Irreversibility, Dissipation, and Its Measure: A New Perspective

Abstract

Dissipation and irreversibility are two central concepts of classical thermodynamics that are often treated as synonymous. Dissipation D is lost or dissipated work $W_{\mathrm{diss}}\ge 0$ but is commonly quantified by entropy generation ${\Delta }_{\mathrm{i}}S$ in an isothermal irreversible macroscopic process that is often expressed as Kullback–Leibler distance $D_{\mathrm{KL}}$ in modern literature. We argue that $D_{\mathrm{KL}}$ is nonthermodynamic, and is erroneously justified for quantification by mistakenly equating exchange microwork ${\Delta }_{\mathrm{e}}{W}_k$ with the system-intrinsic microwork $\Delta W_k={\Delta }_{\mathrm{e}}{W}_k+{\Delta }_{\mathrm{i}}{W}_k$, which is a very common error permeating stochastic thermodynamics as was first pointed out several years ago, see text. Recently, it is discovered that dissipation D is properly identified by ${\Delta }_{\mathrm{i}}W\ge 0$ for all spontaneously irreversible processes and all temperatures T, positive and negative in an isolated system. As T plays an important role in the quantification, dissipation allows for ${\Delta }_{\mathrm{i}}S\ge 0$ for $T>0$, and ${\Delta }_{\mathrm{i}}S<0$ for $T<0$, a surprising result. The connection of D with $W_{\mathrm{diss}}$ and its extension to interacting systems have not been explored and is attempted here. It is found that D is not always proportional to ${\Delta }_{\mathrm{i}}S$. The determination of D requires $d_{\mathrm{i}}{p}_k$, but we show that Fokker-Planck and master equations are not general enough to determine it, which is contrary to the common belief. We modify the Fokker-Planck equation to fix the issue. We find that detailed balance also allows for all microstates to remain disconnected without any transition among them in an equilibrium macrostate, another surprising result. We argue that Liouville’s theorem should not apply to irreversible processes, contrary to the claim otherwise. We suggest to use nonequilibrium statistical mechanics in extended space, where $p_k$’s are uniquely determined to evaluate D.

1. Introduction

Irreversibility, dissipation and the second law (SL) of entropy increase are invariably treated as synonymous in classical and modern nonequilibrium (NEQ) thermodynamics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25], but the significance of spontaneity is mostly overlooked in any modern discussion as if it plays no or only a minor role. The law of increase of entropy is best enunciated by considering an isolated thermodynamic system ${\Sigma }_0$ for which the entropy can never decrease whether it is described classically or quantum mechanically. However, there is an unsolved problem associated with ${\Sigma }_0$, if we treat it as an isolated mechanical system to be denote by ${\Sigma }_0^{\mathrm{D}}$. It is mechanically described by a time-independent deterministic Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right)$, where the dynamical variable $\mathbf{x}\doteq \left\{({\mathbf{r}}_i,{\mathbf{p}}_i)\right\},i=1,2,\cdots ,N$ denotes the set of positions and momenta of N particles and $\mathbf{w}\doteq (N,V,\cdots )$ the fixed set of extensive parameters required to specify the Hamiltonian of the system. We also introduce the set $\mathbf{X}\doteq (E,\mathbf{w})$, which is called an observable since it is controlled from outside the system such as by a medium $\tilde{\Sigma }$ (see Figure 1). It follows from the Liouville theorem that the evolution of isolated systems is not compatible with the second law (SL) as the Gibbs entropy S remains a constant of motion. Liouville’s theorem either applies to a mechanically deterministic ${\Sigma }_0^{\mathrm{D}}$ or a thermodynamic system ${\Sigma }_0$ only in equilibrium (EQ) as is easily verified; see Landau and Lifshitz [15]. This also implies that Liouville’s theorem does not describe spontaneous processes that are a hallmark of the breakdown of time-reversal symmetry, which eventually result in dynamical emergence of EQ in ${\Sigma }_0$, sometimes through thermalization ([26,27,28], and references therein). This make these processes irreversible.

The stochastic notions of entropy S [1,2,29,30,31], temperature T, and macroheat $dQ$ are central to thermodynamics of a system $\Sigma$; macrowork $dW$, in contrast, is purely mechanical; see Equation (14), where they are properly introduced. They are commonly used to distinguish a thermodynamic system $\Sigma$ that is purely stochastic from its purely deterministic mechanical analog system ${\Sigma }^{\mathrm{D}}$ by realizing that T and $dQ$ are applicable to $\Sigma$ but not ${\Sigma }^{\mathrm{D}}$ [32]. The traditional way to describe ${\Sigma }^{\mathrm{D}}$ is using a purely conservative Hamiltonian $\mathcal{H}$. Even though it not usually the case, one can also associate a constant of motion S to ${\Sigma }^{\mathrm{D}}$ [33] as it evolves, which is consistent with the Liouville theorem as expected. In contrast, it usually changes in a thermodynamic process, in which $\mathbf{w}\left(t\right)$ changes with time t. This explains the importance of $\mathbf{w}$ in thermodynamics due to its ability to change. For clarity, we assume $\Sigma$ or ${\Sigma }^{\mathrm{D}}$ to be stationary.

Notation 1.

We partition any extensive process quantity $d\mathfrak{q}$ or its accumulation $\Delta \mathfrak{q}$ in the exchange and internal contributions as follows

$$d\mathfrak{q}=d_{\mathit{e}}\mathfrak{q}+d_{\mathit{i}}\mathfrak{q},\Delta \mathfrak{q}={\Delta }_{\mathit{e}}\mathfrak{q}+{\Delta }_{\mathit{i}}\mathfrak{q},$$

(1)

where $d\mathfrak{q}$ can either denote a macroquantity such as $dW,dS$, etc. or a microquantity (associated with ${\mathfrak{m}}_k$) such as $d{W}_k=d_{\mathit{e}}{W}_k+d_{\mathit{i}}{W}_k,d{p}_k=d_{\mathit{e}}{p}_k+d_{\mathit{i}}{p}_k,$ etc.

Remark 1.

In current literature, ${\Delta }_{\mathit{e}}Q$ and ${\Delta }_{\mathit{e}}W$ are often denoted by Q and W, which we will avoid to remain consistent with the modern notation [3,12,19] shown in Equation (1) according to which the exchange or crossing across the boundary of any extensive process quantity with the medium and its change within the system carry the inner suffix e and i, respectively.

1.1. Goals

One of our goal is to develop a single, unified, approach that applies without exception to an isolated system in Figure 1a that was recently considered [34] as well as an interacting system in Figure 1b that is the main focus of this investigation. This has not always been the case. The development of classical thermodynamics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,29,31] is formulated in terms of exchange or border-crossing quantities $d_{\mathrm{e}}Q$ and $d_{\mathrm{e}}W$ that require the interacting system $\Sigma$ to be embedded in a medium as shown in Figure 1b. These quantities are medium-intrinsic (MI) as they are controlled by the medium $\tilde{\Sigma }$, which is always taken to be in EQ. Thus, there is no analog of the first law, when formulated using these quantities [as in Equation (17)] for an isolated system in Figure 1a, which must be described by system-intrinsic (SI) quantities such as $dQ$ and $dW$ but are not introduced in classical thermodynamics. It is intuitively obvious that the first law must also exist for an isolated system. A thermodynamic process $\mathcal{P}$, which may be reversible (${\mathcal{P}}_{\mathrm{rev}}$) or irreversible (${\mathcal{P}}_{\mathrm{irr}}$), is customarily introduced in classical thermodynamics as manipulating or driving $\Sigma$ between two EQ macrostates by $\tilde{\Sigma }$. Dissipation occurs during ${\mathcal{P}}_{\mathrm{irr}}$ in the form of dissipate (or lost) work $W_{\mathrm{diss}}$ [23] defined by

$$W_{\mathrm{diss}}={\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}-{\Delta }_{\mathrm{e}}W,$$

(2)

where ${\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}$ is the exchange macrowork performed by $\Sigma$ during the reversible processes ${\mathcal{P}}_{\mathrm{rev}}$ associated with a real process $\mathcal{P}$ between the same two equilibrium (EQ) macrostates ${\mathfrak{M}}_{\mathrm{eq}}^{\mathrm{in}}$ and ${\mathfrak{M}}_{\mathrm{eq}}^{\mathrm{fin}}$. Such manipulation is not possible for an isolated system; yet this system certainly undergoes a spontaneous evolution from a nonequilibrium (NEQ) macrostate $\mathfrak{M}$ to its equilibrium (EQ) macrostate ${\mathfrak{M}}_{\mathrm{eq}}$, see Figure 2, which is also irreversible but $W_{\mathrm{diss}}$ in Equation (2) is not meaningful as there is no exchange macrowork in this case and also because there do not exist two terminal EQ macrostates for the process. Thus, the concept of dissipation as dissipated work is developed only for an interacting system in classical thermodynamics. To the best of our knowledge, no attempt has been made to discuss dissipation as “dissipated” work for an isolated system until very recently in [34]. Therefore, it is extremely important to have an approach to discuss both isolated and interacting systems in one unified manner. Our approach that is based on SI-quantities removes this limitation. The resulting NEQ thermodynamics (NEQT) is denoted by MNEQT and the corresponding NEQ statistical mechanics by $\mu$NEQT to distinguish them from their counterparts $\mathring{\mathrm{M}}$NEQT and $\mathring{\mu }$NEQT involving MI-quantities [35].

Remark 2.

It should, however, be noted that the MI-quantities are easy to handle as they show no irreversibility since the medium is taken to be in equilibrium (EQ); the irreversibility if present is always associated with the system Σ [12,15,19,21,25].

Remark 3.

It is much easier to work with microstates directly as they depend only on the work-parameter set and not entropy. It is this approach we take in this investigation.

Dissipation has been defined by many different statements in the literature, see [34] for some of these statements, but they invariably imply it to be always positive, and the source of irreversibility and irreversible entropy $d_{\mathrm{i}}S\ge 0$. It is assumed to originate from dissipative forces such as friction or viscosity, but what causes these forces or what causes the fixed signature of dissipation remains unclear until recently [34]. We refer the reader to this publication for more details on this topic. The focus there has been on an isolated system in Figure 1a.

Our main goal is to extend the previous work on isolated systems to interacting systems in Figure 1b, and properly introduce a definition of dissipation for them so it also covers an isolated system $\Sigma$. Thus, our goal is to have a unified approach that works equally well with both kinds of systems in Figure 1. We also require that it does not introduce any “temperature-like” concept as discussed in [34] but is intimately intertwined with irreversibility and dissipated work. This formulation for interacting systems then allows for understanding its consequence for the generalization (GSL) of the second law (SL), the interplay of (both positive and negative) temperature and SL, and the extension of the concept of lost or dissipated work to processes that are not necessarily between EQ macrostates. This has not been possible to investigate in classical thermodynamics, but we will mostly use the NEQ thermodynamics developed by us [36]. We remind the reader that the concept of negative EQ temperatures is well established in statistical mechanics [37,38,39]. This also helps to shine new light on the quite surprising implications of SL for negative temperatures as discussed elsewhere [34]. We broaden the concept of irreversibility caused by spontaneous and manipulated or driven processes for the unified approach as noted above. For example, we allow for $d_{\mathrm{i}}{W}_k$ to have a fixed sign in any process, but it can be either positive or negative so it may not satisfy SL. This allows us to understand the significance on internal constraints as discussed by Callen [13] extensively as its implication for SL such as in Szilard engine [40,41] has not been properly understood, until clarified recently [42]. There are also interesting attempts to understand dissipation using information theory to understand dissipated heat; we cite a few [43,44].

Other goals we have in this study is to critically examine the roles of some well-known techniques (Liouville Theorem, Fokker-Planck equation, master equations, and Kullback-Leibler distance) often used in modern NEQ thermodynamics at microstate level [45,46] to gauge their success in NEQ thermodynamics.

We introduced thermalization above as a cause of irreversibility. We now list some of the problems we wish to focus on in this study that arise as part of achieving our goals. As mentioned above, we seek their answers that must apply to an isolated system in Figure 1a as well as an interacting system in Figure 1b without exception in our unified approach.

• What makes a thermodynamic process irreversible?
• Does irreversibility always bring a thermodynamic system to EQ or can it take it away from it?
• What is dissipation and how does it depend on the temperature (positive and negative) of the system?
• Is dissipation simply dissipated or lost work $W_{\mathrm{diss}}$ during a process between two EQ macrostates, which is the most common definition of dissipation as was first proposed by Thomson [47]?
• How to describe dissipation between two arbitrary macrostates?
• Does dissipation occur when only heat is transferred from hot to cold?
• How should dissipation be quantified?
• Do current methods of quantification work for both isolated and interacting systems?
• Is dissipation always directly related to the irreversible entropy generation (${\Delta }_{\mathrm{i}}S\ge 0$) as is commonly thought of?
• Is ${\Delta }_{\mathrm{i}}S<0$ a violation of SL?
• Can dissipation be ever negative?

Our aim here is to answer these and other long-standing questions in NEQ thermodynamics and statistical mechanics to fill in the gaps in our understanding and unravel hidden physics behind irreversibility, dissipation, and SL. While the identity

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

(3)

has been known for a while [35,36,48,49], its deep connection with dissipation was clarified only recently [34]. Because of its fundamental importance, we have identified it as the irreversibility principle (Irr-P). It follows from the most fundamental fact

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

(4)

of thermodynamics that no internal processes can change the energy [9]; recall that $d_{\mathrm{i}}E=dE-d_{\mathrm{e}}E$, see Remark 1. The principle, though never appreciated hitherto as fundamental, applies to any $\Sigma$ and $\mathcal{P}$ and is as valid as classical thermodynamics.

1.2. Mechanical and Thermodynamic Uniformity

As dissipation is related to temporal evolution, it requires properly understanding the importance of equilibrium (stable (s) and unstable (u)) points for microstates and macrostates during this evolution. The corresponding systems are identified as ${\Sigma }^{\mathrm{s}}$ and ${\Sigma }^{\mathrm{u}}$, and their macrostates as ${\mathfrak{M}}^{\mathrm{s}}$ and ${\mathfrak{M}}^{\mathrm{u}}$, respectively. What causes the convergence for ${\mathfrak{M}}^{\mathrm{s}}$ is not understood theoretically. The traditional approach in classical thermodynamics is to postulate SL in the form of entropy maximum principle ([13], see for example, Chapter 4), then prove thermodynamic stability [15] and convergence to ${\mathfrak{M}}_{\mathrm{seq}}$ by following ${\mathfrak{M}}^{\mathrm{s}}$-evolution (${\mathfrak{M}}^{\mathrm{s}}\to {\mathfrak{M}}_{\mathrm{seq}}$ as $t\to \infty$) along blue arrows in the state space of $\Sigma$; see Figure 3. However, ${\mathfrak{M}}^{\mathrm{u}}$-evolution is never considered in classical thermodynamics, but we do for the following reason. As SL results in stability, any SL-violation strongly suggests, but not yet verified, thermodynamic instability for its cause so we also consider rarely studied unstable macrostate ${\mathfrak{M}}^{\mathrm{u}}$ emerging out of ${\mathfrak{M}}_{\mathrm{ueq}}$ along red arrows; however, see [38] as exception.

Definition 1.

A Hamiltonian $\mathcal{H}\left(\mathbf{w}\right)$ is said to be uniform if its $\mathbf{w}$ specifies the entire system. In other words, $\mathcal{H}\left(\mathbf{w}\right)$ does not depend on $\mathbf{w}$’s of its different parts that will make it nonuniform.

By definition, therefore, ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ is a uniform macrostate in that all its microstates $\left\{{\mathfrak{m}}_k\left(\mathbf{w}\right)\right\}$ and microenergies $\left\{E_k\left(\mathbf{w}\right)\right\}$ are uniquely determined by its stationary and uniform Hamiltonian $\mathcal{H}\left(\mathbf{w}\right)$. Uniformity of the system or its lack plays an important role in our approach. An EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ is uniform, while a NEQ macrostate $\mathfrak{M}$ is nonuniform [13,15]. All arrows in Figure 3 point towards increasing time t. It is easy to be convinced that EQ macrostates (${\mathfrak{M}}_{\mathrm{seq}}$ or ${\mathfrak{M}}_{\mathrm{ueq}}$) have maximum entropy [34]. Thus, the above evolution describe the behavior of entropy (increasing or decreasing) as ${\mathfrak{M}}^{\mathrm{s}}$ or ${\mathfrak{M}}^{\mathrm{u}}$ becomes less or more nonuniform. Thus, S increases during evolution for ${\mathfrak{M}}^{\mathrm{s}}$ but decreases for ${\mathfrak{M}}^{\mathrm{u}}$. Does it mean that ${\mathfrak{M}}^{\mathrm{s}}$ satisfies SL, ${\mathfrak{M}}^{\mathrm{u}}$ violates SL? We obtained a surprising result in [34] that the analysis requires a reformulation of SL for ${\mathfrak{M}}^{\mathrm{u}}$ as it pertains to negative temperatures.

One of the final conclusions is that dissipation is not always proportional to irreversible entropy generation, another surprising result.

1.3. Layout

We briefly review some fundamental concepts and preliminaries in the next section. It is an important section as it contains many important concepts central to the discussion in this investigation. The first one is the BCGM proposal in Section 2.1 to develop a new approach for NEQ thermodynamics (NEQT) by exploiting the mechanical evolution of microstates and then taking the ensemble average but without requiring SL so it can be violated at this stage of the NEQT development. We then consider two different formulations, cf. Equations (14a) and (17), of the first law based on the use of system-intrinsic (SI) and medium-intrinsic (MI) macroquantities in Section 2.3, which then result in a very important irreversibility principle (Irr-P) ${\Delta }_{\mathrm{i}}W={\Delta }_{\mathrm{i}}Q⪌0$ in Equation (3) that relates to reversibility and time-reversal invariance. We also comment on why Liouville’s theorem fails for a thermodynamic system $\Sigma$. We briefly review dissipation in Section 2.4, to be discussed in detail in Section 5.4. The Kullback-Leibler distance as a measure of dissipation is introduced in Section 2.5 and considered further in Section 6. The use of the principle of detailed balance in fluctuation theorems is briefly described in Section 2.6; the principle is further discussed in detail in Section 7.4. The new NEQ thermodynamic (MNEQT) and statistical mechanics ($\mu$NEQT) using SI quantities but without invoking the second law (SL) in extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ are briefly discussed in Section 3 and Section 4. Various process quantities are allowed to have any signature that may even violate SL. A simple example to show how internal variables that extend the state space to ${\mathfrak{S}}_{\mathbf{Z}}$ from ${\mathfrak{S}}_{\mathbf{X}}$ is given in Section 3.1. Unique microstate probabilities $p_k$ in ${\mathfrak{S}}_{\mathbf{Z}}$ are derived in Section 4.1. After the brief introduction of new thermodynamics, we embark on introducing the mechanical approach based on the BCGM proposal in Section 5. In Section 5.3, we introduce the mechanical equilibrium principle of energy, which is applied to microstates by treating them as a mechanical system ${\Sigma }^{\mathrm{D}}$. This is a very important principle that allows us to demonstrate a generalization of SL (GSL) by exploiting the stability/instability of the equilibrium point of ${\mathfrak{m}}_k$. In the next two subsections, we revisit and critically examine dissipation. We use the irreversible microwork ${\Delta }_{\mathrm{i}}{W}_k$ done by spontaneously generated microforce imbalance in ${\mathfrak{m}}_k$ to obtain a precise formulation of dissipation by dissipated macrowork $D={\Delta }_{\mathrm{i}}W={\Delta }_{\mathrm{i}}Q\ge 0$. We then connect it to $W_{\mathrm{diss}}$ between two EQ macrostates in Section 5.6 to justify that D generalizes the concept of dissipation by Thomson [47] to any thermodynamic process. As D does not require any knowledge of the Temperature, it is valid for positive and negative temperatures without changing the sign of D as discussed in Section 5.7. The fixed signature of D becomes the generalized SL (GSL). Here we learn that stable macrostates ${\mathfrak{M}}^{\mathrm{s}}$ correspond to positive temperatures and unstable macrostates ${\mathfrak{M}}^{\mathrm{u}}$ correspond to negative temperatures. Relation between Kullback-Leibler distance $D_{\mathrm{KL}}$ and ${\Delta }_{\mathrm{i}}S$ is considered in Section 6 and we find that there cannot be any such relation simply because $D_{\mathrm{KL}}$ is not a thermodynamic quantity but ${\Delta }_{\mathrm{i}}S$ is. We point out that the misleading relationship is the result of a very deep misunderstanding that is very common and persists even today, even though the misunderstanding has been known for a while. We then discuss Fokker-Planck equation and master equations in Section 7 that are widely used to determine $d_{\mathrm{i}}{p}_k$ to assess dissipation. We conclude that they are not capable of determining $d_{\mathrm{i}}{p}_k$. Master equations are found to be not applicable to isolated systems. We then follow the discussion of the principle of detailed balance in Section 7.4 and come to a similar conclusion. We conclude that the most successful approach is to use $p_k$ derived in Section 4.1. The final section contains a summary and a brief discussion of new results.

2. Fundamental Concepts and Preliminaries

2.1. BCGM Proposal

Thermodynamic concept of internal energy of a macrostate $\mathfrak{M}$ as an (ensemble) average quantity E is probably the most important mechanical macroquantity of a thermodynamic system $\Sigma$, which requires a mechanical description in terms of its conservative Hamiltonian $\mathcal{H}$ that is deterministic in nature. The latter defines the conservative evolution of a system ${\Sigma }^{\mathrm{D}}$ that is the deterministic analog of the thermodynamic system $\Sigma$. Introduction of stochasticity in ${\Sigma }^{\mathrm{D}}$ by appending probability to it results in $\Sigma$ as was first proposed by Clausius, Maxwell, Boltzmann and Gibbs (the BCGM proposal) [1,5,15,20,29,31,50,51], which is reviewed recently by Gujrati [34], unfortunately, it was called the BCM proposal so it did not give full credit to the contribution of Gibbs and which we follow very closely in this study. The stationary Hamiltonian of ${\Sigma }^{\mathrm{D}}$ is written as $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right)$ [32]. We will usually consider $N\in \mathbf{w}$ fixed in this study so that it can be used to fix the size of the system. Even though it is very common to use time t as a parameter in $\mathcal{H}$, we will always show it explicitly by not including it in $\mathbf{w}$. The microstates ${\mathfrak{m}}_{k\mathrm{eq}}\left(\mathbf{w}\right)$ of microenergy $E_k\left(\mathbf{w}\right)\doteq \mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{w}\right)$ for different ${\mathbf{x}}_k$ denote instantaneous states of the equilibrium (EQ) macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ of $\Sigma$ with probabilities $p_k^{\mathrm{eq}}\left(\mathbf{X}\right)$, where $\mathbf{X}\doteq (E,\mathbf{w})$ is the state variable determining the state space ${\mathfrak{S}}_{\mathbf{X}}$, which is the most important state space where all processes $\mathcal{P}$ are executed by varying $\mathbf{w}\left(t\right)$. As $\mathcal{H}$ depends on $\mathbf{w}$ for the entire ${\Sigma }^{\mathrm{D}}$, we express this fact by saying that $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{w}\right)$ represents a ${\Sigma }^{\mathrm{D}}$ that is uniform in the state space ${\mathfrak{S}}_{\mathbf{X}}$. This uniformity of ${\Sigma }^{\mathrm{D}}$ results in a uniform macrostate, which is the EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ of $\Sigma$. We, therefore, specify ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{w}\right)$ by the collection $\left\{{\mathfrak{m}}_{k\mathrm{eq}},p_k^{\mathrm{eq}}\right\}$; ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ may have an implicit time dependence due to implicit time dependence of $\mathbf{w}\left(t\right)$, but does not have an explicit time dependence. For a time-dependent Hamiltonian $\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$, we obtain a nonequilibrium (NEQ) macrostate $\mathfrak{M}(\mathbf{w},t)$ with an explicit time dependent ${\mathfrak{m}}_k,p_k$ and $E_k$ so that $\mathfrak{M}(\mathbf{w},t)$ is specified by $\left\{{\mathfrak{m}}_k,p_k\right\}$, where $p_k$ is probability of the kth microstate ${\mathfrak{m}}_k$.

Remark 4.

We use the discrete versions [21,52,53] for our systems. Thus, we take the separating boundaries between various systems or subsystems to be sharp. Therefore, the thermodynamics we develop is not locally defined that is required for a continuous description as is, for example, developed in [3,12,18]. Our thermodynamics is over the entire system so all state variables are defined over the whole system.

What distinguishes discrete our approach from the continuous approach is that we proceed using a statistical mechanical approach in which microstates as mechanical systems play the central role to justify the inequality, from which we derive the dissipated work inequality to give rise to the generalized second law [34] as a consequence of analytical mechanics.

We now enunciate the BCGM proposal:

BCGM Proposal: While microstates $\left\{{\mathfrak{m}}_k\right\}$ of a mechanical Hamiltonian are deterministic so they are oblivious to any sense of stochasticity, thermodynamic description of $\Sigma$ is obtained by appending probabilities to them to describe a macrostate $\mathfrak{M}$ specified by $\left\{{\mathfrak{m}}_k,p_k\right\}$.

One can easily identify an extended state space ${\mathfrak{S}}_{\mathbf{Z}}$, in which $\mathbf{Z}\doteq \mathbf{X}\cup \mathsf{\xi }$ is formed by the union of $\mathbf{X}$ with an additional extensive set $\mathsf{\xi }$ of n state variables, known as internal variables [12,19,21,35,54] for both ${\Sigma }^{\mathrm{D}}$ and $\Sigma$. A macrostate $\mathfrak{M}(\mathbf{X},t)$ of $\Sigma$ in ${\mathfrak{S}}_{\mathbf{X}}$ can be uniquely described without any explicit time dependence as a macrostate $\mathfrak{M}\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$. This means that $\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$ is equivalently expressed as $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ with no explicit time dependence; here $\mathbf{W}\doteq \mathbf{w}\cup \mathsf{\xi }$. These macrostates still represent NEQ macrostates in ${\mathfrak{S}}_{\mathbf{Z}}$ as their entropies are strictly less than $S_{\mathrm{eq}}\left(\mathbf{X}\right)$. The set $\mathsf{\xi }$ is not an observable set as it cannot be controlled from outside the system, and keeps changing for $\Sigma$ as it relaxes towards its EQ value, which we take to be ${\mathsf{\xi }}^{\mathrm{eq}}=0$, in time. During this relaxation, $n\to 0$ [55]. Despite this, $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ is a uniform Hamiltonian in ${\mathfrak{S}}_{\mathbf{Z}}$ at each instant for both ${\Sigma }^{\mathrm{D}}$ and $\Sigma$, see Definition 1 with $\mathbf{w}$ replaced by $\mathbf{W}$, but which is not stationary.

By keeping N fixed, there is no chemical micro-/macro-work in $d{W}_k/dW$. To allow for variations in N, we must not keep it fixed. We need some other extensive observable from $\mathbf{X}$ to be kept fixed to fix the size of the system. One possible choice is to keep the volume V fixed. In this case, we can vary N so that its variation will contribute ${\mu }_{k}dN/\mu dN$ to $d{W}_k/dW$. To allow for the possibility of chemical reactions, we consider the partition $dN=d_{\mathrm{e}}N+d_{\mathrm{i}}N$, with $d_{\mathrm{i}}N$ denoting the irreversible change in N due to chemical reactions. Indeed, the possibility of chemical reactions is how affinity and internal variable were first introduced in thermodynamics [3,19].

The change $d\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$ during an infinitesimal part $\delta \mathcal{P}$ of a mechanical process $\mathcal{P}$ results from changes in $\mathbf{w}$ and t [35,36,49]:

$$d\mathcal{H}={\displaystyle \frac{\partial \mathcal{H}}{\partial \mathbf{w}}}·d\mathbf{w}+{\displaystyle \frac{\partial \mathcal{H}}{\partial t}}dt\equiv ({\displaystyle \frac{\partial \mathcal{H}}{\partial \mathbf{w}}}·\stackrel{·}{\mathbf{w}}+{\displaystyle \frac{\partial \mathcal{H}}{\partial t}})dt;$$

(5)

the contribution from the variation of $\mathbf{x}$ vanishes due to Hamilton’s equations of motion. Therefore, we do not need to study this variation in the phase space of the system but only in the state space ${\mathfrak{S}}_{\mathbf{X}}$. The dot over $\mathbf{w}$ denotes the total time derivative. Similarly in ${\mathfrak{S}}_{\mathbf{Z}}$, the change $d\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)\equiv {\displaystyle \frac{\partial \mathcal{H}}{\partial \mathbf{W}}}·d\mathbf{W}$ requires replacing the second term on the right above by

$${\displaystyle \frac{\partial \mathcal{H}}{\partial \mathsf{\xi }}}·d\mathsf{\xi }={\displaystyle \frac{\partial \mathcal{H}}{\partial \mathsf{\xi }}}·\stackrel{·}{\mathsf{\xi }}dt,$$

so that

$$d\mathcal{H}={\displaystyle \frac{\partial \mathcal{H}}{\partial \mathbf{W}}}·d\mathbf{W}\equiv {\displaystyle \frac{\partial \mathcal{H}}{\partial \mathbf{W}}}·\stackrel{·}{\mathbf{W}dt}$$

(6)

Therefore, from now on, we will not be concerned with the $\mathbf{x}$-dependence in $\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$ or $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{W}\right)$ or its variation during $\mathcal{P}$ and its time reversal consequences so we express it simple as $\mathcal{H}(\mathbf{w},t)$. Time-reversal in thermodynamics is, therefore, described by reversing $\mathbf{w}\left(t\right)$ and t only that results in reversing the process $\mathcal{P}$. We will use $\mathfrak{S}$ to denote the two state spaces ${\mathfrak{S}}_{\mathbf{X}}$ and ${\mathfrak{S}}_{\mathbf{Z}}$ if no confusion arises.

Remark 5.

From what is said above, the effect of time reversal on $\mathbf{x}$ in phase space has no effect in determining $d\mathcal{H}$. In particular, its effect on the entropy is of no consequence due to the Liouville theorem [15]. Therefore, the effect of time reversal on microstates, which is central to our investigation, should be restricted only to the state space.

Claim 1.

As we are not concerned in our approach with the temporal evolution of $\mathbf{x}$ in the phase space, which is controlled by the Liouville theorem [15], the latter plays no role in our investigation. As the work-parameter set and the resulting stochasticity caused by the irreversibility principle in Equation (3) are not covered by the Liouville theorem, the latter is not applicable to NEQ thermodynamic evolution. As a side remark, we should recall that the phase space density is not conserved for a damped harmonic oscillator.

The only role of $\mathbf{x}={\mathbf{x}}_k$ is to specify the kth microstate ${\mathfrak{m}}_k$. We denote $d\mathcal{H}$ for ${\mathfrak{m}}_k$ by $d{E}_k(\mathbf{w},t)$ in ${\mathfrak{S}}_{\mathbf{X}}$ or $d{E}_k\left(\mathbf{W}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, which we again simplify by using only $d{E}_k$ in this investigation if no confusion arises. The change determines the microwork

$$d{W}_k\doteq -d{E}_k$$

(7a)

in $\mathfrak{S}$ done by the SI-microforce

$$({\mathbf{f}}_{\mathrm{w}k}=-{\displaystyle \frac{\partial E_k}{\partial \mathbf{w}}},f_{\mathrm{t}k}=-{\displaystyle \frac{\partial E_k}{\partial t}})\mathrm{or}{\mathbf{F}}_{\mathrm{w}k}=-{\displaystyle \frac{\partial E_k}{\partial \mathbf{W}}}$$

(7b)

exerted by ${\mathfrak{m}}_k$ in ${\mathfrak{S}}_{\mathbf{X}}$ and ${\mathfrak{S}}_{\mathbf{Z}}$, respectively. In a thermodynamic work-process ${\mathcal{P}}_{\mathrm{w}}$ determined by the time variation of the observable set $\mathbf{w}\left(t\right)$, ${\mathfrak{m}}_k$ follows a deterministic trajectory ${\gamma }_k$ without altering $p_k$. Only a heat-process ${\mathcal{P}}_{\mathrm{h}}$ (which involves macroheat $dQ$) can change $p_k$ so determining the change $d{p}_k$ becomes the central issue in determining $dQ$. This will become clear soon. Normally, both processes are involved in thermodynamics.

Adopting the BCGM proposal and following Gibbs [31], we give the statistical definition of the macroenergy E and the entropy S as the ensemble or thermodynamic average in $\mathfrak{S}$ over microstates:

$$E={\sum }_k{p}_k{E}_k,S=-{\sum }_k{p}_k{\eta }_k,$$

(8)

where $E_k$ is microstate energy of the kth microstate ${\mathfrak{m}}_k$, and $p_k$ is its probability in the ensemble, and

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

(9)

is the Gibbs probability index [31]. They are two of the state variables, except that E is part of $\mathbf{X}$ or $\mathbf{Z}$ specifying $\Sigma$ in the state space, which we denote by the same notation ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{S}}_{\mathbf{Z}}$ that were introduced above. The EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ is uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$. On the other hand, NEQ macrostate $\mathfrak{M}$ is not uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$ as it lies outside it; see Figure 2; however, it is uniquely defined in an appropriately chosen ${\mathfrak{S}}_{\mathbf{Z}}$.

Remark 6.

The microstates ${\mathfrak{m}}_k(\mathbf{w},t)$ in ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{m}}_k\left(\mathbf{W}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ of ${\Sigma }^{\mathrm{D}}$ are deterministic so they are not affected by their probabilities with which they appear in the (Gibbs) ensemble. This justifies the definitions of E and S in Equation (8) using probability theory.

2.2. Thermodynamic Force

For completeness, we note that the SI-macroforce is the ensemble average

$${\mathbf{f}}_{\mathrm{w}}=-\partial E/\partial \mathbf{w},f_{\mathrm{t}}=-\partial E/\partial t$$

(10a)

in ${\mathfrak{S}}_{\mathbf{X}}$, and

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

(10b)

in ${\mathfrak{S}}_{\mathbf{Z}}$, and the MI-macroforce exerted by $\tilde{\Sigma }$ on $\Sigma$ are denoted by

$${\mathbf{f}}_{\mathrm{w}0}=-\partial \tilde{E}/\partial \tilde{\mathbf{w}},$$

(11)

where ^˜ refers to the medium. Evidently, $f_{\mathrm{t}0}=0$. The imbalance is

$$({\mathbf{f}}_{\mathrm{w}k}-{\mathbf{f}}_{\mathrm{w}0},f_{\mathrm{t}k})\mathrm{or}{\mathbf{F}}_{\mathrm{w}k}-{\mathbf{f}}_{\mathrm{w}0}$$

(12)

for microstates and

$$({\mathbf{f}}_{\mathrm{w}}-{\mathbf{f}}_{\mathrm{w}0},f_{\mathrm{t}})\mathrm{or}{\mathbf{F}}_{\mathrm{w}}-{\mathbf{f}}_{\mathrm{w}0}$$

(13)

for macrostates. They form the mechanical part of the thermodynamic force [12,19]. The thermal part of the thermodynamic force is the thermal force $T-T_0$ and does not represent a mechanical force; here T is the temperature of $\Sigma$ and $T_0$ that of $\tilde{\Sigma }$. We thus see that the thermodynamic force is nothing but the imbalance defined above for any $\mathfrak{M}$ at the macroscopic level. A microstate ${\mathfrak{m}}_k$, for which temperature has no meaning, exerts only the mechanical force ${\mathbf{F}}_{\mathrm{w}k}$ in Equation (7b) and experiences the microforce imbalance in Equation (12). They are ubiquitous even in classical mechanics. A very illustrative example of force imbalance is of a bungee jumper tied to a coiled bungee cord. As the jumper falls under gravity, the only force acting on the jumper is due to gravity as long as the cord remains coiled. Then, the spring force comes into play as the chord uncoils and gradually increases but the two forces are different except at mechanical equilibrium. The concept of ubiquitous microforce imbalance in $\mu$NEQT was first introduced in [56,57]. For irreversibility, it is necessary but not sufficient to have a nonzero (micro)force imbalance. The imbalance is explicitly taken into consideration in the $\mu$NEQT but not in the $\mathring{\mu }$NEQT, which makes $\mu$NEQT superior to investigate irreversible processes.

2.3. First Law Formulations & Irreversibility Principle

As $E_k$ is determined by $\mathcal{H}$ that itself identifies $\Sigma$-or-${\Sigma }^{\mathrm{D}}$, it represents a system-intrinsic (SI) microquantity. Therefore, the state macroquantity E at any point of $\mathcal{P}$ and the process macroquantity $dE$ during $\delta \mathcal{P}$ are also SI-macroquantities, and can be used to identify a SI-version of the first law of thermodynamics in terms of uniquely identified SI-macroheat $dQ$ and SI-macrowork $dW$ as was first proposed in [48]:

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

(14a)

$$\begin{array}{cc}\hfill dQ& ={\sum }_k{E}_{k}d{p}_k,dW=-{\sum }_k{p}_{k}d{E}_k,\hfill \end{array}$$

(14b)

where $dQ$ is the SI-macroheat added to and $dW$ is the SI-macrowork done by the system. In general, our interest is in a system $\Sigma$ in a medium $\tilde{\Sigma }$ forming an isolated system ${\Sigma }_0$ as shown in Figure 1. As is the convention, the medium is extremely large compared to the size of $\Sigma$ so the presence of the latter on the properties of $\tilde{\Sigma }$ can be neglected so the latter can always be taken to be in EQ, which we will always assume to be true. This means that medium-intrinsic (MI) macroquantities are very useful in formulating thermodynamics as it commonly done and was first clarified in [48,58,59]; see also [35,36].

By accumulating various macroquantities, we obtain a formulation of the first law over $\mathcal{P}$. We thus obtain

$$\Delta E=\Delta Q-\Delta W,$$

(15a)

in terms of accumulated macroquantities over entire $\mathcal{P}$. We are able to use this formulation of the first law to obtain self-consistent NEQ thermodynamics (NEQT), which we have called MNEQT (M for macroscopic and using SI-macroquantities). Traditionally, the use of MI-macroquantities is customary and useful for the simple reason that $\tilde{\Sigma }$ is not appreciably affected by $\Sigma$ [35,36,48,58,59]. The exchange macroquantities require the subscript “e” as a reminder of their exchange nature, see Figure 1. The first law over $\mathcal{P}$ takes the form

$$\Delta E={\Delta }_{\mathrm{e}}Q-{\Delta }_{\mathrm{e}}W$$

(15b)

in terms of exchange macroheat ${\Delta }_{\mathrm{e}}Q$ and macrowork ${\Delta }_{\mathrm{e}}W$ over $\mathcal{P}$, see Remark 1, given by the accumulation of

$$d_{\mathrm{e}}Q={\sum }_k{E}_k{d}_{\mathrm{e}}{p}_k,d_{\mathrm{e}}W=-{\sum }_k{p}_k{d}_{\mathrm{e}}{E}_k,$$

(16)

respectively, with $d_{\mathrm{e}}{p}_k$ and $d_{\mathrm{e}}{E}_k$ referring to changes caused by above exchanges in $p_k$ and $E_k$, respectively. The differential form of the first law over $\delta \mathcal{P}$ becomes

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

(17)

Using this a robust macroscopic NEQ thermodynamics (NEQT) has developed, known as classical thermodynamics, which is both highly applied and highly theoretical. We denote this thermodynamics by $\mathring{\mathrm{M}}$NEQT (M for macroscopic and the circle for using exchange quantities).

Let us compare the two versions of the first law in Equations (14a) and (17). By subtraction, we find that

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

Using Equation (1) on both sides for $dQ$ and $dW$ immediately leads to the irreversibility principle (Irr-P) in Equation (3). It intertwines the two independent variations in $\mathfrak{S}$ [34,35,36,48,49]; see Equation (14). The principle does not fix the signature at this stage; however, see Remark 10.

The concept of work is purely mechanical, but that of entropy and heat is intertwined with stochasticity [15,34,36] so they have mutually exclusive, i.e., orthogonal, roots as discussed below. This make Equation (3) very surprising, especially as stochasticity is not an explicit part of applied thermodynamics. As ${\Delta }_{\mathrm{i}}W$ and ${\Delta }_{\mathrm{i}}Q$ are generated internally, one expects T and P to be important. The principle by itself does not restrict the signature of $d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W$.

Remark 7.

The innocuous looking principle Irr-P is consequential in providing a mechanical proof of a generalized second law (GSL) for spontaneous processes as recently shown by in [34], whose approach and motivation is introduced below and briefly reproduced in Section 5.

It is customary to impose the second law (SL) of entropy increase in $\mathring{\mathrm{M}}$NEQT to the MI-intrinsic first law given in the form in Equation (15b) and then prove stability of the thermodynamic system [5,11,13,15,20,31,60]. The MI-intrinsic first law is oblivious to SL as is well known. A different approach is taken by Gujrati [34] in which SL is not enforced but require stability/instability of a stable/unstable mechanical system ${\Sigma }^{\mathrm{D}}$ by imposing the mechanical equilibrium principle of energy (Mec-EQ-P) to each microstate for reasons to be explained below in Section 5.3. Using this mechanical principle for ${\mathfrak{m}}_k$, it is possible to prove a generalized second law (GSL), which is the first ever successful attempt to justify SL based purely on the use of analytical mechanics and to alleviate the concern expressed by Keizer [61] who notes: “… the second law has never been justified on the basis of mechanical laws”.

Consider an EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ in ${\mathfrak{S}}_{\mathbf{X}}$, which evolves in time but always remains in ${\mathfrak{S}}_{\mathbf{X}}$; see the path ${\gamma }_{12}$ in Figure 2, which is a reproduction of Figure 2 in [62]. The temporal evolution is due to the implicit time dependence of $\mathbf{w}$. The Hamiltonian only has implicit time dependence, but no explicit time dependence so the system remains uniform and, therefore, in EQ at all times. We can always reverse the process $\mathcal{P}$ by reversing the way $\mathbf{w}$ changes along $\mathcal{P}$. We denote the reverse form of $\mathbf{w}$ by ${\mathbf{w}}_{\mathrm{r}}$ and the reversed process by ${\mathcal{P}}_{\mathrm{r}}$. We say that this reversal is reversing the flow of time so that the temporal evolution of the system can be reversed so that ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ also reverses its path in ${\mathfrak{S}}_{\mathbf{X}}$. As ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ in ${\mathfrak{S}}_{\mathbf{X}}$ is uniquely specified by $\mathbf{X}$, whether it lies along the forward path $\mathcal{P}$ or along its reverse path ${\mathcal{P}}_{\mathrm{r}}$ on which we denote it by ${\mathbf{X}}_{\mathrm{r}}$, we have

$$S_{\mathrm{eq}}\left(\mathbf{X}\right)=S_{\mathrm{eq}}\left({\mathbf{X}}_{\mathrm{r}}\right),$$

which means that we must have

$$p_k^{\mathrm{eq}}\left(\mathbf{X}\right)=p_k^{\mathrm{eq}}\left({\mathbf{X}}_{\mathrm{r}}\right).$$

The entropy change between two points $\mathbf{X}$ and ${\mathbf{X}}^{\prime }$ satisfy

$$\Delta S_{\mathrm{eq}}(\mathbf{X},{\mathbf{X}}^{\prime })=-\Delta S_{\mathrm{eq}}({\mathbf{X}}_{\mathrm{r}}^{\prime },{\mathbf{X}}_{\mathrm{r}})$$

as expected. We say that such a process in ${\mathfrak{S}}_{\mathbf{X}}$ is reversible, and we denote it by ${\mathcal{P}}_{\mathrm{rev}}$. Thus, we conclude that

Claim 2.

A reversible process ${\mathcal{P}}_{\mathit{rev}}$ along ${\gamma }_{12}$ entirely lies within ${\mathfrak{S}}_{\mathbf{X}}$, and can be reversed so that each microstate, its microenergy and probability are unaffected by reversal. We identify this property as the time-reversal invariance of ${\mathcal{P}}_{\mathit{rev}}$, which only holds in ${\mathfrak{S}}_{\mathbf{X}}$. A process that is not reversible is called irreversible and denoted by ${\mathcal{P}}_{\mathit{irr}}$; it must lie outside of ${\mathfrak{S}}_{\mathbf{X}}$. For the latter, there are nonvanishing thermodynamic forces; see Section 2.2.

Claim 3.

Mathematically, reversibility can be also defined for processes in the extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ by only requiring that there be no thermodynamic force; see Section 2.2.

This time-reversal invariance is a consequence of an explicit time-independence in $\mathcal{H}\left(\mathbf{w}\right)$ of ${\Sigma }^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$. As is well-known, the state of the system ${\Sigma }^{\mathrm{D}}$ is uniform in this case [15] as was also pointed out above. Thus, all macrostates in ${\mathfrak{S}}_{\mathbf{X}}$ represent EQ macrostates. The change $d{E}_k$ results from the variation of $\mathbf{w}$ as seen From Equation (5). The change $d\mathbf{w}$ must occur by manipulating $\mathbf{w}$ by $\tilde{\Sigma }$. This manipulation must be done very very slowly (quasistatically) for the uniformity of the macrostate to remain disturbed.

If, on the other hand, the manipulation of $\mathbf{w}$ by $\tilde{\Sigma }$ is not very slow, the macrostate may not remain uniform, and we obtain a NEQ macrostate $\mathfrak{M}$, which must lie outside of ${\mathfrak{S}}_{\mathbf{X}}$. In this case, the microstates of ${\Sigma }^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ must be described by a Hamiltonian $\mathcal{H}(\mathbf{w},t)$ with explicit time dependence. Different rates of manipulation result in different time dependence. This explicit time dependence results in an explicit time dependence in the macrostate $\mathfrak{M}(\mathbf{w},t)$. However, this time dependence in $\mathfrak{M}(\mathbf{w},t)$ is special in that is due to internal nonuniformity in $\Sigma$ that cannot be controlled from outside the system such as by $\tilde{\Sigma }$. Therefore, reversing $\mathcal{P}$ to ${\mathcal{P}}_{\mathrm{r}}$ by reversing $\mathbf{w}$ to ${\mathbf{w}}_{\mathrm{r}}$ has no effect on the explicit t dependence of how internal nonuniformity is changing. Therefore, the explicit presence of t destroys time-reversal invariance. Such a process ${\mathcal{P}}_{\mathrm{irr}}$ is irreversible as said above. This answers Problem 1 in Section 1.1. A hallmark of an irreversible spontaneous process is the irreversible entropy generation ${\Delta }_{\mathrm{i}}S>0$ over the process [19], whereas it vanishes for a reversible process that lies entirely in ${\mathfrak{S}}_{\mathbf{X}}$.

Remark 8.

As has been shown recently [34] for isolated systems, the inequality ${\Delta }_{\mathit{i}}S>0$ is only true for spontaneous irreversibility at positive temperatures T; for negative T, we must have ${\Delta }_{\mathit{i}}S<0$ for spontaneous irreversibility, a very surprising result; see Section 5.7.

Remark 9.

For interacting systems, the situation changes and the relationship between T and ${\Delta }_{\mathit{i}}S$ is not so direct, as we will see in Section 5.7.

2.4. Thermodynamic Dissipation

Thermodynamic Dissipation is used as a tool to improve energy transformations into useful work [23], and to unravel the mystery of the universal second law of increase of entropy [9,12,15,19] through its axiomatic formulations [7,13,63]. According to the second law, the (irreversible) entropy generation ${\Delta }_{\mathrm{i}}S\doteq \Delta S-{\Delta }_{\mathrm{e}}S\ge 0$ within a system $\Sigma$ puts a restriction on the amount of useful work ${\Delta }_{\mathrm{e}}W$ that can be achieved. As ${\Delta }_{\mathrm{i}}S$ does not appear in the $\mathring{\mathrm{M}}$NEQT, attempt is to derive the useful work by considering dissipated (or lost) work $W_{\mathrm{diss}}$ [23] in Equation (2). The Gouy-Stodola theorem (GST)

$$W_{\mathrm{diss}}=T_0{\Delta }_{\mathrm{i}}S\ge 0$$

(18)

in applied thermodynamics is the theoretical basis for relating the dissipated work $W_{\mathrm{diss}}$ with entropy generation ${\Delta }_{\mathrm{i}}S$, $T_0$ being the constant temperature of the medium $\tilde{\Sigma }$. The theorem only works for a $\mathcal{P}$ between two EQ macrostates in the presence of a $\tilde{\Sigma }$ at constant temperature $T_0$ and pressure $P_0$, and does not involve the instantaneous temperature and pressure $T\left(t\right)$ and $P\left(t\right)$ of $\Sigma$ or the temperatures of other possible heat sources along the process. (We will suppress t from now on unless necessary.) This has made this theorem extremely useful.

In applied thermodynamics textbooks [23], for example, GST is proved using a concurrent series of heat and work (including mass) exchanges, but the conditions under which it holds are usually not clearly expounded. In Equation (18), $T_0$ is for a special heat bath, usually the atmosphere. The absence of T and P along $\mathcal{P}$ means that GST has no explicit dependence on the “thermodynamic forces” $P-P_0$ and $T-T_0$, even though they determine how far $\mathcal{P}$ is from its reversible counterpart ${\mathcal{P}}_{\mathrm{rev}}$.

It is very common in physics to consider $\mathcal{P}$ between two NEQ macrostates or of a spontaneous evolution in an isolated system as it relaxes towards its EQ macrostate, but where GST does not apply as there is no analog of ${\mathcal{P}}_{\mathrm{rev}}$ so $W_{\mathrm{diss}}$ cannot be identified; see Figure 2 for the evolution along red arrow of ${\mathfrak{M}}_0$. For example, for the spontaneous relaxation of an isolated system, ${\Delta }_{\mathrm{i}}S\ge 0$ but there is no analog of $T_0$ so what will be $W_{\mathrm{diss}}$? What will replace GST in that case? Thus, we need to obtain a more general definition of the dissipated work that goes beyond the applicability of GST.

Claim 4.

In general, a reversible process ${\mathcal{P}}_{\mathit{rev}}$ may not exist for a general NEQ process $\mathcal{P}$. Thus, in general, it is not possible to use Equation (2) as ${\left({\Delta }_{\mathit{e}}W\right)}_{\mathit{rev}}$ cannot be determined in this case. Our approach bypasses this construct and is valid for all processes including those that result in stationary states, which would be considered in a separate publication.

This claim answers Problem 8 partially. The rest is answered in Section 7; see Conclusion 6.

We find [34,35,36,48,49] that the irreversible macrowork ${\Delta }_{\mathrm{i}}W$ turns out to be a true measure of irreversibility of the process or machine controlling inert and living matter, and puts a thermodynamic limit on the efficiency of macroscopic energy conversion for any general process which may not even have an analog of ${\mathcal{P}}_{\mathrm{rev}}$ so $W_{\mathrm{diss}}$ cannot be defined as above. This is clear from the fact that reversible macrowork is nothing but the SI-macrowork $\Delta W\equiv {\Delta }_{\mathrm{e}}W+{\Delta }_{\mathrm{i}}W$ performed by $\Sigma$ during $\mathcal{P}$ in which $\Sigma$ is always in EQ with a series of mediums at each instant so at each instant, the macrostate $\mathfrak{M}\left(t\right)$ is uniform and there is no irreversibility. This means that ${\Delta }_{\mathrm{i}}W\equiv 0$ and $\Delta W\equiv {\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}$ under the EQ situation. Being a SI-macroquantity, $\Delta W$ has the same value regardless of the nature of the medium; the latter only determines the MI-macrowork ${\Delta }_{\mathrm{e}}W$. We thus

Claim 5.

$W_{\mathit{diss}}$ is nothing but ${\Delta }_{\mathit{i}}W$ as $W_{\mathit{diss}}=\Delta W-{\Delta }_{\mathit{e}}W={\Delta }_{\mathit{i}}W$.

In other words, ${\Delta }_{\mathrm{i}}W$ provide the generalization of $W_{\mathrm{diss}}$ that is restricted to only ${\mathcal{P}}_{\mathrm{rev}}$ to any possible process $\mathcal{P}$ between any two macrostates; see Figure 2. We prove this claim in Section 5.6 by using Proposition 1 given below, which is founded on experimental observations.

The physicists [6,15] are mostly interested in a microscopic understanding of the source of entropy generation so stochasticity becomes the central ingredient, which is the basis of the BCGM proposal. The focus becomes the total entropy change of the isolated system ${\Sigma }_0\doteq \Sigma \cup \tilde{\Sigma }$, whose statistical mechanics has not attracted much interests among applied thermodynamicists; however, see [36]. Similarly, it is rare to find any substantial coverage of applied thermodynamics in most graduate textbooks in physics, which is unfortunate as both have the same goal to understand dissipation in a process.

The Gouy-Stodola theorem has seen renewed interest ([64,65,66,67], for example) but we still have no understanding of the above perplexing features. One may rationalize that the usual derivation uses Equation (17), which does not involve T, but this cannot be convincing as the law also does not include ${\Delta }_{\mathrm{i}}S$.

Dissipation is always observed to be positive and essentially causes irreversibility and irreversible entropy in a stable (s) macrostate ${\mathfrak{M}}^{\mathrm{s}}$. The basic idea that is commonly accepted is that dissipation brings about irreversible processes in ${\mathfrak{M}}^{\mathrm{s}}$, which asymptotically converges to the stable equilibrium (SEQ) macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ [13]. We express this experimental fact by saying that the spontaneous evolution of ${\mathfrak{M}}^{\mathrm{s}}$ is controlled by the sink ${\mathfrak{M}}_{\mathrm{seq}}$. This experimental observation has been generalized earlier by Gujrati in [36] to also include unstable macrostate ${\mathfrak{M}}^{\mathrm{u}}$ and its spontaneous evolution being controlled by its (unstable) equilibrium macrostate ${\mathfrak{M}}_{\mathrm{ueq}}$, the source. Consequently, it led to the Fundamental Proposition, which also includes the spontaneous evolution of their microstates ${\mathfrak{m}}_{k\mathrm{eq}}$. We reproduce it here as Proposition 1.

Proposition 1.

The spontaneous evolution of $\mathfrak{M}$ (or ${\mathfrak{m}}_k$) is controlled by ${\mathfrak{M}}_{\mathit{eq}}$ (or ${\mathfrak{m}}_{\mathit{keq}}$), where ${\mathfrak{M}}_{\mathit{eq}}$ stands for either ${\mathfrak{M}}_{\mathit{seq}}$ or ${\mathfrak{M}}_{\mathit{ueq}}$ (or ${\mathfrak{m}}_{\mathit{keq}}$ stands for either ${\mathfrak{m}}_{\mathit{kseq}}$ or ${\mathfrak{m}}_{\mathit{kueq}})$.

2.5. Kullback-Leibler Distance and Irreversibility

Recently, there has been a surge in the activity to microscopically investigate $W_{\mathrm{diss}}$ due to its important role in fluctuation theorems of Jarzynski [68] and Crooks [69]; however, see Cohen and Mauzurall [70,71] for their heuristic but very penetrating critique. It has been argued [72] that

$$W_{\mathrm{diss}}=T_0<ln\left(p_{{\gamma }_{\mathrm{f}}}/p_{{\gamma }_{\mathrm{b}}}\right){>}^{\mathrm{cons}}\left\{\begin{array}{c}\hfill >0 \mathrm{for} \mathrm{positive} T_0\\ \hfill <0 \mathrm{for} \mathrm{negative} T_0\end{array}\right.,$$

(19)

where ${\gamma }_{\mathrm{f}}$ and ${\gamma }_{\mathrm{b}}$ are forward and backward trajectories forming some NEQ forward process $\mathcal{P}$ and its reverse process ${\mathcal{P}}_{\mathrm{r}}$, respectively, and $p_{{\gamma }_{\mathrm{f}}}$ and $p_{{\gamma }_{\mathrm{b}}}$ denote their probabilities. The “$constrained$” average ${\langle \rangle }^{\mathrm{cons}}$ is an average using $p_{{\gamma }_{\mathrm{f}}}$ only, see Equation (20) below that introduces the Kullback-Leibler distance or divergence $D_{\mathrm{KL}}(p_{{\gamma }_{\mathrm{f}}}\Vert p_{{\gamma }_{\mathrm{b}}})$, and does not represent an ensemble average; hence, it is not a thermodynamic quantity, which requires a joint probability $p_{{\gamma }_{\mathrm{f}},{\gamma }_{\mathrm{b}}}$ of ${\gamma }_{\mathrm{f}}$ and ${\gamma }_{\mathrm{b}}$ for the ensemble average; see [73] for more details. Therefore, the above identification of $W_{\mathrm{diss}}$, a thermodynamic quantity, is highly suspect and cannot be justified as shown in Section 6. Despite this deficiency, it is a wide-spread practice in theoretical physics to use $D_{\mathrm{KL}}$ as a means to identify irreversible entropy production; see for example a recent publication [74] where it is used. We must also remark that only positive temperatures have been considered in [72], but we have extended it to negative temperatures as we wish to consider dissipation for temperatures of both signs.

The constrained average in Equation (19) represents the Kullback-Leibler distance or divergence $D_{\mathrm{KL}}(p_{{\gamma }_{\mathrm{f}}}\Vert p_{{\gamma }_{\mathrm{b}}})$, which is explicitly defined as

$$D_{\mathrm{KL}}(p\Vert q)\doteq {\sum }_k{p}_{k}ln(p_k/q_k)\equiv {〈ln(p/q)〉}^{\mathrm{cons}}\ge 0$$

(20)

for two sets of probabilities $\left\{p_k\right\}$ and $\left\{q_k\right\}$. The inequality follows from Gibbs’ inequality between the entropy associated with $\left\{p_k\right\}$ and the cross entropy associated with $\left\{p_k\right\}$ and $\left\{q_k\right\}$:

$$-{\sum }_k{p}_{k}ln{p}_k\le -{\sum }_k{p}_{k}ln{q}_k;$$

while the left side is an ensemble average, the right side is not for $p_k\ne q_k$. For q denoting EQ probabilities $\left\{p_k^{\mathrm{eq}}\right\}$, we have

$$D_{\mathrm{KL}}(p\Vert p^{\mathrm{eq}})\ge 0.$$

(21)

It should be noted that the constrained average in $D_{\mathrm{KL}}(p\Vert q)$ in Equation (20) uses the EQ temperature $T_0$ of the medium, and not the NEQ temperature T of the system. This is disconcerting in the same way Cohen and Mauzerall [70,71] have expressed their concern about Jarzynski equality [68], whose shortcomings have been extensively discussed by us also [36,56,57]. Similar issues also arise with $D_{\mathrm{KL}}$ and will be discussed in Section 6.

2.6. Detailed Balance

The principle of detailed balance is formulated by considering possible transitions between different microstates. Let ${\stackrel{·}{T}}_{kl}$ denote the conditional transition probability per unit time for ${\mathfrak{m}}_k$ to transition to ${\mathfrak{m}}_l$. We introduce

$$d{T}_{kl}={\stackrel{·}{T}}_{kl}dt,{\sum }_{l}d{T}_{kl}=1$$

(22)

so that $p_{k}d{T}_{kl}$ is the probability for ${\mathfrak{m}}_k$ to transition to ${\mathfrak{m}}_l$ during time interval $dt$. With this in hand, it is easy to see that the net change $d{p}_k\doteq p_k(t+dt)-p_k\left(t\right)$ in $p_k$ is

$$d{p}_k={\sum }_l{p}_{l}d{T}_{lk}-{\sum }_l{p}_{k}d{T}_{kl}.$$

(23)

The first sum is the probability change $d{p}_k^{\mathrm{in}}$ into k and the second sum is the probability change $d{p}_k^{\mathrm{out}}$ out of k. The stationary condition occurs when $d{p}_k=0,\forall k$. Thus,

$$d{p}_k^{\mathrm{in}}\doteq {\sum }_l{p}_{l}d{T}_{lk}=d{p}_k^{\mathrm{out}}\doteq {\sum }_l{p}_{k}d{T}_{kl}.$$

(24)

This is called the condition of balance; also compare with Equation (76). The detailed balance condition for a pair of microstates refers to

$$p_{l}d{T}_{lk}=p_{k}d{T}_{kl},\forall (k,l),$$

(25)

which is a much stronger condition than the balance condition above. This is the principle of detailed balance (PDB) and is distinct [75] from the condition of microscopic reversibility of Onsager’s reciprocal relations [76,77]

$$d{T}_{lk}=d{T}_{kl},\forall (k,l),$$

(26)

even though many authors make no distinction between them.

PDB is known to only hold for reversible changes as shown by Klein [78,79]; see also Ref. [12] and the footnote 7 in Cohen and Mauzerall [70]. A common usage of the PDB is for heat exchange of a system $\Sigma$ with a medium $\tilde{\Sigma }$. It is well known that an isothermal heat exchange cannot result in any irreversibility [12,15,19,25]. In this case, the only source of irreversibility must be either external macrowork exchange or relaxation of internal variables or both [12,15,19,21,25] during $\mathcal{P}$. This irreversibility results from internal processes caused by thermodynamic forces [12,15,19,21,25,56,57] such as $P-P_0$; see Section 2.2. In their absence, no work-generated irreversibility can occur so the use of the PDB under this scenario implies a reversible process only.

Recently, PDB has been actively used in many NEQ fluctuation theorems (FTs) describing isothermal ($T=T_0$) work processes ${\mathcal{P}}_{\mathrm{iso}}$ [68,69,80,81,82] so the irreversibility must be work-generated. The FTs involve microscopic trajectories along which changes in probabilities are assumed to follow the PDB. They have become central in the modern development of a microscopic theory of nonequilibrium thermodynamics (NEQT) expressed in terms of ${\Delta }_{\mathrm{e}}W$ and ${\Delta }_{\mathrm{e}}Q$ with the medium $\tilde{\Sigma }$. This has made their usage very convenient in the FTs so the latter have attracted a lot of interest over the past three decades or so. This particular thermodynamics has been denoted by $\mathring{\mu }$NEQT, with $\mu$ referring to its microscopic nature and the dot for extending the concept of macroscopic exchange quantities to the microscopic level. It is a direct extension of classical thermodynamics to the microscopic level as the latter also uses only exchange quantities [12,15,19,21,25]. We contrast this with another version, denoted by $\mu$NEQT, which is based on SI quantities [55] like SI macrowork. Here, ${\Delta }_{\mathrm{i}}W$ and ${\Delta }_{\mathrm{i}}Q$ denote irreversible contributions, which vanish in a reversible process. They are incorporated explicitly in the $\mu$NEQT [55,56,57].

We now briefly review MNEQT and $\mu$NEQT in the next two sections.

3. The MNEQT: A Brief Review

Before reviewing the $\mu$NEQT, we briefly review MNEQT that has been recently introduced as a new formulation of NEQT [33,48,55,58,59,62,83,84] in terms of the entropy S of $\Sigma$ that may be far from EQ. To uniquely specify NEQ macrostate $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{Z}}$, we use n independent internal variables denoted by $\mathsf{\xi }=({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)$ [12,18,19,21] in the extended state space ${\mathfrak{S}}_{\mathbf{Z}}$, in which S is a state function of $\mathbf{Z}\doteq (\mathbf{X},\mathsf{\xi })$, which we assume to be continuous; we will use ${\mathfrak{S}}_{\mathbf{W}}$ for the state subspace spanned by work variable $\mathbf{W}\doteq (\mathbf{w},\mathsf{\xi })$. We say that $\Sigma$ is in a state of internal equilibrium (IEQ) in ${\mathfrak{S}}_{\mathbf{Z}}$, which is similar to some extent of $\Sigma$ being in EQ in ${\mathfrak{S}}_{\mathbf{X}}$, except that there is irreversible entropy generation in an IEQ system. As the system gets closer and closer to EQ, n continuously decreases [55] to zero. Thus eventually at EQ, we need the conventional state space ${\mathfrak{S}}_{\mathbf{X}}$ alone; see Figure 2. From continuity, we obtain

$$dS=(\partial S/\partial E)dE+(\partial S/\partial \mathbf{W})·d\mathbf{W},$$

which is inverted to construct $E(S,\mathbf{W})$, from which we obtain the SI-formulation of the first law in Equation (14a), where

$$\begin{array}{}\hfill \mathrm{(27a)}& dW=T(\partial S/\partial \mathbf{W})·dW=PdV+\cdots +\mathbf{A}·d\mathsf{\xi },\hfill \hfill \mathrm{(27b)}& dQ=TdS,\hfill \end{array}$$

where ⋯ refer to other work parameters in $\mathbf{w}$ besides the volume V, and where we have introduced

$$T=\partial E/\partial S,P=-\partial E/\partial V,\cdots ,\mathbf{A}\doteq -\partial E/\partial \mathsf{\xi },$$

(28)

that identify the NEQ thermodynamic temperature, pressure, ⋯, and affinity $\mathbf{A}$ of the system in a state space $\mathfrak{S}(S,\mathbf{W})$ in which E is a state function. It follows from Equation (10a) that ${\mathbf{F}}_{\mathrm{w}}\doteq P,\cdots ,\mathbf{A}$, and T denotes the thermodynamic temperature of a NEQ macrostate $\mathfrak{M}$. Recall that $\tilde{\Sigma }$ is always taken to be in EQ so that its fields are $T_0,P_0,\cdots ,{\mathbf{A}}_0=0$. We have in general,

$$d_{\mathrm{e}}W=P_{0}dV+\cdots ,d_{\mathrm{e}}Q=T_0{d}_{\mathrm{e}}S,$$

(29)

${\mathbf{A}}_0=0$ playing no role for exchange macroquantities, which justifies identifying $\mathsf{\xi }$ as internal variable. We finally have

$$d_{\mathrm{i}}W\doteq (P-P_0)dV+\cdots +\mathbf{A}·d\mathsf{\xi },d_{\mathrm{i}}Q\doteq TdS-T_0{d}_{\mathrm{e}}S$$

(30)

is the internal or irreversible macrowork [48,58,59,62], see Equation (27a); here, $\mathbf{A}$ denotes the affinity of $\mathsf{\xi }$ [19].

By imposing SL in $\mu$NEQT, we have shown that

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

which is consistent with GST in Equation (18) for nonnegativity. However, in our approach as introduced in Section 2.1, we will not impose SL but not only prove it directly but also extend it to negative temperatures T. Indeed, we generalize SL that has been termed Generalized Second Law (GSL) that will be introduced shortly. Therefore,

Remark 10.

We will not require $d_{i}Q=d_{i}W\ge 0$ in μNEQT. In other words, we do not impose any sign restriction on $d_{i}Q=d_{i}W$ anymore as is clear from Equation (3).

In general,

$$d_{\mathrm{i}}S\equiv d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}+d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$$

(31)

has two independent contributions: $d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ due to macroheat exchange with ${\tilde{\Sigma }}_{\mathrm{h}}$ and $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ due to irreversible macrowork $(P-P_0)dV+\cdots$ with ${\tilde{\Sigma }}_{\mathrm{w}}$ and internal macrowork $\mathbf{A}·d\mathsf{\xi }$ within $\Sigma$. The irreversible entropy change $d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ due to macroheat exchange $d_{\mathrm{e}}Q=dQ-d_{\mathrm{i}}Q$ from $T_0$ to T is given by [19]

$$d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}\doteq ({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_0}})d_{\mathrm{e}}Q={\displaystyle \frac{T_0-T}{T}}{d}_{\mathrm{e}}S,$$

(32)

which is determined by the thermal force introduced in Section 2.2. The contribution $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ is determined by

$$d_{\mathrm{i}}W\doteq T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$$

(33)

Remark 11.

Thermal force does not affect the irreversible macrowork $d_{i}W$. It is only determined by the work parameter set $\mathbf{W}$. Its signature for spontaneous processes will be determined in Section 5.4.

This remark is also consistent with Claim 8.

3.1. A Simple Example

We now show by a simple example how internal variables appear in a NEQ state by considering an isolated and composite nonuniform macroscopic system $\Sigma$ having two nonoverlapping but mutually interacting subsystems ${\Sigma }_1$ and ${\Sigma }_2$. Each subsystem is also macroscopically large and that have identical volumes and numbers of particles. The nonuniformity comes from their having different energies $E_1\left(t\right)$ and $E_2\left(t\right)$ so they have different temperatures $T_1\left(t\right)$ and $T_2\left(t\right)$ at any time $t\le {\tau }_{\mathrm{eq}}$ before EQ is reached at $t={\tau }_{\mathrm{eq}}$. A diathermal wall separates them. We assume that $\Sigma$ is not far from EQ so we can treat each subsystem in EQ at each t; their entropies are $S_{1\mathrm{eq}}(E_1\left(t\right),V/2,N/2)$ and $S_{2\mathrm{eq}}(E_2\left(t\right),V/2,N/2)$. However, $\Sigma$ is in a NEQ macrostate at each $t<{\tau }_{\mathrm{eq}}$ for which its entropy S is a function of $E_1\left(t\right),E_2\left(t\right),V$, and N at each t. We form two independent combinations from $E_1$ and $E_2$ as follows: $E=E_1+E_2=const$ and $\xi \left(t\right)=E_1\left(t\right)-E_2\left(t\right)$. Using them, we express the entropy as $S(E,V,N,\xi )$. We identify $\xi$ as an internal variable because it cannot be controlled by the experimenter from outside the system $\Sigma$, which continues to relax towards zero as $\Sigma$ approaches EQ. For given E and $\xi$, $S(E,V,N,\xi )$ has the maximum possible value since both $S_{1\mathrm{eq}}$ and $S_{2\mathrm{eq}}$ have their maximum values at each instant. This is the concept of internal equilibrium (IEQ) in MNEQT, where $S(E,V,N,\xi )$ is a state function in the extended space ${\mathfrak{S}}_{\mathbf{Z}}$. The latter continues to increase as $\xi$ relaxes internally so that S reaches its EQ value $S_{\mathrm{eq}}(E,V,N,0)$.

While equilibrating, $T_1$ and $T_2$ approach towards each other. As $\Sigma$ is in IEQ at each t, we have from Equation (27) the inverse thermodynamic temperature $1/T=\partial S/\partial E$ and the reduced affinity $A/T=\partial S/\partial \xi$ of $\xi$. A simple calculation yields

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

at each instant t; here we have suppressed the time dependence in $T_1$ and $T_2$ for simplicity. As EQ is reached, $T_1\to T_{\mathrm{eq}}$ and $T_2\to T_{\mathrm{eq}}$, the EQ temperature of $\Sigma$, and $A\to A_0=0$. In general, the activity $\mathbf{A}$ controls the behavior of $\mathsf{\xi }$ in a NEQ state and vanishes when EQ is reached.

With the above identification, $dS=dE/T+Ad\xi /T$ or $dE=TdS-Ad\xi$ so the SI-macroheat is $dQ=TdS$, and the SI-macrowork is $dW=Ad\xi$; see Equation (27). In this simple example, there is no macroheat and macrowork exchanges so $dS=d_{\mathrm{i}}S$, and $dW=d_{\mathrm{i}}W$. As $dE=d_{\mathrm{i}}E=0$ here, $dS=d_{\mathrm{i}}S=Ad\xi /T$ is the entropy generation, while $dW=d_{\mathrm{i}}W=$$Ad\xi$ so that

$$d_{\mathrm{i}}W=T{d}_{\mathrm{i}}S=T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)},$$

(34)

which almost seems like GST except that it includes the temperature of the system, see Equation (33), and is applicable to an isolated system so that $d_{\mathrm{e}}Q=0$; hence $d_{\mathrm{i}}S=d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$. The simple example already gives a hint to possible issues with GST. The example also shows how nonzero $d_{\mathrm{i}}W$ emerges from $\xi$ even if $d_{\mathrm{e}}W=0$.

The simple example is easily extended to a $\Sigma$ far from EQ by introducing more $\xi$’s. It is also easily extended to a closed system that undergoes reversible heat exchanges $d_{\mathrm{e}}{Q}_1$ and $d_{\mathrm{e}}{Q}_2$ with ${\Sigma }_1$ and ${\Sigma }_2$ at $T_1$ and $T_2$, respectively. This extension is appropriate for the application a system with two separate heat sources such as a Carnot engine. Using $dE=d_{\mathrm{e}}{Q}_1+d_{\mathrm{e}}{Q}_2$, we have $TdS=d_{\mathrm{e}}{Q}_1+d_{\mathrm{e}}{Q}_2+Ad\xi =T_1{d}_{\mathrm{e}}{S}_1+T_2{d}_{\mathrm{e}}{S}_2+Ad\xi$. Using $dS=d_{\mathrm{e}}S+d_{\mathrm{i}}S$ and $d_{\mathrm{e}}S=d_{\mathrm{e}}{S}_1+d_{\mathrm{e}}{S}_2$, we finally obtain

$$d_{\mathrm{i}}W=T{d}_{\mathrm{i}}S+(T-T_1)d_{\mathrm{e}}{S}_1+(T-T_2)d_{\mathrm{e}}{S}_2,$$

(35)

with $d_{\mathrm{i}}W=Ad\xi$ as above. To provide further support for $T=\partial E/\partial S$ as the thermodynamic temperature, we modify the example by letting the heat exchanges $d_{\mathrm{e}}{Q}_1=T_1{d}_{\mathrm{e}}{S}_1$ and $d_{\mathrm{e}}{Q}_2=T_2{d}_{\mathrm{e}}{S}_2$ at $T_1$ and $T_2$, respectively, occur irreversibly with a $\Sigma$, which is at uniform T. In this case, there is no $Ad\xi$ term to consider. As $d_{\mathrm{e}}W=0$ in this case, we have $dE=TdS-d_{\mathrm{i}}W$ from Equation (14a). It is now easy to see by simple algebra that we obtain the identity in Equation (35). Thus, T can always be assigned to a $\Sigma$, even if it is nonuniform, in the MNEQT. A similar argument by allowing the volumes to change instead of the energy, we can demonstrate that $P=-\partial E/\partial V$ is the NEQ pressure of $\Sigma$; cf. Equation (28).

3.2. General Extension

The above result is easily generalized to having $m+1$ heat sources ${\tilde{\Sigma }}_{\mathrm{h}}^{\left(i\right)},i=0,1,\cdots ,m$ at temperatures $T_i$ (we do not use the additional suffix 0 in $T_i$ to keep similarity with the use of $T_1$ and $T_2$ above), and $l+1$ work sources ${\tilde{\Sigma }}_{\mathrm{w}}^{\left(j\right)},j=0,1,\cdots ,l$ at pressures $P_j$. Then, $d_{\mathrm{e}}Q=T_0{d}_{\mathrm{e}}S$ in Equation (29) should be replaced by

$$d_{\mathrm{e}}Q={\sum }_i{T}_i{d}_{\mathrm{e}}{S}_i,$$

(36a)

where $d_{\mathrm{e}}{S}^{\left(i\right)}$ is the entropy transfer from ${\tilde{\Sigma }}_{\mathrm{h}}^{\left(i\right)}$ at $T_i$, but is received by $\Sigma$ at T. Introducing

$$d_{\mathrm{e}}S={\sum }_i{d}_{\mathrm{e}}{S}_i=dS-d_{\mathrm{i}}S,$$

(36b)

and

$$d_{\mathrm{i}}W={\sum }_j(P_j-P)dV+\mathbf{A}·d\mathsf{\xi },$$

(36c)

we obtain

$$d_{\mathrm{i}}Q=T{d}_{\mathrm{i}}S+{\sum }_i(T-T_i)d_{\mathrm{e}}{S}_i=d_{\mathrm{i}}W,$$

(36d)

which generalizes Equation (35) for $m=1,l=0$ to any arbitrary $m,l$, and we have used Irr-P to set $d_{\mathrm{i}}W\equiv d_{\mathrm{i}}Q$. Using Equation (32), the sum above can be rewritten as

$${\sum }_i(T-T_i)d_{\mathrm{e}}{S}_i\doteq -T{\sum }_i{d}_{\mathrm{i}}{S}_i^{\left(\mathrm{h}\right)}\doteq T{d}_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)},$$

in terms of the net irreversible entropy change due to macroheat exchanges with ${\tilde{\Sigma }}_{\mathrm{h}}^{\left(i\right)}$. Using Equation (31), this allows us to express $d_{\mathrm{i}}W$ in terms of $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ as in Equation (33). It is easy to see that there is no ${\mathcal{P}}_{\mathrm{rev}}$, see Claim 4.

4. NEQ Statistical Mechanics ($\mu$NEQT)

The subspace ${\mathfrak{S}}_{\mathbf{W}}$ spanned by $\mathbf{W}$ determines SI-macrowork, while its orthogonal subspace ${\mathfrak{S}}_S$ spanned by S in $\mathfrak{S}(S,\mathbf{W})$ determines SI-macroheat. This makes them independent variations of E. To clarify the orthogonal distinction between SI-macroheat and macrowork and their components, we consider the statistical definition of E and S in Equation (8). We can now identify [62,83,84]

$$\begin{array}{cc}\hfill dQ& \equiv {\sum }_k{E}_{k}d{p}_k,dW\equiv -{\sum }_k{p}_{k}d{E}_k,\hfill \end{array}$$

(37a)

$$\begin{array}{cc}\hfill dS& \equiv -{\sum }_k({\eta }_k+1)d{p}_k=-{\sum }_k{\eta }_{k}d{p}_k,\hfill \end{array}$$

(37b)

for any arbitrary state. As $dW$ contains fixed $p_k$’s, $dS\equiv 0$ so it is an isentropic quantity [15] in ${\mathfrak{S}}_{\mathbf{W}}$. As $dQ$ contains the changes $d{p}_k$s, it is purely stochastic and occurs as the entropy changes by $dS$ in ${\mathfrak{S}}_S$, so the two quantities must be related. As both quantities are extensive, this relationship must always be linear. This is the physics behind the Clausius equality in Equation (27b) with the intensive field $T\left(t\right)$ identified as the thermodynamic temperature of $\Sigma$. This makes $dQ$ a purely entropic or stochastic quantity in any thermodynamic process. Thus, in general, $dQ$ and $dW$ are due to independent variations in orthogonal subspaces of $\mathfrak{S}(S,\mathbf{W})$ as noted above.

The SI-macrowork as a statistical average of $d{W}_k$ contains in general all kinds of microwork in ${\mathfrak{S}}_{\mathbf{W}}$ including those due to $\mathsf{\xi }$; see Equation (27a). If $d{W}_k$ is only due to volume change $dV$ in ${\mathfrak{S}}_{\mathbf{W}}$ spanned by V alone, then $dW\left(t\right)=P\left(t\right)dV$, where $P\left(t\right)\equiv -{\sum }_k{p}_k{P}_k$ is the average pressure on the walls with a similar complicated dependence through $\left\{p_k\left(t\right)\right\}$; here $P_k\equiv -\partial E_k/\partial V$ is the outward pressure exerted by the kth microstate [85]. Thus, $dW\left(t\right)$ is also linear in $dV\left(t\right)$ as expected. As $T\left(t\right)$ and $\partial E/\partial S$ are the same for a system in internal equilibrium [62], the t-dependence in $T\left(t\right)$ is due to the t-dependence of the state variables, which makes $T\left(t\right)$ a state function. All these comments also apply to $\mathbf{A}\left(t\right)={\sum }_k{p}_k{\mathbf{A}}_k$, where ${\mathbf{A}}_k=-\partial E_k/\partial \mathsf{\xi }$.

Partitioning $d{p}_k=d_{\mathrm{e}}{p}_k+d_{\mathrm{i}}{p}_k$, see Remark 1, we find

$$d_{\mathrm{e}}S\equiv -{\sum }_k{\eta }_k{d}_{\mathrm{e}}{p}_k,d_{\mathrm{i}}S\equiv -{\sum }_k{\eta }_k{d}_{\mathrm{i}}{p}_k;$$

(38)

see Equation (37b). We also have from Equation (37a)

$$\begin{array}{cc}\hfill d_{\mathrm{e}}Q& =-T_0{\sum }_k{\eta }_k{d}_{\mathrm{e}}{p}_k,d_{\mathrm{i}}Q=-{\sum }_k{\eta }_k(Td{p}_k-T_0{d}_{\mathrm{e}}{p}_k),\hfill \end{array}$$

(39a)

$$\begin{array}{cc}\hfill d_{\mathrm{e}}W& =P_{0}dV,d_{\mathrm{i}}W\equiv {\sum }_k{p}_k[(P_k-P_0)dV+\cdots +{\mathbf{A}}_k·d\mathsf{\xi }],\hfill \end{array}$$

(39b)

where we have used the fact that $d_{\mathrm{e}}{E}_k=-d_{\mathrm{e}}{W}_k=-P_{0}dV+\cdots +$, and $d_{\mathrm{i}}{E}_k=-d_{\mathrm{i}}{W}_k=-[(P_k-P_0)dV+\cdots +$${\mathbf{A}}_k·d\mathsf{\xi }]$ to write Equation (39b). We now have statistical definitions of various heats and works in terms of microstates. They show that $d_{\mathrm{e}}W$ and $d_{\mathrm{i}}W$ are isentropic quantities similar to $dW$ so they arise from variations in ${\mathfrak{S}}_{\mathbf{W}}$. They also show that $d_{\mathrm{e}}Q$ and $d_{\mathrm{i}}Q$ are entropic quantities similar to $dQ$, and arise from independent and orthogonal variations in ${\mathfrak{S}}_S$. This finally proves that $d_{\mathrm{i}}Q$ and $d_{\mathrm{i}}W$ have independent roots.

In the MNEQT, it is easy to derive

$$d_{\mathrm{i}}Q=T{d}_{\mathrm{i}}S+(T-T_0)d_{\mathrm{e}}S=T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$$

(40)

by using Equations (27b) and (29), and where we have introduced a new quantity $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$, whose physics become clear in Claim 6. We also observe that both sides are determined by ${\mathfrak{S}}_S$; see Equation (39a). We rewrite Equation (40) by using Irr-P as

$$d_{\mathrm{i}}W=d_{\mathrm{i}}Q=T{d}_{\mathrm{i}}S-T{d}_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}=T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)},$$

(41)

where

$$d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}={\displaystyle \frac{T_0-T}{T}}{d}_{\mathrm{e}}S.$$

(42)

We finally have

$$d_{\mathrm{i}}Q=d_{\mathrm{i}}W=T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}\ne T{d}_{\mathrm{i}}S,$$

(43)

unless $T{d}_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}=0$, which requires no macroheat exchange. The above equation is the same as the very general result in Equation (34).

Claim 6.

$d_i{S}^{\left(w\right)}$ is directly proportional to the irreversible macrowork  $d_{i}W$ due to the mechanical part of the thermodynamic force consisting of the exchange part with ${\tilde{\Sigma }}_w$ and the internal part within $\Sigma$; see Equation (39a).

We recall Claim 2, which does not put any restriction of the signature of any process quantity $d\mathfrak{q}$ that remains unchanged over the entire process $\mathcal{P}$. We closely follow [34] and introduce a class of processes which contains a restriction on the signature over the entire process $\mathcal{P}$.

Definition 2.

By irreversibility of a process $\mathcal{P}$ is meant the fixed signature of a process quantity $d_i\mathfrak{q}$ during any process $\mathcal{P}$. The fixed signature may or may not satisfy SL.

Remark 12.

We have modified the concept of irreversibility in this investigation to allow for the possibility of a violation of SL [34,36]. With this difference, we will establish soon that the generalized version GSL $d_{i}Q\ge 0$ of SL is valid only for spontaneous processes, and not for all irreversible processes.

Claim 7.

A nonspontaneous process is not internally generated and requires an interacting system ${\Sigma }^D$ or $\Sigma$ manipulated by an external medium $\tilde{\Sigma }$. Such a process cannot occur in an isolated ${\Sigma }^D$ or $\Sigma$.

It is convenient to think of $\tilde{\Sigma }$ as a combination of mutually noninteracting working medium ${\tilde{\Sigma }}_{\mathrm{w}}$ and thermal medium ${\tilde{\Sigma }}_{\mathrm{h}}$; however, they interact directly with $\Sigma$ to give rise to exchange macroheat and macrowork, respectively. Internal variables as they equilibrate also perform, what we call internal microwork ${\mathbf{A}}_k·d\mathsf{\xi }$or irreversible macrowork $\mathbf{A}·d\mathsf{\xi }$.

Claim 8.

There is one addition internal variable ${\xi }_{Ek}$ that arises due to nonuniformity of microenergies of different subsystems of $\Sigma$. It does not contribute to $d_i{W}_k$ of ${\mathfrak{m}}_k$ of $\Sigma$ because $E_k$ and ${\xi }_{Ek}$ are independent variables. This has been discussed in [36]. Therefore, corresponding contribution to $d_{i}W$ also vanishes.

It has already been established that Equation (43) remains valid for any number of heat and work mediums as seen Section 3.2. Integrating this along a process $\mathcal{P}$, we obtain the precise formulation of Irr-P, see Equation (3),

$${\Delta }_{\mathrm{i}}W={\int }_{\mathcal{P}}T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}=T^{*}{\int }_{\mathcal{P}}{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}=T^{*}{\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}={\Delta }_{\mathrm{i}}Q,$$

(44)

relating ${\Delta }_{\mathrm{i}}W$ and ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ in ${\mathfrak{S}}_{\mathbf{Z}}$; both quantities are process dependent as the integrals indicate. We have used the mean value theorem of calculus to introduce $T^{*}$, which is some intermediate temperature along $\mathcal{P}$. As the identity will be valid for any n, it justifies Irr-P as a general consequence of MNEQT no matter how far $\Sigma$ is from EQ.

4.1. Microstate Probability $p_k$

The issue of unique microstate probabilities for a system in internal equilibrium has been most recently discussed in [35], but is reviewed here as it is central to our discussion. For the system in IEQ in ${\mathfrak{S}}_{\mathbf{Z}}$, its macrostate is uniquely described. Each microstate in it is also uniquely specified by microenergy $E_k$, micropressure $P_k\doteq -\partial E_k/\partial V,\cdots$, and ${\mathbf{A}}_k\doteq -\partial E_k/\partial \mathsf{\xi }$. The entropy S must be at its maximum at fixed

$$\begin{array}{c}1=\sum _k{p}_k,E\left(t\right)=\sum _k{E}_k{p}_k,\\ P\left(t\right)=\sum _k{V}_k{p}_k,\cdots ,\mathbf{A}\left(t\right)=\sum _k{\mathbf{A}}_k{p}_k,\end{array}$$

(45)

by varying $p_k$ without changing the microstates, i.e., $E_k,P_k,\cdots$, and ${\mathbf{A}}_k$. We use Lagrange multipliers, whose definitions are obvious, to express

$${\eta }_k=-({\lambda }_0+{\lambda }_{\mathrm{E}}{E}_k+{\lambda }_{\mathrm{V}}{P}_k+\cdots +{\mathsf{\lambda }}_{\xi }·{\mathbf{A}}_k),$$

(46)

from which follows $S=-({\lambda }_0+{\lambda }_{\mathrm{E}}E+{\lambda }_{\mathrm{V}}P+\cdots +{\mathsf{\lambda }}_{\xi }·\mathbf{A})$, where ${\lambda }_{\mathrm{E}}=\mathsf{\beta }\doteq 1/T,{\lambda }_{\mathrm{V}}=\beta P,\cdots$, and ${\mathsf{\lambda }}_{\xi }=\beta \mathbf{A}$ are easy to identify so we finally have

$$p_k\left(t\right)=exp\left[\beta \left(t\right)(\widehat{G}\left(t\right)-E_k-P_{k}V-\cdots -{\mathbf{A}}_k·{\xi }_k)\right],$$

(47)

where ${\lambda }_0=-\beta \left(t\right)\widehat{G}\left(t\right)$, where $\widehat{G}\left(t\right)$ is the generalized SI thermodynamic potential defined by

$$\widehat{G}\left(t\right)\doteq E_k-P_{k}V-\cdots -{\mathbf{A}}_k·\xi ,$$

(48)

which then determines the NEQ generalized SI partition function of the system in ${\mathfrak{S}}_{\mathbf{Z}}$,

$$exp(-\beta \left(t\right)\widehat{G}\left(t\right))\equiv \sum _{k}exp[-\beta \left(t\right)(E_k+P_{k}V+\cdots +{\mathbf{A}}_k\left(t\right)·\mathsf{\xi })],$$

(49)

provided constraints in Equation (45) are satisfied. In contrast, the EQ thermodynamic potential in this case is given by

$$G(T_0,P_0,\cdots )=E\left(t\right)-T_{0}S\left(t\right)+P_{0}V\left(t\right)+\cdots ;$$

(50)

see Ref. [58] for more details. It is this potential that is minimized as $\Sigma$ approaches its stable EQ for positive T or maximized at its unstable EQ for negative T as we will discover in Section 5.7. The microstate probability $p_k\left(t\right)$ in Equation (47) clearly shows the effect of irreversibility and is very different from its equilibrium analog $p_k^{\mathrm{eq}}$

$$p_k^{\mathrm{eq}}=exp\left[{\beta }_0(G(T_0,P_0,\cdots )-E_k-P_{k}V-\cdots )\right],$$

(51)

where ${\beta }_0\doteq 1/T_0$.

Remark 13.

The present discussion involving the NEQ SI partition function on the right side of Equation (49) shows its extremely important similarity with EQ partition function [15] in that for stable macrostates ${\mathfrak{M}}^s$ in ${\mathfrak{S}}_{\mathbf{Z}}$ in internal equilibrium, there is an analogous principle of minimization of the thermodynamic potential. According to this principle, the SI-thermodynamic potential is minimum for positive $T\left(t\right)$.

5. Mechanical Foundation of Irreversibility and Dissipation

5.1. Dissipative Dynamics

In [34], we have unraveled the mechanical foundation of irreversibility and resultant dissipation, which we review briefly here. The details can be found in the original publication. The focus in [34] was mostly on isolated systems, see Figure 1a, to understand the mechanical aspect of the irreversibility principle (Irr-P) introduced there for the first time, even though the idea was known for a long time but not its deep physical significance.

Remark 14.

One of the most important observations was that the irreversible macrowork ${\Delta }_{i}W$ between any two arbitrary macrostates can be used to capture dissipation. This has opened a new approach to study irreversibility and dissipation for any system, isolated or interacting. In this investigation, we extend the previous approach to interacting systems in Figure 1b, which brings out new complications that were absent in [34], where the focus was on microstates $\left\{{\mathfrak{m}}_k\right\}$ that play a central role as mechanical systems to determine dissipation, when coupled with stochasticity. Therefore, the current investigation is an extension from isolated to interacting systems. Because of this, there is a certain amount of duplication with [34], which is unavoidable.

It was observed that mechanically, the equilibrium microstate of each system (${\Sigma }^{\mathrm{D}}/\Sigma$) can be classified as stable (s) and unstable (u). The system ${\Sigma }^{\mathrm{D}}/\Sigma$ is denoted by ${\Sigma }^{\mathrm{Ds}}/{\Sigma }^{\mathrm{s}}$ or ${\Sigma }^{\mathrm{Du}}/{\Sigma }^{\mathrm{u}}$, and the corresponding EQ microstate ${\mathfrak{m}}_{k\mathrm{eq}}$ by ${\mathfrak{m}}_{k\mathrm{seq}}$ and ${\mathfrak{m}}_{k\mathrm{ueq}}$, respectively. Their macrostates are denoted by ${\mathfrak{M}}^{\mathrm{s}}$ and ${\mathfrak{M}}^{\mathrm{u}}$ formed by formed by $\left\{{\mathfrak{m}}_k^{\mathrm{s}}\right\}$ and $\left\{{\mathfrak{m}}_k^{\mathrm{u}}\right\}$, respectively. By definition, ${\mathfrak{M}}^{\mathrm{s}}\left(t\right)$ of ${\Sigma }^{\mathrm{s}}$ has a lower bound in energy, see the continuous blue curve in Figure 3, and asymptotically converges along blue arrows to a unique, stationary, and stable equilibrium (SEQ) macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ [20]. The temporal evolution of any arbitrary ${\mathfrak{M}}^{\mathrm{s}}$ due to stability eventually terminates into ${\mathfrak{M}}_{\mathrm{seq}}$, and this dynamics is customarily identified as dissipative [61]. Unstable macrostates and the issue of any possible dissipation have never been discussed in the literature until recently [34].

Recall that $W_{\mathrm{diss}}$ in Equation (2) cannot be defined for a NEQ isolated system. As a consequence, the Gouy-Stodola theorem fails, which is quite disturbing. The new approach salvages the situation. However, the connection between ${\Delta }_{\mathrm{i}}W$ and $W_{\mathrm{diss}}$ was not explored in [34]. Thus, any connection with the concept of dissipation covered in [34] with the original definition proposed by Thomson [75] was left untreated. This shortcoming will also be addressed here.

By definition, see Definition 1, the uniform macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right),\left\{{\mathfrak{m}}_k\left(\mathbf{w}\right)\right\}$, and $\left\{E_k\left(\mathbf{w}\right)\right\}$ are uniquely determined by its Hamiltonian $\mathcal{H}\left(\mathbf{w}\right)$ that is stationary and uniform. In contrast, $\mathfrak{M}(\mathbf{X},t),E(\mathbf{w},t)$, and $S(\mathbf{X},t)$ are explicitly time dependent due to nonuniform that is also reflected in $\left\{{\mathfrak{m}}_k\left(t\right),p_k\left(t\right),E_k\left(t\right)\right\}$.

5.2. $E_w$ and $E_{wk}$

As our interest is in understanding the physics behind ${\Delta }_{\mathrm{i}}W$, it is convenient to focus not on exchange macrowork ${\Delta }_{\mathrm{e}}W$. Therefore, the focus in the original publication was on ${\Delta }_{\mathrm{i}}W$, which we briefly review here for an easy discussion of the missing connection with $W_{\mathrm{diss}}$.

Following [34], we define a cumulative microwork function $W_{k\mathrm{eq}}^{\prime }$ and a microwork function $E_{\mathrm{w}k}$

$$\begin{array}{cc}\hfill W_{k\mathrm{eq}}^{\prime }& \doteq {\int }_{{\mathcal{P}}_{\mathrm{eq}}^{\prime }}{d}_{\mathrm{i}}{W}_k,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill E_{\mathrm{w}k}& \doteq E_{k\mathrm{eq}}-W_{k\mathrm{eq}}^{\prime },d(E_{\mathrm{w}k}-E_{k\mathrm{eq}})\equiv -d_{\mathrm{i}}{W}_k,\hfill \end{array}$$

(53)

along a NEQ process ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$. The process originates from some ${\mathfrak{M}}_{\mathrm{eq}}$ (which explains the suffix “eq”) to some NEQ $\mathfrak{M}$on solid curves in Figure 3. The direction of ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$ is opposite to that of the blue arrow for ${\mathfrak{M}}^{\mathrm{s}}$ but along the red arrow for ${\mathfrak{M}}^{\mathrm{u}}$. Recall that the microwork function $E_{\mathrm{w}k}$ is simply $E_{\mathrm{w}k}\left(\mathsf{\xi }\right)$ of $\mathsf{\xi }$ or an $E_{\mathrm{w}k}\left(t\right)$ of t. This explicit dependence is needed to study irreversibility.

We follow Ramsey [38] and assume for simplicity that $E_{\mathrm{w}k}$ has only a single extremum at $E_{k\mathrm{eq}}$ as shown by the dashed-dotted curves in Figure 3 for ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{m}}_k^{\mathrm{u}}$, each spontaneously evolving to lower microenergies as t increases along the blue and red arrows, respectively. The assumption for ${\mathfrak{m}}_k$ is the extension of the assumption by Ramsey for ${\mathfrak{M}}_{\mathrm{eq}}$ to NEQ situations.

We now define the cumulative macrowork associated with $W_{k\mathrm{eq}}^{\prime }$ by taking the ensemble average of the latter:

$$W_{\mathrm{eq}}^{\prime }\doteq {\int }_{{\mathcal{P}}_{\mathrm{eq}}^{\prime }}{d}_{\mathrm{i}}W\equiv <{\Delta }_{\mathrm{i}}W{>}_{{\mathcal{P}}_{\mathrm{eq}}^{\prime }}.$$

(54a)

We use it to introduce a macrowork function

$$E_{\mathrm{w}}\doteq E_{\mathrm{eq}}-W_{\mathrm{eq}}^{\prime }$$

(54b)

of t or $\mathsf{\xi }$. It is this average $E_{\mathrm{w}}$ that is shown in Figure 3, and differs from $E=E_{\mathrm{eq}}$ by the cumulative macrowork $W_{\mathrm{eq}}^{\prime }$. The solid blue and red curves in Figure 3, respectively, describe the temporal forms of $E_{\mathrm{w}}\left(t\right)=(E_{\mathrm{w}}^{\mathrm{s}}\left(t\right),E_{\mathrm{w}}^{\mathrm{u}}\left(t\right))$. For both of them, we have

$$d{E}_{\mathrm{w}}^{\prime }\doteq d(E_{\mathrm{w}}-E_{\mathrm{eq}})\equiv -d_{\mathrm{i}}W$$

(54c)

along ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$, a result that will be used in deriving GSL below.

We now argue that $W_{\mathrm{eq}}^{\prime }$ is independent of the difference of macroenergies of ${\mathfrak{M}}_{\mathrm{eq}}$ and $\mathfrak{M}$. The difference is basically the exchange energy difference, while $W_{\mathrm{eq}}^{\prime }$ satisfies the Irr-P in Equation (4) so its value is determined by the nature of irreversibility, which may be different for the same energy difference. Therefore,

$$\Delta E_{\mathrm{w}}^{\prime }=-W_{\mathrm{eq}}^{\prime };$$

(54d)

this explains why $E_{\mathrm{w}}$ is called a microwork function.

As discussed in [34], the extremum points of the curves in Figure 3 denote the equilibrium points $E_{\mathrm{seq}}$ (of a stable equilibrium (SEQ) macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ that is stationary and uniform and is normally considered in classical thermodynamics), and $E_{\mathrm{ueq}}$ for $E_{\mathrm{w}}\left(t\right)$ (for an unstable equilibrium (UEQ) macrostate ${\mathfrak{M}}_{\mathrm{ueq}}$ that is uniform but is rarely considered in classical thermodynamics), and $E_{k\mathrm{seq}}$ and $E_{k\mathrm{ueq}}$ for $E_{\mathrm{w}k}\left(t\right)$, respectively. A better understanding of dissipation was obtained by focusing on microstate evolution; the dashed-dot blue and red curves denote $E_{\mathrm{w}k}^{\mathrm{s}}\left(t\right)$ and $E_{\mathrm{w}k}^{\mathrm{u}}\left(t\right)$, respectively. During evolution as $t\to \infty$, $E_{\mathrm{w}}^{\mathrm{s}}\left(t\right)$ runs towards $E_{\mathrm{seq}}$ along the blue arrows for a stable ${\mathfrak{M}}^{\mathrm{s}}$, and $E_{\mathrm{w}}^{\mathrm{u}}\left(t\right)$ runs away from $E_{\mathrm{seq}}$ along the red arrows for an unstable ${\mathfrak{M}}^{\mathrm{u}}$. Similarly, as $t\to \infty$, $E_{\mathrm{w}k}^{\mathrm{s}}\left(t\right)$ runs towards $E_{k\mathrm{seq}}$ along the blue arrows, and $E_{\mathrm{w}k}^{\mathrm{u}}\left(t\right)$ runs away from $E_{k\mathrm{ueq}}$ along the red arrows. refer the reader to [34] for more details.

Remark 15.

A single extremum in $E_w^{\prime }$ will not allow for considering phase transitions, including glass transitions requiring metastable states. While their consideration is also an important part of NEQ statistical mechanics, we will not consider them here.

5.3. Mechanical Equilibrium Principle of Energy (Mec-EQ-P) for $m_k$

In analytical mechanics, ${\mathfrak{m}}_k^{\mathrm{s}}$ is special in that there is no dissipation so its evolution undergoes oscillations about ${\mathfrak{m}}_{k\mathrm{seq}}$ that will persist forever; in contrast, ${\mathfrak{m}}_k^{\mathrm{u}}$ runs away from ${\mathfrak{m}}_{k\mathrm{ueq}}$ along the dashed-dot red curve and terminates in a catastrophe in which ${\mathfrak{m}}_k^{\mathrm{u}}$ becomes extremely nonuniform in ${\mathfrak{S}}_{\mathbf{X}}$ as $t\to \infty$, see [34,36], so there is no oscillation. In the presence of internal variables in ${\mathfrak{S}}_{\mathbf{Z}}$, the catastrophe corresponds to an extremely large $n=n_{\mathrm{cats}}$ as discussed in [34]. Thus, ${\mathfrak{m}}_{k\mathrm{ueq}}$ remains the source for the SI-evolution of ${\mathfrak{m}}_k^{\mathrm{u}}$. As our interest in ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{m}}_k^{\mathrm{u}}$ is in a thermodynamic setting, all thermodynamic processes in ${\mathfrak{M}}^{\mathrm{s}}$ terminate asymptotically at ${\mathfrak{M}}_{\mathrm{seq}}$. Therefore, oscillations in ${\mathfrak{m}}_k^{\mathrm{s}}$ are overlooked as they do not affect $\Delta E_{\mathrm{w}k}^{\mathrm{s}}$ during its SI-evolution to the sink ${\mathfrak{m}}_{k\mathrm{seq}}$.

The mechanical equilibrium point (Mec-EQ) [86,87] in the SI-evolution under the SI-micorforce ${\mathbf{F}}_{\xi k}$ is at the extremum $E_{k\mathrm{eq}}$, where the velocity ${\dot{\mathsf{\xi }}}^{\mathrm{eq}}$ and micorforce ${\mathbf{F}}_{\xi k}^{\mathrm{eq}}$ vanish:

$$\begin{array}{cc}\hfill {\dot{\mathsf{\xi }}}^{\mathrm{eq}}& =0,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill {\mathbf{F}}_{\xi k}^{\mathrm{eq}}& =-\partial E_k/\partial {\mathsf{\xi }}^{\mathrm{eq}}=0.\hfill \end{array}$$

(56)

The mechanical asymptotic stability principle of minimum energy states that the minimum at $E_{k\mathrm{seq}}$ is a mechanically stable EQ point, where internal flows between different parts of the nonuniform system cease as expected at EQ [34].

In contrast, the mechanical instability principle of maximum energy states that the maximum of $E_{\mathrm{w}k}$ at $E_{k\mathrm{ueq}}$ represents a mechanically unstable EQ. In this case, $\left|\mathsf{\xi }\left(t\right)\right|$ behaves similar to t in Figure 3. The two points determine the curvature of $E_k$ at $E_{k\mathrm{eq}}$. The two principles are collectively called Mec-EQ-P [34].

From what we have seen in Equation (6), variation of $\mathsf{\xi }$ can be converted to $dt$ so what is said above is applicable to the system in ${\mathfrak{S}}_{\mathbf{X}}$ with similar conclusions.

We now have an answer to Problem 2 in Section 1.1. The spontaneous processes along solid blue and red arrows discussed above are all irreversible, but those along the blue arrow in ${\mathfrak{M}}^{\mathrm{s}}$ bring it to ${\mathfrak{M}}_{\mathrm{seq}}$ (positive T); those along red arrows take ${\mathfrak{M}}^{\mathrm{u}}$ away from ${\mathfrak{M}}_{\mathrm{ueq}}$ (negative T).

5.4. Dissipation

We are now ready to partially answer Problem 3 and fully answer Problem 4 in Section 1.1. As the extremum at Mec-EQ point represents a uniform ${\mathfrak{m}}_{k\mathrm{eq}}$, the remainder of the dashed-dotted curves denote nonuniform microstates ${\mathfrak{m}}_k$ (${\mathfrak{m}}_k^{\mathrm{s}}$ or ${\mathfrak{m}}_k^{\mathrm{u}}$). According to Proposition 1, ${\mathfrak{m}}_{k\mathrm{seq}}$ controls ${\mathfrak{m}}_k^{\mathrm{s}}$-evolution [88], As shown in Figure 3, $\xi$ keeps decreasing during the evolution and makes ${\mathfrak{m}}_k^{\mathrm{s}}$ more and more uniform with $dS>0$. At ${\mathfrak{m}}_{k\mathrm{ueq}}$ of ${\mathfrak{m}}_k^{\mathrm{u}}$, $E_{k\mathrm{ueq}}$ is maximum; any perturbation away from it spontaneously repels ${\mathfrak{m}}_k^{\mathrm{u}}$ further away from ${\mathfrak{m}}_{k\mathrm{ueq}}$. Again, as shown in Figure 3, $\xi$ keeps increasing during the evolution and makes ${\mathfrak{m}}_k^{\mathrm{u}}$ more and more nonuniform with $dS<0$; see Definition 2. Both are deterministic spontaneous ${\mathfrak{m}}_k$-evolutions, whose directions are controlled by the principle of mechanical equilibrium (Mec-EQ-P).

Remark 16.

In [34], the above comments for $dS$ were directly related to $d_{i}S$ so they immediately provided information about GSL/SL. Because of the interacting nature of the system, we need to also determine $d_{e}S$ to draw any conclusion about $d_{i}S$, which adds to the complication noted above.

It now follows from [34] and Figure 3 that ${\mathfrak{m}}_k$ performs spontaneous nonnegative internal microwork

$${\Delta }_{\mathrm{i}}{W}_k=-{\Delta }_{\mathrm{i}}{E}_k\ge 0.$$

(57)

Performing ensemble average over ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{m}}_k^{\mathrm{u}}$, we conclude that during a spontaneous process $\mathfrak{M}\to {\mathfrak{M}}^{\prime }$ results in the nonnegative spontaneous macrowork

$${\Delta }_{\mathrm{i}}W\doteq <{\Delta }_{\mathrm{i}}W>\equiv -<{\Delta }_{\mathrm{i}}E>\ge 0;$$

(58)

observe that $<{\Delta }_{\mathrm{i}}E>\ne {\Delta }_{\mathrm{i}}E=0$, see Equation (4). This finally proves that dissipation $D\doteq {\Delta }_{\mathrm{i}}W$ can never be negative even for an interacting system as is commonly believed. This conclusion is no difference from that in [34] and answers Problem 11 in Section 1.1. Also, this answers Problems 4 and 5 as ${\Delta }_{\mathrm{i}}W$ is defined between two arbitrary macrostates as seen from Claim 5 and the discussion following it.

5.5. Principle of Dissipation and D

We now answer Problems 7 and 9 in Section 1.1 by introducing the principle of dissipation in our approach. As ${\mathfrak{m}}_k$ is oblivious to the stochasticity in $\Sigma$, validity of Equation (57) follows the validity of analytical mechanics so it cannot be challenged in the BCGM approach. It then follows that the validity of Equation (58) is also beyond suspect. In other words,

Conclusion 1.

As long as thermodynamics is based on analytical mechanics (BCGM proposal), the irreversible macrowork inequality ${\Delta }_{i}W$$\ge 0$ can never be violated in any spontaneous process between any two macrostates, EQ or not, so it holds for an isolated as well as an interacting system. This is the Principle of Dissipation, which finally fixes the signature of ${\Delta }_{i}W={\Delta }_{i}Q$ to be positive and identify it as the dissipated work so that we finally have

$$D\doteq {\Delta }_{i}W={\Delta }_{i}Q=T^{*}{\Delta }_i{S}^{\left(w\right)}\ge 0$$

(59)

from Equation (44). This also formulates the Generalized Second Law (GSL) [34].

According to the principle, the entropic (stochastic) quantity ${\Delta }_{\mathrm{i}}Q$ in ${\mathfrak{S}}_S$ is equal in magnitude to the mechanical quantity ${\Delta }_{\mathrm{i}}W$ in the orthogonal subspace ${\mathfrak{S}}_{\mathbf{W}}$, see the identity (3), even though they are independently generated but must necessarily satisfy the first law. The principle is underpinned by the fundamental thermodynamic fact $d_{\mathrm{i}}E=d_{\mathrm{i}}Q-d_{\mathrm{i}}W=0$; see Equation (4). Therefore, the principle is always valid as long classical thermodynamics holds.

The concepts of SI macrowork and SI macroheat are very different from those considered in [45,46], where these concepts refer to exchange microquantities with a medium, which is absent for the isolated system $\Sigma$, so they vanish. Our concepts are intrinsic to $\Sigma$ and are usually nonzero. Despite this,

Conclusion 2.

According to Irr-P, the SI macrowork produced mechanically by macroforce imbalance and internal affinity is completely converted into the SI macroheat by dissipation. This makes $d_{i}W$ the primary concept that causes the emergence of $d_{i}Q$, which makes the latter a secondary concept. In other words, $d_{i}Q$ cannot be converted into $d_{i}W$.

The fact that ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ does not contribute to D answers Problem 6 in Section 1.1.

5.6. Connection with $W_{\mathit{diss}}$

So far we have treated $E_{\mathrm{w}k}$ having a single extremum as shown in Figure 3. The situation corresponds to some arbitrary NEQ macrostate $\mathfrak{M}$, whose evolution is controlled by ${\mathfrak{M}}_{\mathrm{eq}}$, similar to the one for ${\mathfrak{M}}_0$ in Figure 2 converging to ${\mathfrak{M}}_{0\mathrm{eq}}$ for a stable macrostate. However, we have only considered the case in which $\xi$ continue to either monotonically decrease (${\mathfrak{M}}^{\mathrm{s}}$) or increase (${\mathfrak{M}}^{\mathrm{u}}$). Thus, we either deal with a sink or a source.

To make connection with $W_{\mathrm{diss}}$ in Equation (2), we need to consider a process ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$ replaced by ${\gamma }_{12}$ in Figure 2 between two EQ macrostates ${\mathfrak{M}}_{1\mathrm{eq}}$ and ${\mathfrak{M}}_{2\mathrm{eq}}$, the latter replacing $\mathfrak{M}$ above. As both terminal macrostates are EQ macrostates, this can only happen for stable systems with stable macrostates ${\mathfrak{M}}^{\mathrm{s}}$ that connect ${\mathfrak{M}}_{1\mathrm{seq}}$ and ${\mathfrak{M}}_{2\mathrm{seq}}$, In addition, $\xi$ must vanish, not only for both of them but also for their microstates $\left\{{\mathfrak{m}}_{1k\mathrm{seq}}\right\}$ and $\left\{{\mathfrak{m}}_{2k\mathrm{seq}}\right\}$. This changes the form of $E_{\mathrm{w}k}^{\mathrm{s}}$ and $E_{\mathrm{w}}^{\mathrm{s}}$ dramatically in that their forms must turn back towards $\xi =0$ at their higher values, thus creating a source at their higher ends in each case. This corresponds to basically joining red with blue solid curves for $E_{\mathrm{w}}^{\mathrm{s}}$ and red with blue dashed-dot curves for $E_{\mathrm{w}k}^{\mathrm{s}}$, thus forming closed loops in Figure 3.

Each closed loop has a source at the top and a sink at the bottom. As the system spontaneously move from the top towards the bottom due to the internal force ${\mathbf{F}}_{\xi k}$ or ${\mathbf{F}}_{\xi }$, energy falls so Equations (57) and (58) remain satisfied. Thus,

Conclusion 3.

Changing $\mathfrak{M}$ to ${\mathfrak{M}}_{2\mathit{eq}}$, but keeping ${\mathfrak{M}}_{\mathit{eq}}$ as ${\mathfrak{M}}_{1\mathit{eq}}$ does not invalidate the principle of dissipation encoded in Equation (59).

The above discussion is for the NEQ process ${\mathfrak{M}}_{2\mathrm{seq}}\to {\mathfrak{M}}_{1\mathrm{seq}}$, in which the system from the source at ${\mathfrak{M}}_{2\mathrm{seq}}$ to the sink at ${\mathfrak{M}}_{1\mathrm{seq}}$ under the influence of the medium. For the inverse process ${\mathfrak{M}}_{1\mathrm{seq}}\to {\mathfrak{M}}_{2\mathrm{seq}}$, the source is at ${\mathfrak{M}}_{1\mathrm{seq}}$, which is at higher energy than the sink at ${\mathfrak{M}}_{2\mathrm{seq}}$, which reverses the closed loop. For both processes, $W_{k\mathrm{eq}}^{\prime }$ and $W_{\mathrm{eq}}^{\prime }$ are nonnegative, with the latter measuring the dissipation D.

To connect $W_{\mathrm{eq}}^{\prime }$ with $W_{\mathrm{diss}}$, we use the precise description of the process ${\mathcal{P}}_{12}$ in Figure 2, which must have its source and sink defined in ${\mathfrak{S}}_{\mathbf{X}}$ but elsewhere it must be defined in ${\mathfrak{S}}_{\mathbf{Z}\left(t\right)}$, with $\mathbf{Z}\left(t\right)$ changing along ${\mathcal{P}}_{12}$ so that $\mathbf{Z}\left(t\right)$ reduces to ${\mathbf{X}}_1$ at the sink and ${\mathbf{X}}_2$ at the source. As the medium $\tilde{\Sigma }$ is always in EQ, it is specified in ${\mathfrak{S}}_{\mathbf{X}}$ throughout ${\mathcal{P}}_{12}$.

With these specifications, we are ready to determine ${\Delta }_{\mathrm{e}}W$ and ${\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}$. From Equation (29), we have

$$d_{\mathrm{e}}W=P_{0}dV+\cdots$$

in terms of the fields of the medium. To obtain ${\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}$, we recall that the NEQ-fields of $\Sigma$ are given in Equation (28):

$$P\left(t\right),\mathbf{A}\left(t\right),$$

(60)

along with $T\left(t\right)$. To ensure that ${\mathcal{P}}_{12}$ is carried out reversibly, see Claim 3, we need $\Sigma$ to interact with a series of (NEQ) mediums with fields $T\left(t\right),P\left(t\right),\mathbf{A}\left(t\right)$. Then the exchange macrowork ${\left(d_{\mathrm{e}}W\right)}_{\mathrm{rev}}$ is given by

$${\left(d_{\mathrm{e}}W\right)}_{\mathrm{rev}}=PdV+\cdots +\mathbf{A}·d\mathsf{\xi },$$

which is identical to $dW$ in Equation (27a). It now follows from Equation (2) that

$$d{W}_{\mathrm{diss}}=dW-d_{\mathrm{e}}W\equiv d_{\mathrm{i}}W,$$

(61a)

given in Equation (30). It now follows

$$W_{\mathrm{diss}}\equiv {\Delta }_{\mathrm{i}}W$$

(61b)

after accumulation along ${\mathcal{P}}_{\mathrm{eq},\mathrm{eq}}^{\prime }$, which justifies Claim 5

However, in classical thermodynamics, Equation (2) is applied to a special process ${\gamma }_{12}$, which differs from ${\mathcal{P}}_{12}$ in that ${\gamma }_{12}$ only contains EQ macrostates in ${\mathfrak{S}}_{\mathbf{X}\left(t\right)}$, with $\mathbf{X}\left(t\right)$ changing along ${\gamma }_{12}$ so that it reduces to ${\mathbf{X}}_1$ at the sink and ${\mathbf{X}}_2$ at the source. In this scenario, $\mathbf{A}\left(t\right)\equiv 0$ in Equation (60) so

$${\left(d_{\mathrm{e}}W\right)}_{\mathrm{rev}}=PdV+\cdots ,$$

and

$$d{W}_{\mathrm{diss}}=(P-P_0)dV+\cdots \equiv d_{\mathrm{i}}W,$$

and we again obtain Equation (61b) to justify Claim 5 in this scenario also.

5.7. Positive and Negative T

We now follow the significance of the dissipation measured by D, which can never be negative. However, its relation with ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ depends on $T^{*}$. For positive intermediate temperature $T^{*}$ along $\mathcal{P}$, we have

$${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}\ge 0,T^{*}>0,$$

(62a)

so that GSL is equivalent to SL. For negative intermediate temperature $T^{*}$ along $\mathcal{P}$, we have

$${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}\le 0,T^{*}<0,$$

(62b)

a surprising result, according to which GSL does confirm the conventional formulation of SL of nonnegative irreversible entropy generation. As $D\ge 0$ in both cases, dissipation measured by irreversible macrowork remains nonnegative in both cases. Thus, temperature has no effect on the signature of D but affect ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$. We now conclude that ${\mathfrak{M}}^{\mathrm{s}}$ has positive temperatures, while ${\mathfrak{M}}^{\mathrm{u}}$ has negative temperatures.

Conclusion 4.

The fixed positive sign of D in GSL now becomes the proper statement of dissipation due to spontaneous processes, and must replace the conventional statement ${\Delta }_{i}S\ge 0$ of SL to include both possibilities depending on the sign of the temperature [37,38].

This now completes the answer for the remainder of Problem 3 in Section 1.1. This also answers Problem 10.

5.8. Consequences

By replacing $d_{\mathrm{i}}Q$ by $d_{\mathrm{i}}W$ in Equation (40), we finally justify Equations (33) and (34) with a direct proportionality between $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ in $\mathfrak{S}$. It is their most general relationship over an infinitesimal process $d\mathcal{P}$ belonging to $\mathcal{P}$. In general, $d\mathcal{P}$ need not connect two EQ states. Note that it is not a simple relation between $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}S$ as in GST. As $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ is obtained by subtracting $d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ from $dS$, $d_{\mathrm{i}}W$ depends indirectly also on the temperature difference $T-T_0$ corresponding to irreversible heat exchange through $d_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$, see Equation (32), between T and $T_0$, if written in terms of $dS$. We see that GST ($d_{\mathrm{i}}W=T_0{d}_{\mathrm{i}}S$) holds iff $T=T_0$ (isothermal heat exchange) or $d_{\mathrm{e}}Q=0$ (isolated system) at all t.

The expression in terms of $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ is much more revealing in that the proportionality between $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ is in terms of T of $\Sigma$ only. Moreover, as the derivation is general, it is valid for $d\mathcal{P}$ between any two (EQ or NEQ) states for which S is a state function in ${\mathfrak{S}}_{\mathbf{Z}}$ or E in $\mathfrak{S}(S,\mathbf{W})$. Our statistical analysis also clarifies that the relationship between $d_{\mathrm{i}}W$ and $d_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}$ is possible as a consequence of Equation (4) or Equation (3).

We can integrate Equation (33) along $\mathcal{P}$ to obtain the desired Irr-P given in Equation (44):

$${\Delta }_{\mathrm{i}}W={\int }_{\mathcal{P}}T{d}_{\mathrm{i}}{S}^{\left(\mathrm{w}\right)}={\int }_{\mathcal{P}}T{d}_{\mathrm{i}}S+{\int }_{\mathcal{P}}(T/T_0-1)d_{\mathrm{e}}Q,$$

(63)

where we have allowed variable temperatures T and $T_0$ along $\mathcal{P}$. This is the most general statement of the Irr-P for any arbitrary NEQ process $\mathcal{P}$. Notice that the integration is only over temperatures, and that T does not have to be equal to $T_0$ at the two ends of $\mathcal{P}$.

If $\mathcal{P}$ is between two equilibrium states ${\mathfrak{M}}_{\mathrm{eq}}^{\mathrm{in}}$ and ${\mathfrak{M}}_{\mathrm{eq}}^{\mathrm{fin}}$ at the same constant temperature $T_0$ (their pressures could be different), then

$${\Delta }_{\mathrm{i}}W={\int }_{\mathcal{P}}TdS-{\Delta }_{\mathrm{e}}Q=T^{**}{\Delta }_{\mathrm{i}}S+(T^{**}/T_0-1){\Delta }_{\mathrm{e}}Q,$$

where we have again used the mean value theorem for ${\int }_{\mathcal{P}}TdS$ to introduce $T^{**}$. Only for an isothermal process so that there is no irreversible heat exchange do we obtain the original GST

$${\Delta }_{\mathrm{i}}{W}_{\mathrm{isoth}}=T_0{\Delta }_{\mathrm{i}}S,$$

(64)

since $T^{**}=T_0$. This is a very strong limitation in which $\tilde{\Sigma }$ is at constant $T_0$, and $\Sigma$ remains in continuous contact with it to ensure complete thermal equilibrium at all times.

We finally observe that $D={\Delta }_{\mathrm{i}}W$, see Conclusion 1, can never be negative as we have shown in Equation (58) by following spontaneous evolution. However, one can force a system by overcoming spontaneous forces in ${\mathfrak{m}}_k^{\mathrm{s}}$ to make the microstate “forcefully evolve” in the direction opposite to its spontaneous evolution (blue arrow) by following the dashed-dot green arrow going to the left or right in Figure 3. A similar “forceful evolution” can also be imagined for any unstable microstate ${\mathfrak{m}}_k^{\mathrm{u}}$. In both cases, ${\Delta }_{\mathrm{i}}{W}_k^{\mathrm{forced}}$ can be made negative so that its average ${\Delta }_{\mathrm{i}}{W}^{\mathrm{forced}}$ is also negative. But this is not dissipated or lost work. Another possibility is to “forcefully” create an internal constraint in a stable EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}^{\mathrm{s}}$ in the sense of Callen [13], which reduces the entropy from its EQ value $S\left(\mathbf{X}\right)$. This reduction is also not caused by any spontaneous process so this reduction cannot be identified as negative dissipation.

6. Relation Between Kullback-Leibler Distance ${\mathit{D}}_{\mathrm{KL}}$ and ${\Delta }_{\mathrm{i}}\mathit{S}$

We have already argued that the constrained average in $D_{\mathrm{KL}}$ in Equation (20) is not an ensemble average introduced by Gibbs. Therefore, any connection between it and the SI irreversible entropy generation ${\Delta }_{\mathrm{i}}S$ is quite a suspect. We now revisit the argument that support this conclusion in [72]. We consider the work protocol ${\mathcal{P}}_{\mathrm{J}}$ popularized by Jarzynski [68] in which a system is driven between two macrostates ${\mathfrak{M}}_{\mathrm{in}}(T_0,{\mathbf{w}}_{\mathrm{in}})$ and ${\mathfrak{M}}_{\mathrm{fin}}(T_0,{\mathbf{w}}_{\mathrm{fin}})$ in canonical equilibrium at the same temperature $T_0=1/{\beta }_0$ by varying the work parameter $\mathbf{w}$ from ${\mathbf{w}}_{\mathrm{in}}$ to ${\mathbf{w}}_{\mathrm{fin}}$. We follow the careful discussion of this work protocol ${\mathcal{P}}_{\mathrm{J}}$ as presented for the first time in [56,57], which pointed out the mistaken identity of the microwork that resulted in a logically unsupported equality that is the basis of the above unsubstantiated relationship. We consider a particular microstate ${\mathfrak{m}}_k$, which performs the SI-microwork

$$\Delta W_k=-\Delta E_k\doteq E_{k,\mathrm{in}}-E_{k,\mathrm{fin}}$$

(65)

along its trajectory ${\gamma }_k$ between the two macrostates. Let $p_{{\gamma }_k}$ denote the probability of this trajectory, which results in the ensemble average

$$\Delta W=〈\Delta W〉\doteq {\sum }_{{\gamma }_k}{p}_{{\gamma }_k}\Delta W_k={\Delta }_{\mathrm{e}}W+{\Delta }_{\mathrm{i}}W;$$

(66)

here, ${\Delta }_{\mathrm{e}}W$ is the exchange macrowork and ${\Delta }_{\mathrm{i}}W\ge 0$ is the nonnegative irreversible macrowork over $\mathcal{P}$. We now consider the constrained average of $A_{\mathrm{J}k}\doteq exp({\beta }_0\Delta W_k)$ used by Jarzynski:

$$\begin{array}{cc}\hfill {〈A_{\mathrm{J}}〉}^{\mathrm{cons}}& \doteq {\sum }_k{p}_{k,\mathrm{in}}^{\mathrm{eq}}{A}_{\mathrm{J}k}={\displaystyle \frac{1}{Z_{\mathrm{in}}(T_0,{\mathbf{w}}_{\mathrm{in}})}}{\sum }_{k}exp(-{\beta }_0{E}_{k,\mathrm{fin}})\hfill \\ \hfill & ={\displaystyle \frac{Z_{\mathrm{fin}}(T_0,{\mathbf{w}}_{\mathrm{fin}})}{Z_{\mathrm{in}}(T_0,{\mathbf{w}}_{\mathrm{in}})}},\hfill \end{array}$$

see Equation (11) in [56]; here, Z denotes the partition function. We thus obtain

$${〈e^{{\beta }_0\Delta W_k}〉}^{\mathrm{cons}}=e^{-{\beta }_0\Delta F},$$

(67)

where $\Delta F$$=(F_{\mathrm{fin}}-F_{\mathrm{in}})$. By definition, $dF=-SdT-dW$, so

$$\Delta F=-{\int }_{{\mathcal{P}}_{\mathrm{J}}}SdT-\Delta W,$$

where $\Delta W$ is given in Equation (66).

Using Jensen’s inequality, see also Gujrati [73], for the constrained average on the left in Equation (67), we obtain

$${〈exp({\beta }_0\Delta W)〉}^{\mathrm{cons}}\ge exp\{exp\left({\beta }_0{〈\Delta W〉}^{\mathrm{cons}}\right)\}.$$

If we use it in Equation (67), we conclude the inequality

$${〈\Delta W〉}^{\mathrm{cons}}\le \Delta W+{\int }_{{\mathcal{P}}_{\mathrm{J}}}SdT.$$

If and only if the temperature of the system remained constant at $T_0$ over the entire ${\mathcal{P}}_{\mathrm{J}}$, the integral vanishes identically. Thus, we require ${\mathcal{P}}_{\mathrm{J}}$ to be isothermal throughout the process, not just at its end points. For such an isothermal process, $\Delta F=-\Delta W$, and we conclude that

$$\Delta W-{〈\Delta W〉}^{\mathrm{cons}}\ge 0.$$

(68)

Now, $\Delta W$ is nothing but the reversible macrowork ${\left({\Delta }_{\mathrm{e}}W\right)}_{\mathrm{rev}}$ introduced in Equation (2). One may be inclined to equate this result with the one in Equation (2), but this will be logically unjustifiable. There is no justification to equate the SI-microwork $\Delta W_k$ in Equation (65) or (67) with the exchange microwork ${\Delta }_{\mathrm{e}}{W}_k$ as was first pointed out by Gujrati [56,57], but conjectured earlier by Cohen and Mauzerall [70,71]. Unfortunately, this mistaken identity

$$\Delta W_k\stackrel{?}{=}{\Delta }_{\mathrm{e}}{W}_k⟹{\Delta }_{\mathrm{i}}{W}_k=0$$

continues to persist. The question mark is a reminder of the erroneous assumption that is prevalent in the literature. A most recent usage appears in [89]. A consequence of the mistaken identity is that

$${〈\Delta W〉}^{\mathrm{cons}}\stackrel{?}{=}{〈{\Delta }_{\mathrm{e}}W〉}^{\mathrm{cons}}⟹{〈{\Delta }_{\mathrm{i}}W〉}^{\mathrm{cons}}=0.$$

Furthermore, there is no justification that

$${〈{\Delta }_{\mathrm{e}}W〉}^{\mathrm{cons}}\stackrel{?}{=}〈{\Delta }_{\mathrm{e}}W〉,$$

another very common error in the literature. This issue has also been discussed recently by Gujrati [73]. With this additional error, one incorrectly identifies Equation (68) derived by Jarzynski with Equation (2) and use Equation (18) to claim that $D_{\mathrm{KL}}$ is a direct measure of the irreversible entropy generation ${\Delta }_{\mathrm{i}}S$ [72]. The erroneous claim lacks any justification.

We now give a direct demonstration of the shortcomings of $D_{\mathrm{KL}}$. We take $q_k$ in Equation (20) to denote the EQ probabilities $p_k^{\mathrm{eq}}$ and evaluate the entropy difference of the EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ entropy $S\left(\mathbf{X}\right)$ with its NEQ macrostate $\mathfrak{M}$ entropy $S(\mathbf{X},\mathsf{\xi })$ that requires some internal variable $\mathsf{\xi }$. We will only consider positive temperatures so we are dealing with stable macrostates that spontaneously converges to ${\mathfrak{M}}_{\mathrm{eq}}$. The difference $S\left(\mathbf{X}\right)-S(\mathbf{X},\mathsf{\xi })\ge 0$ is the irreversible entropy that is generated as ${\mathfrak{M}}^{\mathrm{s}}\to {\mathfrak{M}}_{\mathrm{seq}}$. We have

$$\begin{array}{cc}\hfill {\Delta }_{\mathrm{i}}S& =-{\sum }_k{p}_k^{\mathrm{eq}}ln{p}_k^{\mathrm{eq}}+{\sum }_k{p}_{k}ln{p}_k\hfill \\ \hfill & =D_{\mathrm{KL}}(p\Vert p^{\mathrm{eq}})+{\sum }_k(p_k-p_k^{\mathrm{eq}})ln{p}_k^{\mathrm{eq}}\ge 0,\hfill \end{array}$$

which shows that

$${\Delta }_{\mathrm{i}}S\ne D_{\mathrm{KL}}(p\Vert p^{\mathrm{eq}}),$$

(69)

unless $p_k=p_k^{\mathrm{eq}},\forall k$. It is true that both are nonzero, but they are not identical. It is very common to identify a nonnegative quantity such as $D_{\mathrm{KL}}$ with the irreversible entropy generation, but this is not a valid identification, which could result in equating $D_{\mathrm{KL}}(p\Vert p^{\mathrm{eq}})$ with ${\Delta }_{\mathrm{i}}S$. The latter identification is often made in the literature; see ([90], Equation (7)).

7. Determining ${\mathit{dp}}_{\mathit{k}}$ and ${\mathit{d}}_{\mathrm{i}}{\mathit{p}}_{\mathit{k}}$

Determination of D either require evaluating ${\Delta }_{\mathrm{i}}Q$ or ${\Delta }_{\mathrm{i}}W$. For the former, we need to determine the change $d_{\mathrm{i}}{p}_k$; for the latter, we need to determine $d_{\mathrm{i}}{E}_k$. Both are nontrivial and require careful analysis of the physical process $\mathcal{P}$ in ${\mathfrak{S}}_{\mathbf{X}}$.

7.1. Fokker-Planck Equation Not Suitable for ${\mathcal{P}}_{\mathit{irr}}$

Fokker-Planck equation is one of a few celebrated partial differential equation of stochastic probability evolution such as of the one describing Langevin equation for Brownian motion. As such, it is of great importance in statistical mechanics due to its wide applications in stochastic processes. In general, it is not always possible to easily obtain solutions to Fokker-Planck equations, except in a few simple cases. The Fokker-Planck equation [91] can be expressed as a continuity equation

$$d{\rho }_k/dt=\partial {\rho }_k/\partial t+\mathsf{\partial }·{\mathbf{j}}_{\rho k}^{\mathrm{e}}=0,$$

(70a)

where ${\rho }_k(\mathbf{X},t)$ is the probability density in a given region $\Omega \left(\mathbf{X}\right)$ surrounded by its closed boundary $\partial \Omega \left(\mathbf{X}\right)$, and ${\mathbf{j}}_{\rho k}^{\mathrm{e}}$ is the probability current density or flux associated with ${\rho }_k$; the justification for the superscript e will be given shortly. Its integral form is

$$d{p}_k/dt+{\displaystyle \underset{\partial \Omega }{\oint }}{\mathbf{j}}_{\rho k}^{\mathrm{e}}·d\mathbf{f}=0,$$

(70b)

where $d\mathbf{f}$ is the surface element on $\partial \Omega$, which is pointed away from the enclosed region, and where we have used the divergence theorem for $\Omega \left(\mathbf{X}\right)$. The physical implication of the continuity equation is simply a conservation of probability in that the rate at which probability enters or leaves a given region $\Omega$ through its boundary $\partial \Omega$ is equal to the change in the total probability in $\Omega$.

As an example, let us consider the Brownian motion of a single particle of mass m that is described by its coordinate X and momentum P. We follow Sekimoto [46], and identify

$$\begin{array}{cc}\hfill j_{\rho X}& =\left(P/m\right)\rho (X,P,t)\hfill \\ \hfill j_{\rho P}& =\left(-{\displaystyle \frac{\partial U}{\partial X}}-{\displaystyle \frac{P}{m}}\right)\rho (X,P,t)-{\displaystyle \frac{\partial }{\partial P}}\left[\gamma T\rho (X,P,t)\right]\hfill \end{array}$$

by using his notation. Thus, we have

$$\partial \rho /\partial t+\partial j_{\rho X}/\partial X+\partial j_{\rho P}/\partial P=0.$$

(71)

From the structure of the second term in Equation (70b), we observe that it represents the flow of probability across the boundary $\partial \Omega$ so it must refer to the exchange change $d_{\mathrm{e}}{p}_k$ in $d{p}_k$. Thus, the first term should be correctly identified with $d_{\mathrm{e}}{p}_k/dt$ to ensure the correct interpretation of the equation of continuity. This justifies the superscript e in ${\mathbf{j}}_{\rho k}^{\mathrm{e}}$.

Let us apply Equation (70b) to the entire isolated system ${\Sigma }_0$. In this case, there is no flux across the boundary of ${\Sigma }_0$ as there is nothing surrounding it. The same argument also applies to an isolated system $\Sigma$ in Figure 1a. Hence, the integral vanishes identically in both cases, which makes $p_k$ a constant of motion. But this is incorrect as we are dealing with a NEQ macrostate in both cases, unless we are dealing with a reversible process ${\mathcal{P}}_{\mathrm{rev}}$. The conclusion is unavoidable:

Claim 9.

The Fokker-Planck equation as conventionally written cannot be used to determine the irreversible entropy $d_{i}S$ in an irreversible process ${\mathcal{P}}_{\mathit{irr}}$.

However, it is very common to use the conventional Fokker-Planck equation to discuss irreversible processes such as by Tomé and de Oliveira [92], in which the Fokker-Planck equation is used to basically reproduce the Jarzynski equality, which is an invalid equality except for a reversible process as discussed above in Section 6.

Claim 10.

The Fokker-Planck equation must be modified to determine $d_i{p}_k/dt$ and $d_{i}S/dt$ in an irreversible process ${\mathcal{P}}_{\mathit{irr}}$ as follows:

$$d_i{p}_k/dt\doteq d{p}_k/dt+{\displaystyle \underset{\partial \Omega }{\oint }}{\mathbf{j}}_{\rho k}^e·d\mathbf{f}$$

(72)

by adding the source term $d_i{p}_k/dt$.

7.2. Master Equations

There is another widely used approach to determine $d{p}_k$, which uses master equations, which is phenomenological in nature. The approach requires the knowledge of transition rate matrix $d\mathbf{T}$ between different microstates. The name was first proposed by Lamb and Uhlenbeck [93]. The approach determine how $p_k$ evolves in time during a process $\mathcal{P}$; see also Equation (25). The elements of the matrix satisfy

$${\sum }_{l}d{T}_{kl}\left(t\right)=1,$$

(73)

where $T_{kl}$ is the one-step transition probability from ${\mathfrak{m}}_k$ to ${\mathfrak{m}}_l$. It follows from this that

$$d{p}_l\left(t^{″}\right)\doteq p_l\left(t^{″}\right)-p_l\left(t^{\prime }\right)={\sum }_k\left[p_k\left(t^{\prime }\right)d{T}_{kl}\left(t^{\prime }\right)-p_l\left(t^{\prime }\right)d{T}_{lk}\left(t^{\prime }\right)\right]$$

(74)

during a small time interval $dt\doteq t^{\prime \prime }-t^{\prime }$, and where we have neglected terms of higher orders in $dt$. It is evident from this truncation that the one-step transition probability must be a very small quantity proportional to $dt$. The use of the master equation is when we know $d{T}_{kl}\left(t\right)$ so that we can determine $p_l\left(t^{″}\right)$. In this representation, it is then possible to partition

$$d\mathbf{T}=d_{\mathrm{e}}\mathbf{T}+d_{\mathrm{i}}\mathbf{T}$$

(75)

into the exchange and internal contributions in our approach, which will prove useful below. The partition is also similar to that adopted by Klein [79]. However, we will continue to use the notation $d\mathbf{T}$ even though it is not the convention followed by most workers.

The arguments leading to Equation (74) suggests introducing the following notations with clear interpretation:

$$d{p}_l^{\mathrm{in}}(t+dt)\doteq {\sum }_k{p}_k\left(t\right)d{T}_{kl}\left(t\right),d{p}_l^{\mathrm{out}}(t+dt)\doteq {\sum }_k{p}_l\left(t\right)d{T}_{lk}\left(t\right);$$

the summand in the first one is the “inward” contribution from ${\mathfrak{m}}_k$ to ${\mathfrak{m}}_l$ that appears with a positive sign and the summand in the second one is the “outward” contribution from ${\mathfrak{m}}_l$ to ${\mathfrak{m}}_k$ that appears with a negative sign. Their sum over all ${\mathfrak{m}}_k$’s give the “inward” contribution $d{p}_l^{\mathrm{in}}\left(t\right)$ and the “outward” contribution $d{p}_l^{\mathrm{out}}\left(t\right)$ to $d{p}_l\left(t\right)$:

$$d{p}_l(t+dt)\doteq d{p}_l^{\mathrm{in}}(t+dt)-d{p}_l^{\mathrm{out}}(t+dt).$$

Remark 17.

It is the imbalance between the inward and outward contributions $d{p}_l^{\mathit{in}}\left(t\right)$ and $d{p}_l^{\mathit{out}}\left(t\right)$, respectively, that determines the change $d{p}_l\left(t\right)$ during $dt$.

The one-step transition probabilities are usually determined by applying Fermi’s golden rule [85], which requires knowing the perturbation ${\mathcal{H}}_{\mathrm{int}}$ caused by the medium $\tilde{\Sigma }$ on the unperturbed Hamiltonian of the system. Knowing ${\mathcal{H}}_{\mathrm{int}}$ precisely to determine $d\mathbf{T}\left(t\right)$ is an impossible task; the best one can do is to make some simple modeling of it. Even then, the master equation is an incomplete story of the temporal evolution as it does not contain multi-step transitions between ${\mathfrak{m}}_k$ to ${\mathfrak{m}}_l$. As each transition has its own time scale, multi-transitions between ${\mathfrak{m}}_k$ to ${\mathfrak{m}}_l$ are possible during a certain period of time $dt$.

Being a phenomenological approach, Equation (74) has given rise to a variety of master equations based on any number of heuristic assumptions. For example, Kac [94] has shown how molecular chaos assumption in the master equation can lead to the kinetic theory.

Definition 3.

A process is called  balanced and denoted by ${\mathcal{P}}_B$ if it satisfies

$$d{p}_l^{\mathit{in}}(t+dt)=d{p}_l^{\mathit{out}}(t+dt).$$

(76)

Consequently, microstate probabilities $p_k$ become stationary for $\forall k$ so that $d{p}_k\equiv 0$ and $dS\equiv 0$ for ${\mathcal{P}}_{\mathrm{B}}$. It follows from this that

$$d_{\mathrm{i}}S=-d_{\mathrm{e}}S\ge 0.$$

(77)

The magnitude $\left|d_{\mathrm{e}}S\right|$ determines the value of $d_{\mathrm{i}}S$. Thus, the balanced process ${\mathcal{P}}_{\mathrm{B}}$ describes a NEQ steady state process in a macrostate. Our result generalizes Klein’s result [79] that simple balance of ${\mathcal{P}}_{\mathrm{B}}$ in Equation (76) does not always refer to an EQ process ${\mathcal{P}}_{\mathrm{rev}}$.

Remark 18.

For an isolated system, the balanced process ${\mathcal{P}}_B$ corresponds to $d_{e}S\equiv 0$ so it also corresponds to absence of irreversibility ($d_{i}S\equiv 0$).

Definition 4.

A process is called detailed balanced and denoted by ${\mathcal{P}}_{\mathit{DB}}$ if it satisfies

$$p_k\left(t\right)d{T}_{kl}\left(t\right)=p_l\left(t\right)d{T}_{lk}\left(t\right),\forall (k,l).$$

(78)

The condition for detailed balance provides more information that the condition for balance for the same stationary $p_l\left(t\right)$, and makes is simpler to find the solution. We defer the discussion of this solution to Section 7.4.

Remark 19.

The detailed information is thermodynamically not useful as all ensemble averages including the entropy are the same for both so they describe the same macrostate ${\mathfrak{M}}_{\mathit{eq}}\left(t\right)$. Even the detailed imbalance $p_k\left(t\right)d{T}_{kl}\left(t\right)-p_l\left(t\right)d{T}_{lk}\left(t\right)$ and the imbalance $d{p}_l^{\mathit{in}}\left(t\right)-d{p}_l^{\mathit{out}}\left(t\right)$ provide the same $d{p}_k$ so all process quantities remain the same.

7.3. Transition Matrix and Isolated Systems

We now turn to an important issue that, to the best of our knowledge, has never been discussed in the literature as master equations are invariably applied to interacting systems. Therefore, the issue we wish to raise does not seem to have been considered as an issue. For an isolated system with the Hamiltonian $\mathcal{H}(\mathbf{w},t)$, there is no interaction energy to be concerned about. We then

Claim 11.

Transition probabilities between different microstates obtained by using the Fermi’s golden rule must be identically zero.

This is consistent with the fact that the kets of microstates of different microenergies must always remain orthogonal. This remains true even for a time-dependent Hamiltonian since in this case their temporal evolutions are governed by a time-dependent unitary operators that do not destroy their orthogonality. This is very disconcerting as it suggests that the master equation approach is not suitable for describing an isolated system. But that is precisely what is needed to be consider for determining $d_{\mathrm{i}}{p}_k$. Therefore, we conclude that

Claim 12.

No master equation can be used in determining dissipation requiring $d_i{p}_k$ in any NEQ macrostates.

Conclusion 5.

The master equation approach is suitable only for an interacting system in a medium to determine $d_e{p}_k$.

It is a well-known fact that master equations can cast in a form identical to Fokker-Planck equation by truncating to second order the corresponding Kramers-Moyal expansion. This is consistent with Conclusion 5 that master equations can only determine $d_{\mathrm{e}}{p}_k$.

7.4. Principle of Detailed Balance

We consider the defining Equations (22) and (25) for the principle of detailed balance We assume that the stationary condition is the EQ macrostate of the system.

Theorem 1.

The conditional probability ${\stackrel{·}{T}}_{kl}dt$ for the transition $k\to l$ is either (a) the equilibrium probability $p_l^{\mathit{eq}}$ of the arriving microstate $l\ne k$ and $p_k^{\mathit{eq}}$ of not leaving k (b) vanishes for the arriving microstate $l\ne k$ and not leaving with certainty under the assumption of PDB.

Proof. 

According to the condition of the PDB, ${\stackrel{·}{T}}_{kl}\left(t\right)$ satisfies

$$p_k^{\mathrm{eq}}{\stackrel{·}{T}}_{kl}=p_l^{\mathrm{eq}}{\stackrel{·}{T}}_{lk},\forall (i,j).$$

(79)

Its solution are

$$\left(\mathrm{A}\right):{\stackrel{·}{T}}_{kl}dt=p_l^{\mathrm{eq}},\stackrel{·}{T_{lk}}dt=p_k^{\mathrm{eq}},$$

(80a)

which is the most common acceptable solution, and an unexpected solution

$$\left(\mathrm{B}\right):{\stackrel{·}{T}}_{kl}dt\to 0,\stackrel{·}{T_{lk}}dt\to 0,k\ne l.$$

(80b)

We need to ensure the sum rule for $d{T}_{kl}$ in Equation (22), which requires

$${\sum }_l{\stackrel{·}{T}}_{kl}dt=1.$$

Solution (A) obviously satisfies it. For solution (B) to satisfy it requires

$${\stackrel{·}{T}}_{kk}dt=1$$

so that ${\mathfrak{m}}_k$ never leaves, but stays put with certainty.

This completes the proof. □

It should be noted that we have assumed that the rates are time dependent so they keep changing as EQ is reached. But it is also obvious that the solution (A) in Equation (80a) can only refer to a stable macrostate ${\mathfrak{M}}^{\mathrm{s}}$ reaching ${\mathfrak{M}}_{\mathrm{seq}}$ with maximum possible entropy so it is valid only for positive temperatures associated with ${\mathfrak{M}}^{\mathrm{s}}$. As ${\stackrel{·}{T}}_{kk}dt=p_k^{\mathrm{eq}}>0,\forall k$, it means that there is a nonzero probability to remain put in for each microstate. Thus, this solution does not relate to any ${\mathfrak{M}}^{\mathrm{u}}$ running out of ${\mathfrak{M}}_{\mathrm{ueq}}$ catastrophically. The conclusion is that (A) cannot describe ${\mathfrak{M}}^{\mathrm{u}}$ associated with negative temperatures. The solution describes a network of microstates connected with each other by transitions at positive temperatures.

We now consider solution (B), which does not describe a network. In this solution, each microstate does not leave itself with certainty. Despite this difference, both (A) and (B) refer to the same macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ with same $\left\{p_k^{\mathrm{eq}}\right\}$ so they have the same entropy $S_{\mathrm{seq}}$, same energy $E_{\mathrm{seq}}$, and same EQ fluctuations in any quantity of interest. Therefore,

Claim 13.

Thermodynamically, there is no difference between the two solutions so there is no way to argue that (A) has the right physics and (B) has the wrong physics. As a consequence, both solutions must be accepted as correct, making transitions among microstates abstract with no physics. This point needs further investigation.

It should be emphasized that (B) exists for both kinds of system, interacting or isolated. However, as discussed in Section 7.3, this solution does make sense for an isolated system in which transition probabilities vanish, not only under the condition of detailed balance, but even in its absence for an isolated system, such as the entire isolated system in Figure 1b. Thus, we forced to make the same Claim 12 and draw the same Conclusion 5 as in Section 7.3.

Conclusion 6.

We finally conclude that the current popular schemes to determine $d{p}_k$ and $d_i{p}_k$ have serious limitations, some of which have never been reported before to the best of our knowledge. We recommend that we follow MNEQT and $\mu$NEQT that we have briefly describe here since they provide the exact formulation of $p_k$ in an enlarged state space ${\mathfrak{S}}_{\mathbf{Z}}$. From the exact knowledge of $p_k$, we can determine the process operabilities $d{p}_k,d_e{p}_k$, and $d_i{p}_k$.

8. Summary and Discussion

The current study was motivated by looking at the issues of irreversibility, the second law (SL), spontaneity, dissipation and its measure D in interacting systems in Figure 1b, along with various other tangential but extremely pertinent issues that have made the presentation terse and the discussion of topics is spread out over different sections. The study is an extension of the previous work [34] that dealt with isolated systems. The extension creates more complications that have to be dealt with, which we do here. We hope this section will provide a comprehensive view of our results.

In our recent publication [34], we have codified the current approach to thermodynamics in the form of BCGM proposal in Section 2.1, which begins with a mechanical deterministic system ${\Sigma }^{\mathrm{D}}$ with its Hamiltonian, which is then endowed with stochasticity to turn ${\Sigma }^{\mathrm{D}}$ into the thermodynamic system $\Sigma$. The idea in the original publication was to use the principle of mechanical equilibrium (Mec-EQ-P) instead of imposing SL to build a NEQ thermodynamics that allows for ${\Delta }_{\mathrm{i}}S⪋0$ in any process $\mathcal{P}$ and that justifies a generalized second law (GSL) ${\Delta }_{\mathrm{i}}Q\ge 0$, which supersedes SL and which is valid for all spontaneous processes in $\Sigma$ of any size, not just macroscopic as it is built on analytical mechanics, and any temperature, positive or negative, see Section 5.7. This makes the role of T extremely important in NEQ thermodynamics as it determines the signature of ${\Delta }_{\mathrm{i}}S$ in a spontaneous process for which ${\Delta }_{\mathrm{i}}Q$ is never negative. This results in ${\Delta }_{\mathrm{i}}S\le 0$ for negative temperature, see Equation (62b). As GSL can never be violated as long as analytical mechanics is valid, it results in a No-Go theorem for SL violation [95] that states that SL can never be violated.

We now list anddiscuss various new results of this study. We first discuss dissipation.

1. Our interest is more practical here to investigate irreversibility and dissipation in interacting systems to further their understanding. We first consider how a thermodynamic process can be carried out in lab. We conclude that it must be carried out in ${\mathfrak{S}}_{\mathbf{X}}$ by manipulating $\Sigma$ by the observable work parameter set $\mathbf{w}$ as discussed in Claim 2. The process can be reversed by reversing $\mathbf{w}$ to obtain ${\mathbf{w}}_{\mathrm{r}}$ as discussed in Section 2.3. This raises the issue of time reversal invariance, which is usually considered to operate on positions and momenta of particles and the equations of motion.

Here, we take a different perspective. As is clear form Equations (5) and (6), variations of $\mathbf{x}$ in the Hamiltonian has no effect on $d\mathcal{H}$. Thus, reversing time to reverse the equations of motion is not relevant in thermodynamics as the latter is a study of the variations in the energy, which is controlled by $\mathbf{w}$ or $\mathbf{W}$, as the case may be. Whether we consider the temporal evolution of a microstate or a macrostate, it is occurring in ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{S}}_{\mathbf{Z}}$ and not in the phase space. Thus, we conclude that time reversal in thermodynamic setting is reversing the process as noted in Claim 2, and not equations of motion as is commonly believed. For this reason, Liouville’s Theorem has no relevance in NEQ thermodynamics as the former operates in the phase space but does not incorporate microwork $d{W}_k$ in the state space. For this reason, it is oblivious to the irreversibility principle (Irr-P) $d_{\mathrm{i}}W\equiv$$d_{\mathrm{i}}Q$ in Equation (3). This principle is responsible for irreversibility but Liouville’s theorem does not capture it so this makes the theorem irrelevant.

2. Irr-P covered in Section 2.3 is the most important concept of the new thermodynamics (no-imposed SL) in [34], which allows any sign of $d_{\mathrm{i}}W$ or $d_{\mathrm{i}}Q$ as shown in Equation (3). It is only when Mec-EQ-P is brought into the mechanical analysis and is coupled with Proposition 1 that we are able to fix their signature and to obtain the generalized second law (GSL) given in Equation (59). We conclude that Equations (57) and (58) are also valid for interacting systems.

3. As dissipation D is determined by ${\Delta }_{\mathrm{i}}W$, see Conclusion 1, there is no contribution to D from irreversible macroheat transfer in interacting systems. This requires subtracting the irreversible entropy generation ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ as is clearly seen from Equations (41) and (44). Thus, irreversible macroheat transfer generates irreversible entropy generation ${\Delta }_{\mathrm{i}}{S}^{\left(\mathrm{h}\right)}$ but no dissipated work as $D=0$.

Iff we have an isothermal macroheat transfer, do we have D directly proportional to ${\Delta }_{\mathrm{i}}S$ but not otherwise as we see from Equation (64). This is the formulation of the Gouy-Stodola theorem (GST) in Equation (18).

4. In the current study, the irreversible macrowork ${\Delta }_{\mathrm{i}}W$ is identified with the dissipated work in the original spirit of Thomson [75] in Section 5.6. This requires modifying the forms of $E_{\mathrm{w}k}$ and $E_{\mathrm{w}}$ that now take the shape of closed loops, each with a source at a higher work function and a sink at a higher work function. Despite the loop formation, $D={\Delta }_{\mathrm{i}}W$ remains the measure of dissipation; see Conclusion 3.

We now discuss new results about various techniques to measure D that are normally used.

5. One of them is the Kullback-Leibler distance $D_{\mathrm{KL}}$ that we introduced in Section 2.5. It has been argued to be a measure of D by assuming it to be proportional to ${\Delta }_{\mathrm{i}}S$. This seems a surprising relationship. Here, we follow the original approach of Cohen and Mauzerall [70] by wondering if a non-thermodynamic quantity is attempted to describe a thermodynamic quantity. Their concern was about Jarzynski equality [68], but the approach and criticism are as valid here. We find the definition of $D_{\mathrm{KL}}$ in Equation (19) is non-thermodynamic, while ${\Delta }_{\mathrm{i}}S$ is a thermodynamic quantity. We have carefully examined this issue logic in Section 6 and demonstrate that it is indeed a non-thermodynamic quantity and that there is no justification to relate it to ${\Delta }_{\mathrm{i}}S$ even if the process is spontaneous and purely isothermal. In addition, as $D_{\mathrm{KL}}$ is a nonnegative quantity, it cannot be related to ${\Delta }_{\mathrm{i}}S<0$ for negative temperatures. The proper limitation of $D_{\mathrm{KL}}$ is captured in Equation (69), which establishes their non-equivalence explicitly.

6. The next one is the Fokker-Planck equation, which is nothing but a continuity equation for the conservation of probability as we discuss in Section 7.1. We use this continuity equation to argue that the equation can only determine the exchange probability change $d_{\mathrm{e}}{p}_k$ as it flows through the surface. Thus, the equation cannot determine $d_{\mathrm{i}}{p}_k$ that is needed to determine $d_{\mathrm{i}}S$, see Equation (38); see Claim 9. In Claim 10, we propose to generalize the Fokker-Planck equation to capture $d_{\mathrm{i}}{p}_k$ by adding a source term, but without modifying the probability current term.

7. We have also consider master equations in Section 7.2. We find that balanced master equations cannot be applied to a NEQ isolated system; see Remark 18. We also argue that detailed balanced master equations are not thermodynamically superior to balanced master equations; see Remark 19. We find that master equations cannot apply to isolated system as they cannot allow for change in $p_k$. This point is better illustrated in the next subsection dealing with detailed balance; see Equation (80b) for solution (B) and its discussion. This solution shows that the probabilities of each microstate remains unchanged as there are no transitions between microstates. But both solutions describe the same physics. This is a very disturbing feature; see Claim 13.

Conclusion 7.

The conclusion is that we must used thermodynamics in extended state space ($\mu$NEQT), where $p_k$’s are uniquely determined; see Conclusion 6.