{w}k}·d\mathbf{W},{\mathbf{F}}_{\mathrm{w}k}\doteq -\partial E_k/\partial \mathbf{W}.$$

(149)

denotes the Fl-generalized microwork produced by the Fl-generalized microforce ${\mathbf{F}}_{\mathrm{w}k}$; see Definition 17. These are SI-microquantities. As $E_k$ is uniquely determined by $\mathbf{W}$, the microforce is a deterministic and continuous function of $\mathbf{W}$; see below. The  SI-microwork $d{W}_k$ is mechanically defined work as $\mathbf{W}$ is varied, which explains why $\mathbf{W}$ is identified as the work parameter in $\mathcal{H}$. The variation $d\mathbf{Z}\left(t\right)\doteq \left(dE\right(t),d\mathbf{W}(t\left)\right)$ in time defines a thermodynamic process $\mathcal{P}$. The trajectory ${\gamma }_k$ in ${\mathfrak{S}}_{\mathbf{Z}}$ followed by ${\mathfrak{m}}_k$ during $\mathcal{P}$ as a function of time will be called the Hamiltonian trajectory. Being purely mechanical in nature, the trajectory is completely deterministic and cannot describe the evolution of the thermodynamic macrostate $\mathfrak{M}$ during $\mathcal{P}$ unless supplemented by thermodynamic stochasticity over $\mathcal{P}$; see Claim 1. This is accounted for by the variation in $p_k\left(\mathfrak{M}\right)$ as $\mathfrak{M}$ changes, and is determined by some stochastic perturbation such as the random interaction with $\tilde{\Sigma }$ [33,59]; see Definition 25. We discuss the origin of this stochasticity in Section 7, which will allow us to introduce heat and temperature.

Since ${\mathfrak{m}}_k$ and $p_k\left(\mathfrak{M}\right)$ are independent of each other, we can treat them separately. This provides a major simplification, as described below, for studying the process $\mathcal{P}$ in terms of a Hamiltonian trajectory ${\gamma }_k$. We study the mechanical evolution of ${\mathfrak{m}}_k$ along ${\gamma }_k$ without being concerned about the probabilities. The effect of the probability can then be supplemented by an appropriate probability. This will lead to the introduction of the concept of SI-microheat; see Section 10, where we investigate this concept in detail for the first time.

6.3. SI-Legendre Transform

We can alternatively consider the case with $\left\{{\mathbf{W}}_k\right\}$ as the Fl-parameter. In that case, we will be dealing with $\mathbb{W}$ as a random variable with outcome ${\mathbf{W}}_k$; see Claim 3. Let us clarify the significance of Equation (18) by considering ${\mathbf{F}}_{\mathrm{w}k}=(P_k,A_k)$ defined above, and show how we ensure a fixed P and A by considering Fl-${\mathbf{W}}_k=(V_k,{\xi }_k)$. We consider a ${\mathfrak{m}}_k$ with microenergy $E_k(V,\xi )$, from which we obtain $P_k(V,\xi )$ and $A_k(V,\xi )$. They are functions of two variables, and we look for their crossing ${\mathbf{W}}_k=(V_k,{\xi }_k)$ with a plane $\Pi$ defined by ${\mathbf{F}}_{\mathrm{w}}=(P,A)$ to determine ${\mathbf{W}}_k$. We now do this for every k using the same plane $\Pi$. Using these crossings, we have

$$\forall k,P=-\partial E_k/\partial V_k,A=-\partial E_k/\partial {\xi }_k.$$

(150a)

As the two derivatives have fixed values for every k, their averages are also the same fixed values in ${\mathbf{F}}_{\mathrm{w}}=(P,A)$ as required in Equation (18). The crossings ${\mathbf{W}}_k$ give the fluctuating $(V_k,{\xi }_k)$.

Alternatively, we can easily determine $(V_k,{\xi }_k)$ by considering an NEQ SI-Legendre transform $E_k^{\mathrm{L}}$ of $E_k$, defined as

$$E_k^{\mathrm{L}}(P,A)\doteq E_k(V_k,{\xi }_k)+P{V}_k+A{\xi }_k,$$

(150b)

which is a function of P and A, but not of $V_k$ and ${\xi }_k$, since $\partial E_k^{\mathrm{L}}/\partial V_k=0,\partial E_k^{\mathrm{L}}/\partial {\xi }_k=0$, as is easily seen using Equation (150a). We now have

$$V_k=\partial E_k^{\mathrm{L}}/\partial P,{\xi }_k=\partial E_k^{\mathrm{L}}/\partial A.$$

(150c)

After averaging over microstates in ${\mathfrak{M}}_{\det }$, we obtain

$$E^{\mathrm{L}}(P,A)\doteq E(V,\xi )+PV+A\xi .$$

(151)

Remark 50. 

$E^{\mathrm{L}}(P,A)$ must not be confused with the NEQ enthalpy $H=E(V,\xi )+P_{0}V$.

We can generalize the above discussion for the general case of NFl ${\mathbf{F}}_{\mathrm{w}}$ or Fl $\left\{{\mathbf{W}}_k\right\}$. We first define the SI-Legendre-transformed Hamiltonian

$${\mathcal{H}}^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)\doteq \mathcal{H}\left(\mathbf{W}\right)+\Phi ({\mathbf{F}}_{\mathrm{w}},\mathbf{W}),$$

(152a)

in terms of $\Phi ({\mathbf{F}}_{\mathrm{w}},\mathbf{W})$ introduced in Equation (23b). Its microenergy $E_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)$ is the SI-Legendre transform of $E_k\left({\mathbf{W}}_k\right)$, and is given by

$$E_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)\doteq E_k\left({\mathbf{W}}_k\right)+\Phi ({\mathbf{F}}_{\mathrm{w}},{\mathbf{W}}_k);$$

(153)

compare with $E_k^{\mathrm{L},\mathrm{Fl}}\left({\mathbf{F}}_{\mathrm{w}}\right)$ in Equation (22a). We are suppressing the suffix NFl, as it is clear from the dependence on ${\mathbf{F}}_{\mathrm{w}}$ that we are dealing with Fl ${\mathbf{W}}_k$; see Claim 5. For $E_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)$, ${\mathbf{F}}_{\mathrm{w}}$ plays the role of the (Legendre-transformed) NFl “work” parameter ${\mathbf{W}}^{\mathrm{L}}$($={\mathbf{F}}_{\mathrm{w}}$) so that the generalized (Legendre-transformed) Fl “microforce” ${\mathbf{F}}_{\mathrm{w}k}^{\mathrm{L}}$ is given by

$${\mathbf{F}}_{\mathrm{w}k}^{\mathrm{L}}\doteq -\partial E_k^{\mathrm{L}}/\partial {\mathbf{F}}_{\mathrm{w}}=-{\mathbf{W}}_k,$$

(154)

which should be compared with the second equation in Equation (149); note the presence of the negative sign above on the right side. The extension of the generalized microwork given in the first equation in Equation (149) to this case is the Fl Legendre-transformed microwork

$$d{W}_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)=-{\mathbf{W}}_k·d{\mathbf{F}}_{\mathrm{w}},$$

(155)

so that

$$d{E}_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)\equiv -d{W}_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right),$$

(156)

which is identical in form with Equation (148).

For the medium $\tilde{\Sigma }$, we have

$$d{\tilde{W}}_{\tilde{k}}^{\mathrm{L}}\left({\tilde{\mathbf{f}}}_{\mathrm{w}}\right)=-{\tilde{\mathbf{w}}}_{\tilde{k}}·d{\tilde{\mathbf{f}}}_{\mathrm{w}},$$

which, after reduction, yields

$$d{\tilde{W}}_k^{\mathrm{L}}\left({\tilde{\mathbf{f}}}_{0\mathrm{w}}\right)=d{\tilde{W}}^{\mathrm{L}}\left({\tilde{\mathbf{f}}}_{0\mathrm{w}}\right)=-\tilde{\mathbf{w}}·d{\tilde{\mathbf{f}}}_{0\mathrm{w}}=-d_{\mathrm{e}}{W}^{\mathrm{L}}\left({\tilde{\mathbf{f}}}_{0\mathrm{w}}\right),$$

(157)

where we have replaced ${\tilde{\mathbf{f}}}_{\mathrm{w}}$ by ${\tilde{\mathbf{f}}}_{0\mathrm{w}}$ of ${\Sigma }_0$, and used Equation (64a).

The average of $E_k^{\mathrm{L}}\left({\mathbf{F}}_{\mathrm{w}}\right)$ is given by

$$E^{\mathrm{L}}(S,{\mathbf{F}}_{\mathrm{w}})\doteq E(S,\mathbf{W})+{\mathbf{F}}_{\mathrm{w}}·\mathbf{W}$$

(158)

(compare with Equation (152a)), while other microquantities have their averages given by

$$\begin{array}{cc}\hfill {\mathbf{F}}_{\mathrm{w}}^{\mathrm{L}}& \doteq -\partial E^{\mathrm{L}}(S,{\mathbf{F}}_{\mathrm{w}})/\partial {\mathbf{F}}_{\mathrm{w}}=-\mathbf{W},\hfill \\ \hfill d{W}^{\mathrm{L}}(S,{\mathbf{F}}_{\mathrm{w}})& =-\mathbf{W}·d{\mathbf{F}}_{\mathrm{w}},\hfill \\ \hfill d{E}^{\mathrm{L}}(S,{\mathbf{F}}_{\mathrm{w}})& \equiv -d{W}^{\mathrm{L}}(S,{\mathbf{F}}_{\mathrm{w}}),\hfill \end{array}$$

(159)

as is expected from the above discussion.

As considering Fl-$\mathbf{W}$ creates no additional complication, we will mostly deal with NFl-$\mathbf{W}$ in this review.

For completeness and later usage in Section 12.2, we also introduce another Legendre transform in the case that $\mathbf{W}$