Mechanical Foundations of the Generalized Second Law and the Irreversibility Principle

Abstract

We follow the Boltzmann-Clausius-Maxwell (BCM) proposal to establish the generalized second law (GSL) that is applicable to a system of any size, including a single particle system as our example establishes, and that supercedes the celebrated second law (SL) of increase of entropy of an isolated system. It is merely a consequence of the mechanical equilibrium (stable or unstable) principle (Mec-EQ-P) of analytical mechanics and the first law. We justify an irreversibility priciple that covers all processes, spontaneous or not, and having both positive and negative nonequilibrium temperatures temperatures T defined by ${(dQ/dS)}_E$. Our novel approach to establish GSL/SL is the inverse of the one used in classical thermodynamics and clarifies the concept of spontaneous processes so that $dS\ge 0$ for $T>0$ and $dS<0$ for $T<0$. Nonspontaneous processes such as creation of internal constraints are not covered by GSL/SL. Our demonstration establishes that Mec-EQ-P controls spontaneous processes, and that temperature (positive and negative) must be considered an integral part of dissipation.

1. Introduction

1.1. Nonequilibrium Entropy and the BCM Proposal

The concepts of entropy S (first introduced by Clausius) [1,2], temperature T, and heat $dQ$ play a central role in thermodynamics of a system $\mathsf{\Sigma }$. As such, it is very common to use them to distinguish a thermodynamic system $\mathsf{\Sigma }$ that is purely stochastic from its mechanical counterpart system ${\mathsf{\Sigma }}^{\mathrm{D}}$ that is purely deterministic by recognizing that the concepts of T and $dQ$ are novel to thermodynamics but are not applicable to the mechanical system ${\mathsf{\Sigma }}^{\mathrm{D}}$ [3]. The latter is traditionally taken to be described by a purely conservative Hamiltonian $\mathcal{H}$. While it is not usually the case, one can also associate an entropy to ${\mathsf{\Sigma }}^{\mathrm{D}}$ [4] that, however, always remains a constant of motion as it evolves. On the other hand, it usually changes in a thermodynamic process undertaken by the system $\mathsf{\Sigma }$, which explains that its importance in thermodynamics is due to its ability to change. We will always take $\mathsf{\Sigma }$ or ${\mathsf{\Sigma }}^{\mathrm{D}}$ to be stationary.

Although S plays important roles in diverse fields ranging from classical thermodynamics of Clausius [1,2,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26], quantum mechanics and uncertainty [27,28,29], black holes [30,31], coding and computation [32,33,34], to information technology [35,36,37,38], it does not seem to have a standard definition in all cases, even though it is well defined under equilibrium (EQ) conditions as extensively discussed in the literature; see, for example [8,12,39,40,41], where EQ entropy $S_{\mathrm{eq}}$ is uniquely determined as a state function of observables that are state variables introduced below, and is stationary in the sense that it has no explicit time dependence; observables that are state variables are allowed to have implicit time dependence in a process $\mathcal{P}$ so that $S_{\mathrm{eq}}$ can have an implicit time dependence during $\mathcal{P}$. Whether S has any physical significance for a NEQ macrostate $\mathfrak{M}$ of $\mathsf{\Sigma }$ has been a topic of extensive debate; see for example [39,40,41]. What is undisputable is that such an entropy must have an explicit time dependence.

Definition 1.

A sytem-intrinsic (SI) quantity of ${\mathsf{\Sigma }}^D$ or Σ depends only on the system such as its Hamiltonian, energy, entropy, temperature, etc.

We usually specify ${\mathsf{\Sigma }}^{\mathrm{D}}$ or $\mathsf{\Sigma }$ by SI-observables that include the energy E, the volume V, the number of particles $N$, etc. These observables are extensive and can be controlled from outside the system. We denote them by the set $\mathbf{X}=(E,V,N,\cdots )$ that identify the state space ${\mathfrak{S}}_{\mathbf{X}}$ formed by the observables. It defines the set of state variables of ${\mathsf{\Sigma }}^{\mathrm{D}}$, and also plays the same role for the corresponding thermodynamic system $\mathsf{\Sigma }$. We also introduce the observable work set $\mathbf{w}=(V,N,\cdots )$ by deleting E from $\mathbf{X}$. The equilibrium macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ and its entropy $S_{\mathrm{eq}}\left(\mathbf{X}\right)$ of $\mathsf{\Sigma }$ are uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$ so they have no explicit time dependence as noted above.

The way to relate $\mathsf{\Sigma }$ and ${\mathsf{\Sigma }}^{\mathrm{D}}$ is to allow stochasticity in the former by appending to ${\mathsf{\Sigma }}^{\mathrm{D}}$ probability as was first proposed by Clausius, Maxwell, and Boltzmann (the BCM proposal) [1,7,8,11,12,13,42,43], which we now describe. The stationary Hamiltonian of ${\mathsf{\Sigma }}^{\mathrm{D}}$ is written as $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right)$, where $\mathbf{x}\doteq \left\{({\mathbf{r}}_i,{\mathbf{p}}_i)\right\},i=1,2,\cdots ,N$ denotes the set of positions and momenta of N particles and the work set $\mathbf{w}$ acts as the parameter set specifying the Hamiltonian [3]. While time t can be a parameter in $\mathcal{H}$, we will always show it explicitly by not including in $\mathbf{w}$. The microstates ${\mathfrak{m}}_{k\mathrm{eq}}\left(\mathbf{w}\right)$ of $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathbf{w}\right)$ for different ${\mathbf{x}}_k$ denote instantaneous states of the macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ of $\mathsf{\Sigma }$ with probabilities $p_k^{\mathrm{eq}}$. We express this relationship as ${\mathfrak{M}}_{\mathrm{eq}}$ denoting the collection $\left\{{\mathfrak{m}}_{k\mathrm{eq}},p_k^{\mathrm{eq}}\right\}$.

Definition 2.

A system ${\mathsf{\Sigma }}^D$ or Σ is said to be uniform in the state space ${\mathfrak{S}}_{\mathbf{X}}$, if it is uniquely specified by its Hamiltonian with no explicit time dependence in that state space. If not, it is said to be nonuniform in ${\mathfrak{S}}_{\mathbf{X}}$ and requires explicit time dependence. Uniformity in ${\mathfrak{S}}_{\mathbf{X}}$ should not be confused with homogeneity in the phase space of or the real and momentum spaces occupied by ${\mathsf{\Sigma }}^D$ or Σ.

As $\mathbf{w}$ in $\mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right)$ refers to the entire system and not various parts of it, the Hamiltonian and its various microstates $\left\{{\mathfrak{m}}_k\left(\mathbf{w}\right)\right\}$ having microenergies $\left\{E_k\left(\mathbf{w}\right)\right\}$ describe a uniform ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$. If the Hamiltonian $\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$ has an explicit time dependence, it, $\left\{{\mathfrak{m}}_k(\mathbf{w},t)\right\}$, and $\left\{E_k(\mathbf{w},t)\right\}$ describe a nonuniform ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$; see Section 4 for justification, where an extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ is introduced, with $\mathbf{Z}$ containing additional state variables to $\mathbf{X}$; the corresponding work set is denoted by $\mathbf{W}$ which contains $\mathbf{w}$ as part of it.

The problem for NEQ entropy S arises because it is not clear how $\mathfrak{M}$ can be uniquely identified in terms of extensive state variables, compactly denoted by $\mathbf{Z}=\mathbf{Z}\left(t\right)$.

Definition 3.

All quantities pertaining to $\mathfrak{M}$ are identified as macroquantities, while those pertaining to ${\mathfrak{m}}_k$ are identified as microquantities and always have the index k of ${\mathfrak{m}}_k$.

Because of the lack of uniqueness, introducing $S\left(\mathfrak{M}\right)$ as a state function becomes nontrivial. Recently, we have been able to extend the classical concept of Clausius entropy from EQ states to NEQ states where irreversible entropy is generated [39,40]. That approach is an outgrowth of an earlier review [4] dealing with a possible source of stochasticity that is required in a thermodynamic system, even though its mechanics is completely deterministic so that heat and temperature have no mechanical analog.

Following the BCM proposal, we consider a Gibbs ensemble in which ${\mathfrak{m}}_k$ appears with probability $p_k$ to form a macrostate $\mathfrak{M}\doteq \left\{{\mathfrak{m}}_k,p_k\right\}$ in both cases. The entropy S in both cases is given by its Gibbs formulation [8,36] for $\mathsf{\Sigma }$ as the ensemble average of $(-ln{p}_k)$,

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

(1)

which is easy to verify, see for example [44], for any arbitrary collection $\left\{p_k\right\}$. The validity of the above S is based on the use of Stirling’s approximation. We will not be concerned about this error and accept the formula to be the definition of S. Different choices for $\left\{p_k\right\}$ result in different macrostates and averages for the same microstates.

Definition 4.

As constant $p_k$ refer to a purely mechanical system, stochasticity for us will always mean changing $p_k$ so that it will always refer to nonvanishing $\left\{d{p}_k\right\}$ and nonvanishing $dQ$ as discussed in the text. We use dissipation to always mean converting $dW\ge 0$ to $dQ\ge 0$, but use irreversibility to imply any signature of $dW$.

A constant $p_k,\forall k$, describes a pure mechanical system with constant E and S.

Remark 1.

The prefix micro- is used for quantities pertaining to microstates, while macro- is used for quantities pertaining to macrostates.

Consider uniform microstates $\left\{{\mathfrak{m}}_k\left(\mathbf{w}\right)\right\}$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$. We construct different macrostates $\mathfrak{M}$ of $\mathsf{\Sigma }$ in ${\mathfrak{S}}_{\mathbf{X}}$ by appending different choices of $\left\{p_k\right\}$. Only one choice $\left\{p_k^{\mathrm{eq}}\left(\mathbf{X}\right)\right\}$ gives the maximum entropy $S_{\mathrm{eq}}\left(\mathbf{X}\right)$ and identifies the EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ of $\mathsf{\Sigma }$ in ${\mathfrak{S}}_{\mathbf{X}}$. It is a unique and stationary macrostate. All other macrostates must denote nonequilibrium (NEQ) macrostates that are still identified as uniform macrostates because of $\mathsf{\Sigma }$; see Definition 2. Assuming time-dependent $p_k$ Definition 4 to allow for their temporal evolution, they must be expressed as $\mathfrak{M}(\mathbf{X},t)$ and having entropy $S(\mathbf{X},t)$. In addition to these uniform macrostates, there are other NEQ macrostates that originate from $\left\{{\mathfrak{m}}_k(\mathbf{w},t)\right\}$ and various possible sets $\left\{p_k\right\}$, all written as $\mathfrak{M}(\mathbf{X},t)$ with entropy $S(\mathbf{X},t)$. They are identified as nonuniform macrostates. Both macrostates are uniquely defined in ${\mathfrak{S}}_{\mathbf{X}}$ because of time dependence.

We will discover later in Section 6.2, see Remark 12, that uniform NEQ macrostates are thermodynamically irrelevant as they violate energy conservation during their temporal evolution for arbitrary $\left\{p_k\left(t\right)\right\}$. Therefore, from now on, such macrostates are not considered in this investigation. Only nonuniform macrostates associated with nonuniform $\mathsf{\Sigma }$ need to be considered that satisfy energy conservation, i.e., the first law. Therefore, such macrostates are simply identified as NEQ macrostates and denoted by $\mathfrak{M}(\mathbf{X},t)$ or simply $\mathfrak{M}$ from now on.

Remark 2.

The explicit time dependence in NEQ macrostates and their properties are due to explicit time dependence in nonuniform microstates ${\mathfrak{m}}_k(\mathbf{w},t)$; probabilities $p_k\left(t\right)$ may also have explicit time dependence.

We will see later that it is possible to identify an extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ formed by a set $\mathsf{\xi }$ of additional state variables, known as internal variables [10,26,45,46,47], with $\mathbf{Z}\doteq \mathbf{X}\cup \mathsf{\xi }$ so that $\mathfrak{M}(\mathbf{X},t)$ can be uniquely described as a macrostate $\mathfrak{M}\left(\mathbf{Z}\right)$ without any explicit time dependence in ${\mathfrak{S}}_{\mathbf{Z}}$ (Section 4). This requires that the corresponding Hamiltonian $\mathcal{H}(\left.\mathbf{x}\right|\mathbf{w},t)$ 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 $S_{\mathrm{eq}}\left(\mathbf{X}\right)$. As is well known, all NEQ macrostates are governed by the second law (SL), which is introduced below.

From what is said above, ${\mathcal{H}}_{\mathrm{eq}}\doteq \mathcal{H}\left(\left.\mathbf{x}\right|\mathbf{w}\right),{\mathfrak{m}}_{k\mathrm{eq}}\doteq {\mathfrak{m}}_k\left(\mathbf{w}\right)$ and $E_{k\mathrm{eq}}\doteq E_k\left(\mathbf{w}\right)$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$ that are uniform in $\mathbf{w}$ can also be labeled with the suffix “eq” for simplicity, even though the concept of equilibrium in thermodynamics is usually considered an average one over microstates. This should not cause any confusion, and is in the spirit of and consistent with the well-known result of equilibrium thermodynamics [11] that an EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}$is uniform in the state variable set $\mathbf{X}\doteq (E,\mathbf{w})$ that defines the EQ state space ${\mathfrak{S}}_{\mathbf{X}}$. As a consequence of uniformity, ${\mathfrak{M}}_{\mathrm{eq}}$and ${\mathfrak{m}}_{k\mathrm{eq}}$ can neither perform any mechanical work nor generate any mechanical power. Various SI EQ functions including entropy $S_{\mathrm{eq}}=S\left(\mathbf{X}\right)$ of ${\mathfrak{M}}_{\mathrm{eq}}=\mathfrak{M}\left(\mathbf{X}\right)$ remain time independent. (From now on, we will mostly suppress $\mathbf{w}$ for ${\mathsf{\Sigma }}^{\mathrm{D}}$ and $\mathsf{\Sigma }$ unless necessary as it is fixed.) In a NEQ macrostate $\mathfrak{M}$ of $\mathsf{\Sigma }$ that we are interested in, all SI-quantities acquire an explicit time dependence; see, for example, Equation (15) that captures this observation for microwork $d{W}_k\left(t\right)$ and micropower ${\mathsf{P}}_k\left(t\right)$ by nonuniform microstate ${\mathfrak{m}}_k\left(t\right)$ (Remark 1). This time-dependence is a result of nonuniformity among various disjoined but mutually interacting uniform subsystems $\left\{{\mathsf{\Sigma }}_l\right\}$ with Hamiltonians $\left\{{\mathcal{H}}_{l\mathrm{eq}}\doteq {\mathcal{H}}_l\left(\left.{\mathbf{x}}_l\right|{\mathbf{w}}_l\right)\right\}$ in terms of work variables ${\mathbf{w}}_l,l=1,2,\cdots$ as discussed in Section 4; nonuniformity results in internal flows among $\left\{{\mathsf{\Sigma }}_l\right\}$ that affect ${\mathbf{X}}_l\doteq (E_l,{\mathbf{w}}_l)$ so a NEQ $\mathfrak{M}\left(t\right)$ and ${\mathfrak{m}}_k\left(t\right)$ are individually nonuniform in $\left\{{\mathbf{X}}_l\right\}$ and $\left\{{\mathbf{w}}_l\right\}$, respectively. We also show there that nonuniformity in ${\mathfrak{m}}_k\left(t\right)$ can be described precisely in terms of a set $\mathsf{\xi }$ of internal variables [10,26,45,46,47] if we restrict the minimum size ${\lambda }_{\mathrm{E}}$ of subsystems to satisfy quasi-additivity of their energies, ${\lambda }_{\mathrm{E}}$ being determined by the range of inter-particle interactions; see the discussion following Equation (20b). This requires imposing minimum size restriction on $\mathsf{\Sigma }$ or ${\mathsf{\Sigma }}^{\mathrm{D}}$, which is denoted by ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$, for which $\mathbf{W}\doteq \left(\mathbf{w},\mathsf{\xi }\right)$ actsas the work variable and $\mathbf{Z}\doteq (E,\mathbf{W})$ as the state variable that determines a NEQ state space ${\mathfrak{S}}_{\mathbf{Z}}$ for ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$. Thus, we see that the explicit time dependence of $\mathfrak{M}\left(t\right)$ for nonuniform $\mathsf{\Sigma }$ in ${\mathfrak{S}}_{\mathbf{X}}$ is fully equivalent to the implicit dependence through $\mathsf{\xi }\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, see Equation (19); $p_k\left(t\right)$ is allowed to have explicit time as noted above in Remark 2.

To make subsystems’ entropies additive, we require S to be a state function $S\left(\mathbf{Z}\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$, which requires a further restriction on the minimum size as macroscopic (M) to capture quasi-independence of subsystems. This macroscopic system is identified as ${\mathsf{\Sigma }}_{\mathrm{M}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$ that require the state space ${\mathfrak{S}}_{\mathbf{Z}}$ to be uniquely specified; in contrast, $\mathsf{\Sigma }$ (or ${\mathsf{\Sigma }}^{\mathrm{D}}$) and ${\mathsf{\Sigma }}_{\mathrm{E}}$ (or ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$) are not uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$ and ${\mathfrak{S}}_{\mathbf{Z}}$, respectively.

For the benefits of readers, we collect various acronyms used in the text in the following table (

Table 1. Acronyms.

BCM	: Boltzmann-Clausius-Maxwell
Cl-Th	: Classical Thermodynamics for ${\mathsf{\Sigma }}_{\mathrm{M}}$
EQ	: Equilibrium
Gen-Th	: General Thermodynamics ($dW=dQ⋛0$) for $\mathsf{\Sigma }$
Gen-GSL-Th	: Gen-Th satisfying GSL ($dW=dQ\ge 0$) for $\mathsf{\Sigma }$
GSL	: Generalized second law $dQ\ge 0$ for $\mathsf{\Sigma }$
IC	: Internal constraint
Irr-P	: Irreversible Principle ($dW=dQ$) for $\mathsf{\Sigma }$
Mech-Eq-P	: Mechanical Equilibrium Principle for ${\mathsf{\Sigma }}^{\mathrm{D}}$
MicroBCM	: BCM applied to microstates of ${\mathsf{\Sigma }}^{\mathrm{D}}$
NEQ	: Nonequilibrium
Rest-Th	: Restriction Thermodynamics for ${\mathsf{\Sigma }}_{\mathrm{M}}$
SEQ	: Stable Equilibrium
SI	: System-intrinsic
SL	: Second Law $dS\ge 0$
UEQ	: Unstable Equilibrium
VGSL	: Violation of GSL ($dQ<0$) for $\mathsf{\Sigma }$
VSL	: Violation of SL ($dS<0$) for ${\mathsf{\Sigma }}_{\mathrm{M}}$
Viol-Th	: Gen-Th Violating SL ($dS<0$) for ${\mathsf{\Sigma }}_{\mathrm{M}}$
Viol-GSL-Th	: Gen-Th Violating GSL ($dQ<0$) for $\mathsf{\Sigma }$

).

1.2. The Second Law

The mechanical view of the universe cannot be reconciled with the facts of reality, which invariably consists of irreversible spontaneous processes controlled by the celebrated second law (SL) first proposed by Clausius, according to which there exists irreversible entropy generation, see Figure 1,

$$d_{\mathrm{i}}S\ge 0.$$

(2)

The stochasticity is appended in ${\mathsf{\Sigma }}^{\mathrm{D}}$ by introducing probability following the BCM proposal to capture thermodynamic irreversibility. Einstein [48] was convinced that classical thermodynamics (Cl-Th) as “…the only physical theory of universal content…will never be overthrown”, and Eddington [49] considered the fundamental axiom of the second law (SL) in terms of entropy S for an isolated system $\mathsf{\Sigma }$ as its cornerstone holds “…the supreme position among the laws of Nature”. There are two approaches to introduce SL in the BCM proposal. The first one is to “prove” it as was first done by Boltzmann through the H-theorem [43,50,51] involving molecular chaos assumption (Stosszahlansatz) or by using master equations [7,52] since then, both of which have their limitations [53]. The other one is to accept SL as an axiom in the axiomatic formulation of Cl-Th; see Callen [16] so there is no need to prove SL, but it does not help to unravel if SL ever fails. The violation of SL (VSL) is a topic of interest in this article as xpained later. We emphasize that both approaches deal only with macroscopically (M) large systems, which we always denote by ${\mathsf{\Sigma }}_{\mathrm{M}}$ [7,11,16,41] carrying the suffix M. It is a macroscopic system for which SL is assumed to be valid, but its violation (VSL) can occur for finite size systems. On the other hand, we will also be interested in a system of any size, even a system of a single particle, for reasons that will become clear below. We will use $\mathsf{\Sigma }$ or ${\mathsf{\Sigma }}^{\mathrm{D}}$ to denote systems with no size restriction so it includes ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ of an intermediate minimum size, and ${\mathsf{\Sigma }}_{\mathrm{M}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$ of a minimum macroscopic size. Thus, all results valid for $\mathsf{\Sigma }$ or ${\mathsf{\Sigma }}^{\mathrm{D}}$ also hold for intermediate and macroscopic systems.

Remark 3.

We use $\overline{\mathsf{\Sigma }}$ or ${\overline{\mathsf{\Sigma }}}^D$ to denote any of their three versions and their possible subsystems when we are not interested in specifying the state space ${\mathfrak{S}}_{\mathbf{X}}$ or ${\mathfrak{S}}_{\mathbf{Z}}$.

It is customary to claim SL to be a consequence of our “ignorance” of the intricate dynamics of constituent particles of the system that are too numerous for our gross senses to observe [54]. However, this idea is in some sense defeated by Maxwell’s demon [42], who overcomes this “ignorance” by a precise control of these particles and hopefully restore reversibility by decreasing entropy by violating SL and dethroning the latter as a supreme law of Nature. This has created an active industry of demolishing SL [55,56,57,58,59,60,61,62,63,64,65,66,67,68] by following the challenge proposed by Maxwell [42]. These attempts in $\mathsf{\Sigma }$, whose size ranges from mesoscopic to cosmological scales, cast doubt on Cl-Th of a NEQ macrostate $\mathfrak{M}\left(t\right)\doteq \left\{{\mathfrak{m}}_k\left(t\right),p_k\left(t\right)\right\}$; here, ${\mathfrak{m}}_k\left(t\right)$ is a microstate of the Hamiltonian $\mathcal{H}\left(t\right)$ with microenergy $E_k\left(t\right)$ appearing with probability $p_k\left(t\right),k=1,2,\cdots$. The t-dependence is made explicit (for reasons that will be clarified later), which we will suppress in the following unless needed for clarity. These recent attempts span widely different fields from information to biological thermodynamics. If correct, these violations suggest a new law, viz. the violation of SL (VSL), separate from SL. As physics is an experimental science, these violations not only pose a challenge to the current understanding of SL but give hope that we might be able to unravel the mystery behind SL as its root cause has never been formally identified so far, making it only an empirical law. As SL and VSL hold in Nature, each must hold under certain conditions, which are not known a priori and need to be identified. These conditions must be distinct to ensure that both laws are not obeyed simultaneously within the same process. This will make SL not a fundamental law of Nature that is obeyed under all conditions so its supremacy becomes questionable.

The current understanding of SL is that it applies to processes in a macroscopic system ${\mathsf{\Sigma }}_{\mathrm{M}}$, whether isolated as shown in Figure 1a or interacting with a medium as shown in Figure 1b. In the former case, the extensive change $dZ$ in any thermodynamic process $\mathcal{P}$ is solely due to internal processes so it is written as $d_{\mathrm{i}}Z$ to emphasize it. In the latter case, the extensive change $dZ\equiv d_{\mathrm{e}}Z+d_{\mathrm{i}}Z$ is partly due to exchange processes ($d_{\mathrm{e}}Z$) between $\mathsf{\Sigma }$ and the surrounding medium $\tilde{\mathsf{\Sigma }}$, and partly due to internal processes ($d_{\mathrm{i}}Z$); this partitioning also remains valid for $\mathsf{\Sigma }$ of any size. It is found experimentally [26] that $d_{\mathrm{i}}Z$ is always found to be of a fixed sign

$$d_{\mathrm{i}}Z\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\ge 0}{\le 0}}\right.,$$

(3)

depending on its definition. Once the definition is accepted, the signature cannot change during any real process. It is this fixed signature that is a signal of irreversibility of a process $\mathcal{P}$, see Definition 6, and results in SL during $\mathcal{P}$ as seen in Equation (2) for the extensive change $dS$. For a reversible process ${\mathcal{P}}_{\mathrm{rev}}$, inequalities in (3) turn into an equality [26]. It is clear, therefore, that we need to understand the rational for this fixed signature to justify the universality of SL.

Remark 4.

It should be emphasized that, being universal, SL is supposed to apply to all systems, regardless of their physical and chemical compositions. Therefore, we only consider a generic $\overline{\mathsf{\Sigma }}$ without paying any attention to its physical or chemical properties. This means that all we require is the existence of some deterministic Hamiltonian for ${\overline{\mathsf{\Sigma }}}^D$ but without specifying its form.

As $d_{\mathrm{i}}Z$ is the additional contribution to $d_{\mathrm{e}}Z$ and is determined by process irreversibility, it is most easily understood by focusing only an isolated system $\mathsf{\Sigma }$ as all processes associated with it are purely internal so they are not controllable from any outside agent. (However, there can also be irreversible contributions due to exchanges in an interacting systems as briefly discussed in Section 8.5). This extremely simplifies the discussion so our main focus in this article will be the isolated $\mathsf{\Sigma }$ for which the following Remark holds.

Remark 5.

As all exchanges $\left\{d_{\mathrm{e}}Z\right\}$for isolated Σ vanish identically,

$$d_{\mathrm{e}}Z\equiv 0,dZ\equiv d_{\mathrm{i}}Z,$$

(4)

so we avoid complications introduced by various exchanges. Accordingly, Equation (2) is expressed as

$$\left.\begin{array}{c}dS\ge 0\\ dE\equiv 0\end{array}\right\}\mathit{for}\mathit{isolated}\mathsf{\Sigma }.$$

(5)

Remark 6.

One additional benefit of considering isolated systems is that numerical simulations of isolated systems are probably the simplest possible ones, as one is not encumbered by interactions from the outside. It is easy to verify if the microstate energy $E_k$ determined by the Hamiltonian one chooses decreases to its minimum or increases to its maximum as the NEQ system evolves in time; see Figure 2.

The conventional SL, i.e., the inequality $dS\ge 0$ in Equation (5), as a nonstrict inequality does not require any knowledge of thermodynamic temperature T of the system. Therefore, the conventional wisdom [69] (see, for example) is that SL must not change regardless of whether T is positive or negative. This explains the common belief that SL is a universal law of Nature as discussed above so that a spontaneous process with $dS<0$ will never occur for any $T\gtrless 0$. We believe that this conventional wisdom has never been questioned, so we pose the following

Problem 1.

Do positive  and negative temperatures both satisfy SL as is commonly believed [69]?

We wish to emphasize that negative temperatures are not physically impossible [11,69,70,71] even in EQ situations, where they are defined by the standard thermodynamic relation

$$T\doteq \partial E/\partial S$$

(6)

in a region where E decreases with S or S decreases with E. This definition is later extended to NEQ macrostates in Equation (39).

1.3. Spontaneous Processes

We now discuss the important issue of spontaneity, which results in a spontaneous process. Its absence results in a nonspontaneous process controlled by an outside agent.

Definition 5.

A spontaneous process in Σ occurs naturally (on its own) without any assistance from the outside of Σ. In this case, Σ is said to possess spontaneity.

To clarify the meaning of a spontaneous process, we first remark that the sudden expansion of a gas, which is initially confined in a given volume $V_1$ in a cylinder with a removable partition, to a larger volume $V_2>V_1$ by abruptly removing the partition outside the cylinder, is such a process. Once the partition is removed, there is no way to internally control expansion unless the partition is moved back by an external agent such as a hand to a volume between $V_1$ and $V_2$. Another version undergoing a similar process is a system ${\mathsf{\Sigma }}_{\mathrm{ic}}$ consisting of two subsystems ${\mathsf{\Sigma }}_1$ and ${\mathsf{\Sigma }}_2$ that share a common wall shown by the red partition in Figure 3a. First consider an inert partition so that ${\mathsf{\Sigma }}_1$ and ${\mathsf{\Sigma }}_2$ remain noninteracting. If initially they are not in equilibrium with each other, they will remain so permanently due to the inert partition that creates an internal constraint (IC) [16]. We denote this nonuniform state of $\mathsf{\Sigma }$ by ${\mathsf{\Sigma }}_{\mathrm{ic}}$ to remind us of the above IC. As soon as the partition (IC) is removed, there will be some internal flow that is generated spontaneously to make $\mathsf{\Sigma }$ more and more uniform until finally, it asymptotically becomes fully uniform and time independent. The latter is an example of the stable equilibrium (SEQ) macrostate. Once $\mathsf{\Sigma }$ arrives in this macrostate, it will remain there forever. Any thermodynamic fluctuation will bring it back to it due its stability by the spontaneous flow, which is itself intrinsic to $\mathsf{\Sigma }$, and is driven by SL.

All processes in $\mathsf{\Sigma }$ are system-intrinsic (SI) and generated internally to change $E_k$, and must be governed by dynamics within the system. They are spontaneous Remark 7 and are governed by SL in Cl-Th, which is only satisfied by a macroscopic system ${\mathsf{\Sigma }}_{\mathrm{M}}$ [7,11,16,41]. Despite this, as Keizer [72] notes: “...the second law has never been justified on the basis of mechanical laws”. We wish to overcome this shortcoming so we focus on dynamical evolution in this study.

Remark 7.

All processes in Σ are synonymously identified as spontanious, internal, or SI processes in that none of them is affected by anything exterior to it. Thus, we use these terms interchangeably.

Any SL-violation, being uncommon and controversial, will most probably result in puzzling outcomes, not all of which are recognized or discussed so far in the current literature. Also, there is no serious inquiry into violation thermodynamics (Viol-Th) supporting it because the root cause of SL and any special stochasticity Definition 4 for its validity are not understood; however, see [69]. A nonspontaneous process is not internally generated and requires external agent $\tilde{\mathsf{\Sigma }}$ so cannot occur in $\mathsf{\Sigma }$. A nonspontaneous process such as the intervention carried out by Maxwell’s demon [42] that is external to ${\mathsf{\Sigma }}^{\mathrm{D}}$ or $\mathsf{\Sigma }$ only creates an internal constraint of Callen [16] as discussed later, whose removal, however, is controlled by SL. On the other hand, a nonspontaneous process is not controlled by SL.

The investigation of situations with $dS<0$ that results in VSL has become an important activity recently. Such violations will dethrone SL from being a universal law of Nature for macroscopically large ${\mathsf{\Sigma }}_{\mathrm{M}}$; see Remark 4. In order to carefully understand dethroning, we need to determine conditions necessary for $dS>0$ and $dS<0$, respectively. Thus, we will need to construct a new thermodynamics, to be called general thermodynamics (Gen-Th) in which the second law is not enshrined. In contrast, it is enshrined in Cl-Th by making it an axiom [16] or by assuming it by “proving” it as discussed above.

Various forms of thermodynamics that we consider in this study are listed in the following table and properly explained at appropriate places (

Table 2. Various Forms of Thermodynamics in the text.

Gen-Th	(S1), (S2), (S3) for $\overline{\mathsf{\Sigma }}$, $dW=dQ⋛0$
Cl-Th	Gen-Th and SL axiom for ${\mathsf{\Sigma }}_{\mathrm{M}}$
Rest-Th	Gen-Th for ${\mathsf{\Sigma }}_{\mathrm{M}}$ with state function S
Viol-Th	Gen-Th for $\overline{\mathsf{\Sigma }}$ with $dS<0$
Gen-GSL-Th	Gen-Th and (S4) for $\overline{\mathsf{\Sigma }}$ with $dQ\ge 0$
Viol-GSL-Th	Gen-Th and (S4) for $\overline{\mathsf{\Sigma }}$ with $dQ<0$

).

1.4. SL, Stability and Dissipation

There are indirect but strong arguments for a deep connection of SL with thermodynamic stability in a stable (s) system ${\mathsf{\Sigma }}^{\mathrm{s}}$ [7,8,11,13,16,24,41,73,74,75,76,77], in which a stable NEQ macrostate ${\mathfrak{M}}^{\mathrm{s}}\left(t\right)$ with a lower bound in energy, the continuous blue curve in Figure 2, asymptotically converges along blue arrows to a unique and stable equilibrium (SEQ) macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ [13] that is stationary. According to SL, temporal evolution of any arbitrary ${\mathfrak{M}}^{\mathrm{s}}\left(t\right)$ will eventually terminate into ${\mathfrak{M}}_{\mathrm{seq}}$ in such systems. As the mechanical evolution of a microstate ${\mathfrak{m}}_k^{\mathrm{s}}$ of ${\mathsf{\Sigma }}^{\mathrm{sD}}$ is controlled by (deterministic) Hamiltonian equations of motion, it always undergoes oscillations about its minimum ${\mathfrak{m}}_{k\mathrm{seq}}$. The dynamics of ${\mathfrak{M}}^{\mathrm{s}}\left(t\right)$ in ${\mathsf{\Sigma }}^{\mathrm{s}}$, therefore, cannot be deterministic [72] as noted above. It is customary to call this dynamics dissipative, but the term does not seem to have a well-defined meaning, and most certainly has not been clearly defined for an isolated system. Various statements are available in the literature to describe dissipation that was first introduced by Thomson [78], some of which are:

• It is the conversion of mechanical energy of the system into thermal energy with an associated increase in entropy.
• It is decoherence in energy, i.e., conversion of coherent or directed energy flow into an indirected or more isotropic distribution of energy.
• Its main effect is an increase in the temperature of the system.
• It is often used to describe ways in which energy is wasted (lost to $\tilde{\mathsf{\Sigma }})$ in the form of heat.
• Planck [79], instead of defining it, cites friction as its prime example.

All these statements, some of which may be extended to isolated systems with some leap of imagination, assume that dissipation, which is always positive, essentially causes irreversibility and produces entropy as noted above. 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. What distinguishes ${\mathsf{\Sigma }}^{\mathrm{s}}$ from ${\mathsf{\Sigma }}^{\mathrm{sD}}$ is the presence of stochasticity in the former so dissipation is an average concept over stochasticity, not applicable to microstates that evolve under deterministic Hamiltonian equations of motion. It is not clear how a dissipative force can be identified in a spontaneous heat flow (from hot to cold), which is also irreversible. As dissipation is related to some form of energy transformation, there is no direct connection between it and entropy, which is not a form of energy. That necessarily requires introducing some sort of temperature in the discussion, which is not properly defined for nonequilibrium macrostates in above statements. As is known, there are many different kinds of NEQ temperature definitions available in the literature, depending on the school of NEQ thermodynamics; see for example [53] (Section 1) on this point. However, the main drawbacks of these statements are that they are not applicable to an isolated system for which $dE=0$; see Equation (5). For example, if E is treated as the mechanical energy, then no part of it can be converted into thermal energy as $dE=0$. There cannot be a directed flow into the isolated system. Thus, the conventional understanding of dissipation is not only not precise but also not very general, even though the basic idea is commonly accepted that dissipation brings about irreversible processes in ${\mathfrak{M}}^{\mathrm{s}}$ so it asymptotically converges to the stable equilibrium macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ in time [16]. This is an experimental fact, which we express by saying that the spontaneous evolution of ${\mathfrak{M}}^{\mathrm{s}}$ is controlled by the sink ${\mathfrak{M}}_{\mathrm{seq}}$. We generalize this experimental observation to also include unstable macrostate ${\mathfrak{M}}^{\mathrm{u}}$, whose spontaneous evolution is controlled by its (unstable) equilibrium macrostate ${\mathfrak{M}}_{\mathrm{ueq}}$, the source, to be introduced later and shown in Figure 2, in the form of the Fundamental Proposition, see Proposition 1, which also includes the spontaneous evolution of their microstates ${\mathfrak{m}}_{k\mathrm{eq}}$, and will be justified in analytical mechanics later.

Proposition 1.

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

Dissipation in an isolated system is further considered in Section 6, and for an interacting system in the form of Claim 26 is considered in Section 8.5.

1.5. Equilibrium Point, Uniformity and Temporal Evolution

By definition, ${\mathfrak{M}}_{\mathrm{eq}}\left(\mathbf{X}\right)$ 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 Hamiltonian $\mathcal{H}\left(\mathbf{w}\right)$; see Definition 2. In contrast, any NEQ macrostate $\mathfrak{M}(\mathbf{X},t)$, its macroenergy $E(\mathbf{w},t)$, and entropy $S(\mathbf{X},t)$ have explicit time dependence that is also reflected in the set $\left\{{\mathfrak{m}}_k\left(t\right),p_k\left(t\right),E_k\left(t\right)\right\}$ as discussed above. As we will see in Section 4, the explicit time dependence is due the NEQ system being nonuniform.

We will suppress $\mathbf{X}$ in the following unless clarity is needed. Temporal evolution of $\mathfrak{M}\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{X}}$ is controlled by ${\mathfrak{M}}_{\mathrm{eq}}$ of energy E and macrowork function $E_{\mathrm{w}}\left(t\right)$ defined later in Equation (36b); the temporal form of $E_{\mathrm{w}}\left(t\right)=(E_{\mathrm{w}}^{\mathrm{s}}\left(t\right),E_{\mathrm{w}}^{\mathrm{u}}\left(t\right))$ is shown by the solid blue and red curves in Figure 2, respectively. The microstate energies $E_k^{\mathrm{s}}\left(t\right)$ and $E_k^{\mathrm{u}}\left(t\right)$ are also shown by the dashed-dot blue and red curves in Figure 2, respectively. The extremum points of these curves represent the equilibrium points $E_{\mathrm{seq}}$ and $E_{\mathrm{ueq}}$ for $E_{\mathrm{w}}\left(t\right)$, and $E_{k\mathrm{seq}}$ and $E_{k\mathrm{ueq}}$ for $E_{k\mathrm{w}}\left(t\right)$, respectively. During evolution as $t\to \infty$, $E_{\mathrm{w}}^{\mathrm{s}}\left(t\right)$ runs towards $E_{\mathrm{seq}}$ along the blue arrows, and $E_{\mathrm{w}}^{\mathrm{u}}\left(t\right)$ runs away from $E_{\mathrm{ueq}}$ along the red arrows. The former is the behavior of the energy for a stable but NEQ macrostate that terminates in a stable equilibrium (SEQ) macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ that is stationary and uniform, and is normally considered in Cl-Th. The latter is that of an unstable and nonuniform macrostate, which runs away from an unstable equilibrium (UEQ) macrostate ${\mathfrak{M}}_{\mathrm{ueq}}$ that is uniform but is rarely considered. Similarly, as $t\to \infty$, $E_k^{\mathrm{s}}\left(t\right)$ runs towards $E_{k\mathrm{seq}}$ along the blue arrows, and $E^{\mathrm{u}}\left(t\right)$ runs away from $E_{k\mathrm{ueq}}$ along the red arrows.

Definition 6.

By irreversibility of a process $\mathcal{P}$ is meant the fixed signature of $dZ$ during any process $\mathcal{P}$.

Claim 1.

A nonspontaneous process is not internallygenerated and requires an external medium $\tilde{\mathsf{\Sigma }}$ so cannot occur in ${\mathsf{\Sigma }}^D$ or Σ.

1.6. Our Goals

1.6.1. Generalized Second Law (GSL) for $\mathsf{\Sigma }$

In NEQ thermodynamics, ${\mathfrak{S}}_{\mathbf{X}}$ is the most convenient state space as it requires no knowledge of any internal nonuniformity or stochasticity that is hidden from the observer and may not always be known, but requires an explicit time dependence such as in $E\left(t\right)$ or $E_{\mathrm{w}}\left(t\right)$. Despite this, the choice of ${\mathfrak{S}}_{\mathbf{Z}}$ helps to explain not only the origin of explicit time dependence resulting from nonuniformity but also to justify the nonstandard definition of T in Equation (7) as it becomes identical to the conventional definition in Equation (39) of NEQ temperature in the extended state space ${\mathfrak{S}}_{\mathbf{Z}}$. The first law and macroheat $dQ$, see Section 6, can be introduced for any size $\overline{\mathsf{\Sigma }}$, and play a central role in proving the generalized second law (GSL); both can also be directly introduced without identifying T. But we need T to relate $dQ$ in GSL to $dS$ to also directly prove the second law (SL) $dS\ge 0$, and to shed light on the possible significance of $dS<0$ in its violation, which is one of our goals. In most cases of interest shown by the solid blue curve ${\mathfrak{M}}^{\mathrm{s}}$ [11], nonuniform $\overline{\mathsf{\Sigma }}$ attempts to thermodynamically minimize nonuniformity in $\left\{{\mathbf{X}}_l\right\}$ by reducing internal flows and become uniform as S continue to increase as explained below.

We prove GSL, which must not be confused with the one proposed by Beckenstein [80,81] for black holes, by considering $\mathfrak{M}$ in an isolated NEQ discrete system $\overline{\mathsf{\Sigma }}$ [82,83] of any size, not necessarily ${\mathsf{\Sigma }}_{\mathrm{M}}$. It unravels the mystery of and provides deeper insight into the root cause of SL ($dS\ge 0$) and its apparent violation $dS<0$. We make the following important

Remark 8.

The formulations of SL by Clausius and by Kelvin-Planck are not applicable to isolated systems as they cannot perform any cyclic operation of an engine.

The equality for $dS$ in Equation (5) occurs in the equilibrium (EQ) macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ having maximum value of entropy for fixed observable $\mathbf{X}$ in the state space ${\mathfrak{S}}_{\mathbf{X}}$, with S always given by Equation (1). The intermediate size of the system ${\mathsf{\Sigma }}_{\mathrm{E}}$ is determined by the interaction range ${\lambda }_{\mathrm{E}}$ to ensure $E_k$-additivity in a nonuniform $\mathfrak{M}$ in the state space ${\mathfrak{S}}_{\mathbf{X}}$; see Section 4. A NEQ $\mathfrak{M}$ can always be described in ${\mathfrak{S}}_{\mathbf{X}}$ by $S(\mathbf{X},t),E(\mathbf{w},t),E_{\mathrm{w}}(\mathbf{w},t)$, etc. all having explicit t dependence, but there is another ways to specify nonuniform $\mathfrak{M}$ using the extended state space ${\mathfrak{S}}_{\mathbf{Z}}$ spanned by $\mathbf{Z}$ [40,44,45,46,53,84]. Here, one uniquely specifies $\mathfrak{M}=\mathfrak{M}(\mathbf{w},\mathsf{\xi })=\mathfrak{M}\left(\mathsf{\xi }\right),E(\mathbf{w},\mathsf{\xi })=E\left(\mathsf{\xi }\right),E_{\mathrm{w}}(\mathbf{w},\mathsf{\xi })=E_{\mathrm{w}}\left(\mathsf{\xi }\right)$, etc., where $\mathsf{\xi }$ contains $\iota$ components, with $\iota$ increasing and S decreasing with nonuniformity of $\mathfrak{M}\left(\mathsf{\xi }\right)$. There is no explicit time dependence in $S\left(\mathbf{Z}\right),E\left(\mathsf{\xi }\right),E_{\mathrm{w}}\left(\mathsf{\xi }\right)$, etc. in ${\mathfrak{S}}_{\mathbf{Z}}$ that provides some benefit. We define $\mathsf{\xi }$ so that $\mathsf{\xi }$ $=0$ in ${\mathfrak{M}}_{\mathrm{eq}}$ for which $\iota =0$. We also assume that $\mathsf{\xi }$ is the same for all ${\mathfrak{m}}_k$’s; extension to different ${\mathsf{\xi }}_k$ is easily done. As explicit time dependence for nonuniform $\mathsf{\Sigma }$ in ${\mathfrak{S}}_{\mathbf{X}}$ is equivalent to the implicit time dependence in ${\mathsf{\Sigma }}_{\mathrm{E}}$ and ${\mathsf{\Sigma }}_{\mathrm{M}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$, we can adopt either approach. We mostly use $\mathsf{\Sigma }$ and the first approach for its simplicity to desribe $\mathfrak{M}$. We use $\overline{\mathsf{\Sigma }}$ for any system when the state space is not specified as stated earlier.

1.6.2. NEQ Temperature for $\mathsf{\Sigma }$

As E is fixed, we introduce a NEQ temperature as the ratio $dQ/dS$ in any infinitesimal internal process within $\overline{\mathsf{\Sigma }}$; see Equation (7). While this definition is unconventional, it is indeed the conventional definition ${\left(\partial E/\partial S\right)}_{\mathsf{\xi }}$ for ${\mathsf{\Sigma }}_{\mathrm{M}}$ as seen later from Equations (30) and (39). Thus, we take T above to represent the temperature for any number of particles in $\overline{\mathsf{\Sigma }}$.

1.6.3. Dissipation

One of our goals is to properly define dissipation in the isolated system $\mathsf{\Sigma }$, which does not require introduction of any “temperature-like” concept but is intimately intertwined with irreversibility. Dissipation for an interacting system is postponed to Section 8.5. What causes the convergence for ${\mathfrak{M}}^{\mathrm{s}}$ is not understood theoretically as mentioned above. The traditional approach in Cl-Th is to postulate the existence of ${\mathfrak{M}}_{\mathrm{seq}}$ in the state space of ${\mathsf{\Sigma }}_{\mathrm{M}}$, postulate SL in the form of entropy maximum principle [16] (see for example, Chapter 4]), and then prove thermodynamic stability [11] by following ${\mathfrak{M}}^{\mathrm{s}}$-evolution (${\mathfrak{M}}^{\mathrm{s}}\to {\mathfrak{M}}_{\mathrm{seq}}$ as $t\to \infty$) described above. However, ${\mathfrak{M}}^{\mathrm{u}}$-evolution is never considered in Cl-Th, which we overcome in this study for the following reason. As SL results in stability, a SL-violation strongly suggests, but not yet verified, thermodynamic instability for its cause so we also consider rarely studied unstable macrostate ${\mathfrak{M}}^{\mathrm{u}}$emergingout of ${\mathfrak{M}}_{\mathrm{ueq}}$ along red arrows; however, see [69] as exception. All arrows in Figure 2 point towards increasing time t, during which we need to investigate the behavior of entropy (increasing or decreasing) as ${\mathfrak{M}}^{\mathrm{s}}$ or ${\mathfrak{M}}^{\mathrm{u}}$ becomes less or more nonuniform. This will allow us to identify the conditions required for SL and VSL in the isolated system $\mathsf{\Sigma }$, which is our main goal.

1.7. New Results

The following results relate to an isolated system ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ and $\overline{\mathsf{\Sigma }}$, which can be of any size. We follow the BCM proposal, which requires ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ to follow analytical mechanics [85,86]. We denote a macroscopic thermodynamic system by a suffix M as ${\overline{\mathsf{\Sigma }}}_{\mathrm{M}}$.

R1

The Hamiltonian $\mathcal{H}$ of a nonuniform system ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ has explicit time dependence as shown in Section 4; the time dependency is absent for a uniform system, which is described by a stationary $\mathcal{H}$.

R2

The Hamiltonian $\mathcal{H}$ of a nonuniform system ${\mathsf{\Sigma }}^{\mathrm{D}}$ does not carry explicit time dependence if it is expressed in a suitably chosen ${\mathfrak{S}}_{\mathbf{Z}}$ despite being nonuniform as shown in Section 4.

R3

A nonuniform microstate ${\mathfrak{m}}_k$ and its microenergy $E_k$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ have explicit time dependence; the time dependency is absent for a uniform system. It follows from R1.

R4

A nonuniform microstate ${\mathfrak{m}}_k$ and its microenergy $E_k$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$ do not carry explicit time dependence if they are expressed in a suitably chosen ${\mathfrak{S}}_{\mathbf{Z}}$ despite nonuniformity. It follows from R2.

R5

A nonuniform macrostate $\mathfrak{M}$, its macroenergy E and entropy S of $\mathsf{\Sigma }$ in ${\mathfrak{S}}_{\mathbf{X}}$ have explicit time dependence; the time dependence disappears for a uniform system. It follows from R1.

R6

A nonuniform macrostate $\mathfrak{M}$, its macroenergy $E$and entropy S of $\mathsf{\Sigma }$ do not carry explicit time dependence if they are expressed in a suitably chosen ${\mathfrak{S}}_{\mathbf{Z}}$ despite nonuniformity. It follows from R2.

R7

The first law $dE=dQ-dW$ applies to any thermodynamic system $\overline{\mathsf{\Sigma }}$ of any size, not necessarily macroscopic, as discussed in Section 6.

R8

An isolated system satisfies the irreversibility principle $dQ\equiv dW$ so that the sign of $dQ$ is determined by the mechanical work $dW$ during evolution, as discussed in Section 6.

R9

During evolution following analytical mechanics, $dW\ge 0$, which proves the generalized second law (GSL)

$$dQ\ge 0$$

for $\mathsf{\Sigma }$, which include ${\mathsf{\Sigma }}_{\mathrm{E}}$ and ${\mathsf{\Sigma }}_{\mathrm{M}}$. GSL cannot be violated without violating analytical mechanics.

R10

Introducing a NEQ temperature

$$T\doteq {\left.dQ/dS\right|}_E,$$

(7)

we see that

$$TdS\ge 0$$

so $dS\ge 0$ for $T>0$ and $dS<0$for $T<0$ for $\overline{\mathsf{\Sigma }}$ of any size, including ${\overline{\mathsf{\Sigma }}}_{\mathrm{M}}$. We thus obtain the extension of SL for $\overline{\mathsf{\Sigma }}$ in terms of this temperature, which is always defined for and which covers both positive and negative temperatures. We express this extension by GSL/SL for $\overline{\mathsf{\Sigma }}$.

R11

Thus, $dS<0$ is not a violation of GSL for $\overline{\mathsf{\Sigma }}$ and, therefore, SL for ${\overline{\mathsf{\Sigma }}}_{\mathrm{M}}$ as $dQ\ge 0$ for $T<0$.

1.8. Layout

Section 2 introduces our new approach that is reverse of the tradition approach of Cl-Th. It describes how the new approach allows us to prove a generalization of SL in the form of GSL that is valid for $\overline{\mathsf{\Sigma }}$ of any size, not necessarily a macroscopic one, but also covers both positive and negative temperatures introduced in Equation (7). Section 3, Section 4 and Section 5 are based on analytical mechanics only, and deal with microstates $\left\{{\mathfrak{m}}_k\left(t\right)\right\}$ of a nonuniform deterministic Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|t\right)$. Section 3 deals with it in ${\mathfrak{S}}_{\mathbf{X}}$ to follow the consequences of the explicit time dependence in $\left\{E_k\left(t\right)\right\}$, which is related to the implicit time-dependence of $\mathsf{\xi }\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ introduced in Section 4. Section 5 introduces the Mechanical Equilibrium Principle (Mec-EQ-P) of Energy [85], which extends the analytical mechanics aspect of the BCM proposal to microstates. $\mathfrak{M}\left(t\right)$ and their thermodynamics in ${\mathfrak{S}}_{\mathbf{X}}$ and${\mathfrak{S}}_{\mathbf{Z}}$ are considered next. In Section 6, the stochastic aspect of the BCM proposal is introduced with any arbitrary $p_k$ to identify a general NEQ macrostates $\mathfrak{M}$, and develop its first law for any $\overline{\mathsf{\Sigma }}$. This results in the General thermodynamics (Gen-Th) but without invoking Mec-EQ-P for any possible $\mathfrak{M}$ so Gen-Th allows for both $dQ\ge 0$ and $dQ<0$. An important irreversibility principle (Irr-P) is introduced in Section 6.2, where we also introduce the interesting macrowork function $E_{\mathrm{w}}$ in Equation (36b) depicted in Figure 2 to provide a macroscopic understanding of GSL. The Irr-P forms the heart of GSL, which is formulated in Section 8.3 and is encoded by Lemma 1 and Theorem 2 by focusing on microsctates. A simpler, alternative derivation of GSL also follows from considering $E_{\mathrm{w}}$; however, $E_{\mathrm{w}}$ is not necessary in proving GSL but helps in a macroscopic appreciation of GSL. The temperature plays an important role in connecting GSL with SL to $\overline{\mathsf{\Sigma }}$. We find that with GSL, we are able to extend classical formulation of SL to $\overline{\mathsf{\Sigma }}$ of any size. A simple example of a single particle having two microstates is considered to demonstrate how Gen-Th applies in Section 6.4. Section 7 puts macroscopic size restriction and considers ${\mathsf{\Sigma }}_{\mathrm{M}}$ to ensure that S is additive and a state function in ${\mathfrak{S}}_{\mathbf{Z}}$. We also consider various scenarios of SL for positive and negative temperatures. Section 8 contains a discussion of new results, extension to interacting system in a medium, and a brief summary. One of the final conclusions is that dissipation is not proportional to irreversible entropy generation, a surprising result.

2. Reversing Traditional Approach

In order to determine the required conditions for SL and its violation in ${\mathsf{\Sigma }}_{\mathrm{M}}$, we need to go beyond Cl-Th, and construct a general thermodynamics (Gen-Th), which will be unconstrained by SL so that it allows both SL and VSL. To understand the importance of the size of the system, we do not impose the macroscopic constraint required for ${\mathsf{\Sigma }}_{\mathrm{M}}$, and focus on $\mathsf{\Sigma }$ so that it can denote any isolated system with any number of particles $N=1,2,\cdots$. The corresponding mechanical system ${\mathsf{\Sigma }}^{\mathrm{D}}$ similarly has no size restriction. A macroscopically large system is always denoted by ${\mathsf{\Sigma }}_{\mathrm{M}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$ as the case may be.

To unravel various conditions for SL and VSL in ${\mathsf{\Sigma }}_{\mathrm{M}}$, we reverse the traditional approach taken in Cl-Th, see Section 1.6. The new approach is fully described below, and can be easily contrasted with that in Cl-Th as follows

$$\begin{array}{c}\mathrm{Cl}\text{-}\mathrm{Th}\left({\mathsf{\Sigma }}_{\mathrm{M}}\right):(\mathrm{S}1,\mathrm{S}2,\mathrm{S}3) \mathrm{and} \mathrm{SL}⟹\mathrm{Stability}\hfill \\ \mathrm{Gen}\text{-}\mathrm{Th}\left(\mathsf{\Sigma }\right):(\mathrm{S}1,\mathrm{S}2,\mathrm{S}3) \mathrm{and} \mathrm{Stability}⟹\mathrm{GSL}\hfill \end{array},$$

(8)

with (S1, S2, S3) being the parts of the four steps that form the core of our analysis. We see that the reversal is between the roles played by SL and Stability. We we now describe the four steps.

(S1) Describe ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ mechanically by its deterministic $\mathcal{H}$ by specifying $\left\{{\mathfrak{m}}_k,E_k\right\}$ and SI-microwork $\left\{d{W}_k\right\}\doteq -\left\{d{E}_k\right\}$ as described in Section 3. This is the first half of the BCM proposal.

(S2) Introduce stochasticity by appending a probability $p_k$ to ${\mathfrak{m}}_k$, which is the second half of the BCM proposal to capture dissipation in $\mathfrak{M}$ of $\overline{\mathsf{\Sigma }}$, and to identify various ensemble averages $〈•〉$

$$〈•〉\doteq {\sum }_k{p}_k{(•)}_k,$$

(9)

where ${(•)}_k$ refers to the microstate value. An example is the energy E

$$E\doteq {\sum }_k{p}_k{E}_k,$$

(10)

The Gibbs formulation [8] for S in Equation (1) is another ensemble average.

Remark 9.

It should be emphasized that the ensemble average is defined for any Σ, regardless of its size. For example, it is as valid for the toss of a single coin that is repeated many number of times; the many attempts form the ensemble. This observation will prove useful when we discuss the first law in Section 6.

(S3) Introduce the first law$dE=dQ-dW$ in Section 6 during any infinitesimal change of $\overline{\mathsf{\Sigma }}$ to determine the allowed change $\left\{d{p}_k\right\}$ and to identify the physics behind $dQ\doteq {\sum }_k{E}_{k}d{p}_k$, a stochastic quantity Definition 4, and $dW\doteq -{\sum }_k{p}_{k}d{E}_k={\sum }_k{p}_{k}d{W}_k$, a mechanical quantity. The law immediately leads to the irreversibility principle (Irr-P) [40,44] expressed by the identity

$$dQ\equiv dW⪌0$$

(11)

that intertwines these seemingly unrelated macroquantities similar to intertwining of electric and magnetic fields in the Maxwell theory; see also Section 8.3. While they have the same sign, their sign is not fixed yet. We use (S1), (S2), the first law involving $dQ$ and $dW$, and $dS$ to formulate a generic version of NEQ statistical thermodynamics to be called General Thermodynamics (Gen-Th) for any possible $\mathfrak{M}$ by taking arbitrary $\left\{p_k\right\}$ for $\overline{\mathsf{\Sigma }}$; see Equation (13).

(S4) Invoke the mostly overlooked fundamental principle Mechanical Equilibrium Principle of Energy (Mec-EQ-P) [85,86] introduced in Section 5 that controls (spontaneous) evolution of ${\mathfrak{m}}_k,\forall k$, in $\mathsf{\Sigma }$ Remark 7. This extends the BCM proposal to microstates $\left\{{\mathfrak{m}}_k\right\}$ (MicroBCM) that fixes the nonnegativesignature of $d{W}_k$ in Lemma 1, and to directly establish the generalized second law (GSL) that controls dissipationin $\overline{\mathsf{\Sigma }}$ in Theorem 2:

$$dQ=dW\ge 0;d{W}_k\ge 0,\forall k.$$

(12)

It requires $\left\{d{p}_k\right\}$ to take a special form $\left\{d{p}_k^{\mathrm{GSL}}\right\}$ that not only determines $dS$ but also ensures $dQ\ge 0$; any form different from $\left\{d{p}_k^{\mathrm{GSL}}\right\}$ such as $\left\{-d{p}_k^{\mathrm{GSL}}\right\}$ does not satisfy GSL and must be rejected under MicroBCM proposal. The nonnegative signature of $dW$ after including (S4) in Gen-Th determines the (spontaneous) evolution of $\mathfrak{M}$, and the resulting thermodynamics is denoted by Gen-GSL-Th. We combine Equations (11) and (12) as

$$TdS=dW\left\{\begin{array}{c}⪌0 \mathrm{in} \mathrm{Gen}\text{-}\mathrm{Th}\hfill \\ \ge 0 \mathrm{in} \mathrm{Gen}\text{-}\mathrm{GSL}\text{-}\mathrm{Th}\hfill \end{array}\right.$$

(13)

in terms of the SI-temperature T for any $\overline{\mathsf{\Sigma }}$ that is given in Equation (7). For ${\mathsf{\Sigma }}_{\mathrm{M}}$, it becomes the standard definition ${\left(\partial E/\partial S\right)}_{\mathsf{\xi }}$; see Conclusion 1 and the arguments leading to it.

In Cl-Th, we supplement (S1), (S2), (S3) by the SL axiom for ${\mathsf{\Sigma }}_{\mathrm{M}}$ [16] instead of (S4) and then prove its thermodynamic stability as shown in the first line in Equation (8). Instead of proving stability from SL, we allow stability as a mechanical requirement of Mec-EQ-P instead of SL in the reverse approach to construct Gen-Th. We use Gen-Th to successfully prove GSL for any spontaneous process in $\overline{\mathsf{\Sigma }}$ for any $\left\{p_k\right\}$ as a direct consequence of ${\mathfrak{m}}_k$-evolution in (S4). We then prove that this condition is sufficient to prove SL by following the BCM proposal and start with $m_k$ of a deterministic mechanical system ${\mathsf{\Sigma }}^{\mathrm{D}}$, and impose stochasticity on its evolution.

Using stability that follows from Mec-EQ-P, we demonstrate the existence of ${\mathfrak{M}}_{\mathrm{seq}}$ in the state space of ${\mathsf{\Sigma }}_{\mathrm{M}}$. However, we also allow for the existence of ${\mathfrak{M}}_{\mathrm{ueq}}$ as suggested above by allowing instability. This results in the existence of ${\mathfrak{M}}_{\mathrm{eq}}$. From this, we establish the existence of ${\mathfrak{m}}_{k\mathrm{eq}}$ for every microstate ${\mathfrak{m}}_k$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$, which becomes the focus in constructing Gen-Th.

Our proof of GSL should be contrasted with the current situation of proving $dS\ge 0$ without ever mentioning T; one usually recourses to ad-hoc assumptions like molecular chaos or master equations [52]. The real significance of our reverse approach becomes very transparent and unravels many mysteries of SL by recognizing fluctuating $d{W}_k$ as the primitive mechanical concept that completely captures the stochasticity in $dQ$ and the importance of T for SL. Their intertwining provides the basis for the famous fluctuation-dissipation theorem [87,88]. To paraphrase Kubo [88], the internal relationship between the frictional force in Brownian motion resulting in $dQ$ and the random microscopic driving forces resulting in $\left\{d{W}_k\right\}$ is, “… in fact, a very general matter, which is manifested in the so-called fluctuation-dissipation theorem … because both come from the same origin”. We also observe that the concepts of generalized macrowork and macroheat are very different from those considered in [89,90], where these concepts refer to exchange microquantities with a medium, which is absent for our isolated system $\mathsf{\Sigma }$, so they vanish. Our concepts are intrinsic to $\mathsf{\Sigma }$ and are usually nonzero. Nevertheless, the generalized macrowork according to Irr-P is completely converted into generalized macroheat but not the other way around as our microscopic approach will justify. This makes $dW$ the primary concept and $dQ$ the secondary concept.To our knowledge, our reverse approach with MicroBCM proposal has never been used before to directly prove SL ($dS\ge 0$) for $T\ge 0$ and its extension requiring a reformulation to $dS<0$ for $T<0$; the latter contradicts the conventional wisdom [69]. In addition, GSL also disproves that instability causes SL violation.

Remark 10.

We use compact notation $\mathsf{Q}$ for (${\mathsf{Q}}^s$,${\mathsf{Q}}^u$), and ${\mathsf{Q}}_{\mathit{eq}}$ for (${\mathsf{Q}}_{\mathit{seq}}$,${\mathsf{Q}}_{\mathit{ueq}}$). Thus, we say that ${\mathfrak{M}}_{\mathit{eq}}$ controls $\mathfrak{M}:$(${\mathfrak{M}}^s$,${\mathfrak{M}}^u$)-evolution along continuous curves, and ${\mathfrak{m}}_{k\mathit{eq}}$ controls ${\mathfrak{m}}_k:$(${\mathfrak{m}}_k^s$,${\mathfrak{m}}_k^u$)-evolution along dashed-dot trajectories. We also let $\mathsf{Q}$ denote $E_k,E_{k\mathit{eq}}$, and ${\mathbf{F}}_k$, the last two defined later.

3. Mechanical Systems ${\overline{\mathsf{\Sigma }}}^{\mathbf{D}}$

We first consider a nonuniform mechanical system ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ that is specified by $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|t\right)$ given in terms of ${\mathbf{x}}_k\doteq \left\{({\mathbf{r}}_{ki},{\mathbf{p}}_{ki})\right\}$ and with explicit time dependence due to nonuniformity; see Section 4 for an alternative explanation. Let ${\mathfrak{m}}_k\left(t\right)\doteq {\mathbf{x}}_k$ (we set $2\pi \hslash =1$ for simplicity) denote one of its microstates with microenergy $E_k\doteq \mathcal{H}\left(\left.{\mathbf{x}}_k\right|t\right)$ so that

$$d{E}_k={\displaystyle \frac{\partial E_k}{\partial {\mathbf{x}}_k}}·d{\mathbf{x}}_k+{\displaystyle \frac{\partial E_k}{\partial t}}dt={\displaystyle \frac{\partial E_k}{\partial t}}dt$$

(14)

as the first term vanishes due to Hamilton’s equations of motion. As this term includes the effect of interparticle potentials, we are not interested in those potentials so we will no longer exhibit ${\mathbf{x}}_k$, which is explicitly identified by the suffix k. Instead, we are interested in $d{E}_k$ that is determined solely by varying the parameter t [3]. We write $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|t\right)$ as $E_k\left(t\right)$ or simply $E_k$; similarly, we simply write ${\mathfrak{m}}_k$ for ${\mathfrak{m}}_k\left(t\right)$ unless clarity is needed. The SI-microwork $d{W}_k\left(t\right)$ can be expressed in terms of instantaneous power ${\mathsf{P}}_k\left(t\right)$ in ${\mathfrak{S}}_{\mathbf{X}}$ as

$$d{W}_k\left(t\right)=-d{E}_k\left(t\right)={\mathsf{P}}_k\left(t\right)dt.$$

(15)

Similarly, the mechanical system ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in a microstate ${\mathfrak{m}}_k\left(\mathsf{\xi }\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ is specified by $\mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathsf{\xi }\right)$, which now depends on the set $\mathsf{\xi }$ of work parameters of ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$, see Section 4, that determines the microwork $d{W}_k\left(\mathsf{\xi }\right)$ as we now explain. We remark that $\mathsf{\xi }$ is assumed to be the same for all microstates so it does not carry the suffix k, while the microforce ${\mathbf{F}}_k$ in Equation (17) and $d{W}_k$ do so; the latter are fluctuating (over ${\mathfrak{m}}_k$). We will not consider here the case when $\mathsf{\xi }$ is fluctuating (over ${\mathfrak{m}}_k$), to which the present method is easily extended [53,91]. The significance of $\mathsf{\xi }$ becomes clear when we consider the microenergy $E_k\left(\mathsf{\xi }\right)\equiv \mathcal{H}\left(\left.{\mathbf{x}}_k\right|\mathsf{\xi }\right)$ of ${\mathfrak{m}}_k\left(\mathsf{\xi }\right)$, for which

$$d{E}_k={\displaystyle \frac{\partial E_k}{\partial \mathsf{\xi }}}·d\mathsf{\xi }$$

(16)

by using the same argument as used in Equation (14). As above, we simply denote $E_k\left(\mathsf{\xi }\right)$ and ${\mathfrak{m}}_k\left(\mathsf{\xi }\right)$ by $E_k$ and ${\mathfrak{m}}_k$, respectively, unless clarity is needed. The SI-microforce, also known as the generalized microforce, in ${\mathfrak{m}}_k$ is given by

$${\mathbf{F}}_k\doteq -\partial E_k/\partial \mathsf{\xi },$$

(17)

and the SI-microwork done by it is given by

$$d{W}_k\left(\mathsf{\xi }\right)={\mathbf{F}}_k·d\mathsf{\xi }=-d{E}_k,$$

(18)

which clarifies the significance of $\mathsf{\xi }$ as the work parameter for ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$. [In NEQ thermodynamics, the macroscopic analog (ensemble average $〈•〉$, see Equation (9)) $\mathbf{F}=〈\mathbf{F}〉$ of ${\mathbf{F}}_k$ determines the affinity [10,26] associated with $\mathsf{\xi }$ for ${\mathsf{\Sigma }}_{\mathrm{E}}$]. We will see that $\mathsf{\xi }$ describes nonuniformity and changes spontaneously with time in accordance with the principle of mechanical equilibrium, called Mec-EQ-P [85] (p. 99) that is introduced in Section 5. Thus, $E_k\left(\mathsf{\xi }\right)$ continues to change with $\mathsf{\xi }$ and results in the above microwork. The corresponding instantaneous microstate power in ${\mathfrak{S}}_{\mathbf{Z}}$ is

$${\mathsf{P}}_k\left(t\right)\doteq {\mathbf{F}}_k·\dot{\mathsf{\xi }},$$

(19)

which can be used in Equation (15) to obtain $d{W}_k\left(t\right)$. We thus see that implicit time-dependence in $\mathsf{\xi }$ for ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ is equivalent to the explicit time dependence in ${\mathfrak{S}}_{\mathbf{X}}$ without any $\mathsf{\xi }$ for ${\mathsf{\Sigma }}^{\mathrm{D}}$. The main difference is that for ${\mathsf{\Sigma }}^{\mathrm{D}}$ not as big as ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$, its nonuniformity cannot be described by $\mathsf{\xi }$ and must be captured by time dependence only.

The variation of $E_k\left(\mathsf{\xi }\right)$ for an isolated system ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ as shown in Figure 2 with $\mathsf{\xi }$ can also be treated as time variation of $E_k\left(t\right)$ for ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$. This is also consistent with the way a time-dependent Hamiltonian is treated in mechanics [3] by considering t as a parameter in the Hamiltonian so we can use t instead of $\mathsf{\xi }$, in which case Equation (18) reduces to Equation (15) obtained in ${\mathfrak{S}}_{\mathbf{X}}$, where ${\mathsf{\Sigma }}^{\mathrm{D}}$ has no restriction on its size. We will not be concerned with the actual time dependence of $\mathsf{\xi }$ in ${\mathfrak{S}}_{\mathbf{Z}}$ in this investigation; all we need to remember is that it can be accounted for by an explicit time dependence in ${\mathsf{P}}_k\left(t\right)$ and other quantities but only for ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$.

According to the discussion above, we can consider either $\mathsf{\xi }$ explicitly (to be specified in Section 4) and use the state space ${\mathfrak{S}}_{\mathbf{Z}}$, or consider t instead in the state space ${\mathfrak{S}}_{\mathbf{X}}$ without any specification of $\mathsf{\xi }$. In the former case, microstates are specified in ${\mathfrak{S}}_{\mathbf{Z}}$, which restricts ${\mathsf{\Sigma }}^{\mathrm{D}}$ to ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$.

4. Internal Variables for $\left\{{\mathfrak{m}}_{\mathit{k}}\right\}$ of ${\mathsf{\Sigma }}_{\mathbf{E}}^{\mathbf{D}}$

We now turn to the definition of internal variables [10,26,45,47] for a deterministic nonuniform mechanical system ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$, but denote it by ${\mathsf{\Sigma }}^{\mathrm{D}}$ in this section to simplify notation unless clarity is needed. For simplicity, we consider ${\mathsf{\Sigma }}^{\mathrm{D}}$ to be formed by two different and disjoined but mutually interacting subsystems ${\mathsf{\Sigma }}_1^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$ that are each uniform in ${\mathfrak{S}}_{\mathbf{X}}$. The internal variables are required to uniquely specify $\left\{{\mathfrak{m}}_k\right\}$ in ${\mathfrak{S}}_{\mathbf{Z}}$; they are not specified uniquely in ${\mathfrak{S}}_{\mathbf{X}}$. We consider ${\mathsf{\Sigma }}^{\mathrm{D}},{\mathsf{\Sigma }}_1^{\mathrm{D}}$, and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$ in microstate ${\mathfrak{m}}_k,{\mathfrak{m}}_{k_1}$, and ${\mathfrak{m}}_{k_2}$ of energy $E_k,E_{k_1}$, and $E_{k_2}$, respectively. We also introduce $n_1=N_1/N$ and $n_2=N_2/N$. If we neglect the energy $\delta E_k$ due to the interface between ${\mathsf{\Sigma }}_1^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$, the Hamiltonians of the three bodies (we do not show their arguments ${\mathbf{x}}_k,{\mathbf{x}}_{k_1}$ and ${\mathbf{x}}_{k_2}$, respectively, but use microstate suffixes) are related by

$${\mathcal{H}}_k(\mathbf{w},\mathsf{\xi })\approxeq {\mathcal{H}}_{k_1}\left({\mathbf{w}}_1\right)+{\mathcal{H}}_{k_2}\left({\mathbf{w}}_2\right),$$

(20a)

which ensures microenergy quasi-additivity

$$E_k\approxeq E_{1k_1}+E_{2k_2};$$

(20b)

it can be justified only if we restrict the minimum sizes of ${\mathsf{\Sigma }}_1^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$ to be some ${\lambda }_{\mathrm{E}}$ that itself is determined by the range of inter-particle interactions. This explains why ${\mathsf{\Sigma }}^{\mathrm{D}},{\mathsf{\Sigma }}_1^{\mathrm{D}}$, and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$ must be denoted by ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}},{\mathsf{\Sigma }}_{1\mathrm{E}}^{\mathrm{D}}$, and ${\mathsf{\Sigma }}_{2\mathrm{E}}^{\mathrm{D}}$. We also have ${\mathfrak{m}}_k\approxeq {\mathfrak{m}}_{k_1}\otimes {\mathfrak{m}}_{k_2}$.

The discussion in Section 3 applies to any ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$. In general, ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ is much smaller in size than ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$. The additivity of extensive (which t is not) work parameters ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$ in Equation (20c) is not affected by the presence or absence of $\delta E_k$ so that we can introduce the extensive work parameter set

$$\mathbf{w}\doteq {\mathbf{w}}_1+{\mathbf{w}}_2.$$

(20c)

We introduce a new extensive internal variable set ${\mathsf{\xi }}_k\doteq ({\xi }_{k\mathrm{E}},\mathsf{\xi })$, where

$${\xi }_{k\mathrm{E}}\doteq E_{1k_1}/n_1-E_{2k_2}/n_2,\mathsf{\xi }\doteq {\mathbf{w}}_1/n_1-{\mathbf{w}}_2/n_2,$$

(21)

and the extensive work parameter

$$\mathbf{W}\doteq (\mathbf{w},\mathsf{\xi })$$

(22)

are independent variables for ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in that $\left(E_k,{\xi }_{k\mathrm{E}}\right)$ and $\left(w_l\in \mathbf{w},{\xi }_l\in \mathsf{\xi }\right)$ form pairs of independent variables. The introduction of ${\mathsf{\xi }}_k$ is to ensure that the number of variables on both sides in Equation (20a) are equal as they must be for an equality. We also have

$$E_{1k_1,2k_2}=n_{1,2}(E_k\pm n_{2,1}{\xi }_{k\mathrm{E}}),{\mathbf{w}}_{1,2}=n_{1,2}(\mathbf{w}\pm n_{2,1}\mathsf{\xi }).$$

(23)

We can easily extend the above discussion to any numbers m of subsystems $\left\{{\mathsf{\Sigma }}_l^{\mathrm{D}}\right\},$$l=1,2,\cdots ,m$ forming ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$, each specified by its own observable set ${\mathbf{w}}_l$ to allow for a complex form of nonuniformity in terms of uniform subsystems; Equation (22) remains valid for any m. It is easy to verify that all internal variables in $\mathsf{\xi }$, whose number we denote by $\iota$, can be expressed as a linear combinations of $\left\{{\mathbf{w}}_l\right\}$ [47,53], with m and $\iota$ increasing with the degree of nonuniformity of ${\mathfrak{m}}_k\left(\mathsf{\xi }\right)$. Even though the definition of ${\mathsf{\xi }}_k$ is not unique, we choose it for convenience so that it vanishes when ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ is uniform as defined in Equation (21).

The set $\left(E_k,{\mathsf{\xi }}_k\right)$ forms the complete set of variables to uniquely specify ${\mathfrak{m}}_k$ of ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$. It is easily verified that

$$F_{k\mathrm{E}}\doteq -\partial E_k/\partial {\xi }_{k\mathrm{E}}=0,$$

(24)

which is a consequence of independent $E_k$ and ${\xi }_{k\mathrm{E}}$ so the variation of ${\xi }_{k\mathrm{E}}$ does not generate any SI-microwork for any $\left\{{\mathbf{w}}_m\right\}$. This explains why it is not included in $\mathbf{W}$, in which $\mathbf{w}$ remains constant as ${\mathsf{\Sigma }}^{\mathrm{D}}$ evolves in time; however, ${\mathsf{\xi }}_k$ continues to change due to internal flows.

Let us focus on two ($m=2$) subsystems ${\mathsf{\Sigma }}_1^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$, for simplicity, each of which is uniform (no internal variables for them) but ${\mathsf{\Sigma }}^{\mathrm{D}}$ is not. We let $E_{k_l},{\mathbf{w}}_l\doteq (N_l,V_l)$ specify uniform microstates${\mathfrak{m}}_{k_l},$$l=1,2$ in ${\mathfrak{S}}_{\mathbf{X}}$. We now focus on the nonuniform microstate ${\mathfrak{m}}_k$ of ${\mathsf{\Sigma }}^{\mathrm{D}}$, which require internal variables along with $E_k$ and $\mathbf{w}\doteq (N,V)$ to be specified in ${\mathfrak{S}}_{\mathbf{Z}}$. We keep observables $E_k$ and $\mathbf{w}\doteq (N,V)$ of ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ fixed along with $N_1$ and $N_2$ of ${\mathsf{\Sigma }}_{1\mathrm{E}}^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_{2\mathrm{E}}^{\mathrm{D}}$, respectively. Following Equation (21), we have

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

(25)

to identify $\mathbf{Z}=(E_k,N,V,{\xi }_{k\mathrm{E}},{\xi }_{\mathrm{V}})$ for ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$.

Densities in ${\mathsf{\Sigma }}_{1\mathrm{E}}^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_{2\mathrm{E}}^{\mathrm{D}}$ are equal in a uniformmicrostate ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$ of ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ so ${\mathsf{\xi }}_k=0$ and need not be considered. This is consistent with the fact that uniform ${\mathfrak{m}}_{k\mathrm{eq}}$ is uniquely specified in ${\mathfrak{S}}_{\mathbf{X}}$. In this case, we have a trivial additivity of the Hamiltonians in Equation (20a) given by

$${\mathcal{H}}_k\left(\mathbf{w}\right)\simeq {\mathcal{H}}_{k_1}\left({\mathbf{w}}_1\right)+{\mathcal{H}}_{k_2}\left({\mathbf{w}}_2\right)$$

(26)

with no internal flows between subsystems. In a nonuniform microstate, ${\mathsf{\xi }}_k\ne 0$. We recall that $\mathbf{w}$ is fixed for ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$ but ${\mathbf{w}}_1$ and ${\mathbf{w}}_2$ can change due to possible transfers (internal flows) between ${\mathsf{\Sigma }}_{1\mathrm{E}}^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_{2\mathrm{E}}^{\mathrm{D}}$ with $d{\mathbf{w}}_1=-d{\mathbf{w}}_2$. It follows from Equation (23) that

$${\dot{E}}_{1k_1,2k_2}=n_{1,2}({\dot{E}}_k\pm n_{2,1}{\dot{\xi }}_{k\mathrm{E}}),{\dot{\mathbf{w}}}_1=-{\dot{\mathbf{w}}}_2=n_1{n}_2\dot{\mathsf{\xi }}$$

(27)

for ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$.

As $\mathbf{W}$, i.e., $\mathsf{\xi }$ in Equation (22) for ${\mathsf{\Sigma }}^{\mathrm{D}}$ is the work variable in ${\mathcal{H}}_k\left({\mathsf{\xi }}_k\right)$ (we suppress $\mathbf{w}$ as it it constant), $E_k\left({\mathsf{\xi }}_k\right)$ corresponding to ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$ is only a function of $\mathsf{\xi }$ due to Equation (24). This is shown in Figure 2. As ${\mathsf{\xi }}_k$ continuously changes due to transfers between ${\mathsf{\Sigma }}_1^{\mathrm{D}}$ and ${\mathsf{\Sigma }}_2^{\mathrm{D}}$, this causes variations in $E_k\left(\mathsf{\xi }\right)$. The variation is similar to the variation in $E_k\left(t\right)$ for a ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ as discussed in Section 3.

Different components of ${\mathsf{\xi }}_k$ take different times when they vanish. They are called relaxation times so the components can be ordered according to them [44] as fast and slow. As ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}$ evolves in time, they disappear at different times, making ${\mathfrak{m}}_k$ more or less uniform, a point that is discussed in Section 5. For a single $\xi$ in the Figure 2, ${\mathfrak{M}}^{\mathrm{s}}$ (${\mathfrak{M}}^{\mathrm{u}}$) becomes more and more (less and less) uniform as t increases so that $dS>$ (<) 0 during any infinitesimal change $dt$.

In summary, as long as an internal variable is used, the system is restricted to be at least ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ in size and requires ${\mathfrak{S}}_{\mathbf{Z}}$ for unique specification. There is no explicit time dependence in this state space. If considerations of subsystems are not important in any discussion then quasi-additivity is not an issues as is the case for the the discussion of Equation (15). In that case, ${\mathsf{\Sigma }}^{\mathrm{D}}$ can be considered as a whole in ${\mathfrak{S}}_{\mathbf{X}}$; there is no need to consider $\delta E_k$ separately, which is included in ${\mathcal{H}}_k(\mathbf{w},t)$. Thus, the mechanical situation here is that of Section 3.

5. Mechanical Equilibrium Principle of Energy (Mec-EQ-P) for ${\mathit{m}}_{\mathit{k}}$

Time-independent microstates $\left\{{\mathfrak{m}}_{k\mathrm{eq}}\right\}$ of a Hamiltonian system ${\mathsf{\Sigma }}^{\mathrm{D}}$, being uniform as follows from the Uniformity Theorem 1, play an important role in our approach as we explain now. Any nonuniformity in ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ endows $\left\{{\mathfrak{m}}_k\right\}$ with explicit time dependence as $\left\{{\mathfrak{m}}_k\left(t\right)\right\}$ for ${\mathsf{\Sigma }}^{\mathrm{D}}$ or implicit time dependence through ${\mathsf{\xi }}_k$ as ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$ for ${\mathsf{\Sigma }}_{\mathrm{E}}^{\mathrm{D}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}^{\mathrm{D}}$ as shown by $\left\{{\mathfrak{m}}_k^{\mathrm{s}}\right\}$ and $\left\{{\mathfrak{m}}_k^{\mathrm{u}}\right\}$; see dashed-dotted blue and red curves, respectively, in Figure 2; the directions of blue and red arrows on them denote increasing t during their temporal SI-evolution controlled by internally generated processes that are mechanically spontaneous and generate micropower ${\mathsf{P}}_k\left(t\right)$. In analytical mechanics, there is no dissipation in evolution but ${\mathfrak{m}}_k^{\mathrm{s}}$ is special in that its evolution will normally undergo oscillations in $\mathsf{\xi }$ about ${\mathfrak{m}}_{k\mathrm{seq}}$ that will persist forever as manifested by the dashed-dot blue curve; in contrast, the evolution of ${\mathfrak{m}}_k^{\mathrm{u}}$ has no oscillation, see the dashed-dot red curve there, as ${\mathfrak{m}}_k^{\mathrm{u}}$ runs away from ${\mathfrak{m}}_{k\mathrm{ueq}}$ but terminates is a catastrophe in which ${\mathfrak{m}}_k^{\mathrm{u}}$ becomes extremely nonuniform, which for ${\mathsf{\Sigma }}_{\mathrm{E}}$ or ${\mathsf{\Sigma }}_{\mathrm{M}}$ corresponds to an extremely large $\iota ={\iota }_{\mathrm{cats}}$; see Section 7. Thus, ${\mathfrak{m}}_{k\mathrm{ueq}}$ is the source for the SI-evolution of ${\mathfrak{m}}_k^{\mathrm{u}}$.

As we are eventually interested in ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{m}}_k^{\mathrm{u}}$ in a thermodynamic setting, there will be macroheat that we discuss in Section 6 to be simultaneously considered. As is well known, its presence gives rise to dissipation so all thermodynamic processes in stable ${\mathfrak{M}}^{\mathrm{s}}$ terminate asymptotically at ${\mathfrak{M}}_{\mathrm{seq}}$ as shown by the solid blue curve in the figure. Keeping this in hindsight, we intentionally overlook oscillations in ${\mathfrak{m}}_k^{\mathrm{s}}$ as they do not affect the change in $E_k^{\mathrm{s}}$ during its SI-evolution to ${\mathfrak{m}}_{k\mathrm{seq}}$ only, making the latter as the sink for the SI-evolution of ${\mathfrak{m}}_k^{\mathrm{s}}$. This is a useful strategy, since oscillations become thermodynamically irrelevant as ${\mathfrak{M}}^{\mathrm{s}}$ asymptotically approaches ${\mathfrak{M}}_{\mathrm{seq}}$; see below. This also clarifies the importance of the uniform microstate ${\mathfrak{m}}_{k\mathrm{eq}}$; see the Uniformity Theorem 1.

We first treat ${\mathsf{\Sigma }}_{\mathrm{E}}$, whose results pave the way for a clear understanding of what to expect for $\mathsf{\Sigma }$. In analytical mechanics, the temporal evolution of ${\mathsf{\xi }}_k$ and ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$ in ${\mathfrak{S}}_{\mathbf{Z}}$ is controlled by ${\mathfrak{m}}_{k\mathrm{eq}}$, and is governed by the emergent SI-microforce ${\mathbf{F}}_k$ in Equation (17) by controlling internal flows within ${\mathfrak{m}}_k\left({\mathsf{\xi }}_k\right)$. The emergent processes in ${\mathsf{\Sigma }}_{\mathrm{E}}$ resulting in the SI-microwork in Equation (18) are commonly identified as spontaneous since they are internally controlled by SI-microforce ${\mathbf{F}}_k$ and not from the outside by any nonsystem microforce. The mechanical equilibrium (Mec-EQ) point at $E_k^{\mathrm{eq}}$ is the equilibrium point [85] in the SI-evolution under the SI-micorforce ${\mathbf{F}}_k$, where not only the SI-internal velocity ${\dot{\mathsf{\xi }}}_k^{\mathrm{eq}}$ but also the SI-micorforce ${\mathbf{F}}_k^{\mathrm{eq}}$ vanish:

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

(28a)

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

(28b)

It follows from Equation (28b) that this point represents an extremum in $E_k$; see Figure 2. Its minimum at $E_k^{\mathrm{seq}}=E_k^{\mathrm{eq}}$ representing a mechanically stable EQ point enunciates the mechanical asymptotic stability principle of minimum energy. In thermodynamics, stability refers to asymptotic stability in that the system must eventually approach stable equilibrium so that ${\mathsf{\xi }}_k\left(t\right)\to 0$ as $t\to \infty$ [16]. In this sense, we see that the absolute value $\left|{\mathsf{\xi }}_k\left(t\right)\right|$ behaves similar to $1/t$ in the figure.

Remark 11.

The asymptotic approach is a stronger requirement than just imposing stability in which the system never strays far away from stable EQ point.

In contrast, the maximum of $E_k$ at $E_k^{\mathrm{ueq}}=E_k^{\mathrm{eq}}$ representing a mechanically unstable EQ point enunciates the mechanical instability principle of maximum energy. In this case, $\left|{\mathsf{\xi }}_k\left(t\right)\right|$ behaves similar to t in the figure. Both kinds of points determine the curvature of $E_k$ at $E_k^{\mathrm{eq}}$. The two principles are collectively called Mec-EQ-P in this investigation.

We now prove the following important theorem emphasizing the physical significance of uniformity of the Mec-EQ point ${\mathfrak{m}}_{k\mathrm{eq}}$ for thermodynamic stability, which must exist only in ${\mathfrak{S}}_{\mathbf{X}}$ at ${\mathsf{\xi }}_k=0$, where ${\mathfrak{M}}_{\mathrm{eq}}$ exists.

Theorem 1.

Uniformity Theorem of ${\mathfrak{m}}_{k\mathit{eq}}$: The Mec-EQ point ${\mathfrak{m}}_{k\mathit{eq}}$ of ${\mathfrak{m}}_k$ is the stationary and uniform microstate with no internal flows between the microstates of its uniform subsystems.

Proof. 

We begin by considering a microstate ${\mathfrak{m}}_k\left(t\right)$ for ${\mathsf{\Sigma }}^{\mathrm{D}}$ in ${\mathfrak{S}}_{\mathbf{X}}$ to establish that ${\mathfrak{m}}_{k\mathrm{eq}}:({\mathfrak{m}}_{k\mathrm{seq}},{\mathfrak{m}}_{k\mathrm{ueq}})$ is stationary. The asymptotic convergence of ${\mathfrak{m}}_k^{\mathrm{s}}$ to ${\mathfrak{m}}_{k\mathrm{seq}}$ as $t\to \infty$ means that ${\mathfrak{m}}_{k\mathrm{seq}}$ must be stationary so it must be a uniform microstate of the time-independent Hamiltonian $\mathcal{H}\left(\mathbf{x}\right)$. The instability of ${\mathfrak{m}}_{k\mathrm{ueq}}$ is due to the instability of the $\mathcal{H}\left(\mathbf{x}\right)$ in this case so it is also uniform.

As ${\mathfrak{S}}_{\mathbf{X}}$ contains no information about the internal structure of ${\mathsf{\Sigma }}^{\mathrm{D}}$, it cannot be used to understand internal flows for which we need to consider ${\mathfrak{S}}_{\mathbf{Z}}$ and at least ${\mathsf{\Sigma }}_{\mathrm{E}}$ to justify the remainder part of the theorem. From Equation (28a), we obtain ${\dot{\xi }}_{k\mathrm{E}}^{\mathrm{eq}}=0$ and ${\dot{\mathsf{\xi }}}^{\mathrm{eq}}=0$. Using first ${\dot{\mathsf{\xi }}}^{\mathrm{eq}}=0$ in Equation (27), we conclude that ${\dot{E}}_k^{\mathrm{eq}}=0$, which follows directly from Equation (26). Using then ${\dot{\xi }}_{k\mathrm{E}}^{\mathrm{eq}}=0$ in Equation (27) along with ${\dot{E}}_k^{\mathrm{eq}}=0$, we conclude that ${\dot{E}}_{1k_1,2k_2}^{\mathrm{eq}}=0$. Together, they show that all flows cease at $E_k^{\mathrm{eq}}$, which makes ${\mathfrak{m}}_{k\mathrm{eq}}$ stationary and, therefore, uniform as above. Any nonzero ${\mathsf{\xi }}_k$ is due to nonuniformity, and affects $E_k\left(\mathsf{\xi }\right)\equiv E_k\left({\mathsf{\xi }}_k\right)$ as shown in Figure 2. This completes the proof of the theorem. □

It should be clear from the above mechanical proof of the theorem that any time-dependent ${\mathfrak{m}}_k:({\mathfrak{m}}_k^{\mathrm{s}},{\mathfrak{m}}_k^{\mathrm{u}})$ must be uniform and stationary only at the equilibrium point ${\mathfrak{m}}_{k\mathrm{eq}}:({\mathfrak{m}}_{k\mathrm{seq}},{\mathfrak{m}}_{k\mathrm{ueq}})$ for $\mathsf{\Sigma }$. Away from it, ${\mathfrak{m}}_k$ must be nonuniform.

We assume for simplicity that $E_k$has only a single extremum at $E_{k\mathrm{eq}}$ as shown by the dashed-dotted curves in Figure 2 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 [69] for ${\mathfrak{M}}_{\mathrm{eq}}$ to NEQ situation. From the Uniformity Theorem 1, the extremum represents a uniform microstate ${\mathfrak{m}}_{k\mathrm{eq}}$ so the rest of the dashed-dotted curves denote nonuniform microstates ${\mathfrak{m}}_k$ (${\mathfrak{m}}_k^{\mathrm{s}}$ or ${\mathfrak{m}}_k^{\mathrm{u}}$). Then, ${\mathfrak{m}}_{k\mathrm{seq}}$is the (asymptotically) stable equilibrium of ${\mathfrak{m}}_k^{\mathrm{s}}$, where $E_{k\mathrm{seq}}$ is minimum. Any perturbation away from ${\mathfrak{m}}_{k\mathrm{seq}}$ always restores ${\mathfrak{m}}_k^{\mathrm{s}}$ back to ${\mathfrak{m}}_{k\mathrm{seq}}$ so that $E_k^{\mathrm{s}}\to E_{k\mathrm{seq}}$ spontaneously as shown by the blue arrow. Thus, following Remark 10, ${\mathfrak{m}}_{k\mathrm{seq}}$ controls ${\mathfrak{m}}_k^{\mathrm{s}}$-evolution [88] during which ${\mathfrak{m}}_k^{\mathrm{s}}$ becomes more and more uniform with $dS>0$. At a mechanically unstable equilibrium ${\mathfrak{m}}_{k\mathrm{ueq}}$ of ${\mathfrak{m}}_k^{\mathrm{u}}$, $E_{k\mathrm{ueq}}$ is maximum; any perturbation from it to ${\mathfrak{m}}_k^{\mathrm{u}}$ spontaneously repels it further away from ${\mathfrak{m}}_{k\mathrm{ueq}}$. As ${\mathfrak{m}}_k^{\mathrm{u}}$ never returns to ${\mathfrak{m}}_{k\mathrm{ueq}}$ with $E_k^{\mathrm{u}}$ running away from $E_{k\mathrm{ueq}}$, ${\mathfrak{m}}_{k\mathrm{ueq}}$ becomes a source controlling ${\mathfrak{m}}_k^{\mathrm{u}}$-evolution along the red arrow during which ${\mathfrak{m}}_k^{\mathrm{u}}$ becomes more and more nonuniform with $dS<0$; see Remark 10. Both are deterministic spontaneous ${\mathfrak{m}}_k$-evolutions, whose directions are controlled by the principle of mechanical equilibrium (Mec-EQ-P).

6. General Thermodynamics (Gen-Th) for $\overline{\mathsf{\Sigma }}$

So far, we have only considered the step (S1). We now take the step (S2) and introduce stochasticity by considering an ensemble of $\overline{\mathsf{\Sigma }}$ in which ${\mathfrak{m}}_k$ appears with probability $p_k$, but without invoking Mec-EQ-P discussed in Section 5. It should be stressed that the deterministic Hamiltonian $\mathcal{H}\left(\left.\mathbf{x}\right|t\right)$ is oblivious to $p_k$ so ${\mathfrak{m}}_k\left(t\right)$ is also oblivious to $p_k$ in that all its microquantities such as $E_k\left(t\right)$ do not change with $p_k$. We put no restrictions on possible sets $\left\{p_k\right\}$ so the resulting macrostates may have nothing to do with what we encounter in classical thermodynamics (Cl-Th), see Table 2. The resulting thermodynamics that is called general thermodynamics (Gen-Th) may not always satisfy the second law $(dS\ge 0)$ that is a fundamental axiom (or assumption) in Cl-Th [16]. Making $\left\{p_k\right\}$ arbitrary will help us to determine the root cause of the second law of Cl-Th and its possible violation in the violation thermodynamics (Viol-Th), both of which are contained in Gen-Th. For it to be thermodynamics, we will need to introduce T following Equation (7) and S following Equation (1) in the corresponding ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$.

6.1. The First Law

The macroenergy E of $\overline{\mathsf{\Sigma }}$ in Gen-Th is given by the ensemble average $〈E〉$

$$E=〈E〉\doteq {\sum }_k{p}_k{E}_k,$$

which is valid for any $N\ge 1$ as observed earlier with respect to the definition in Equation (9), see Remark 9. It follows from this that

$$dE={\sum }_k{E}_{k}d{p}_k+{\sum }_k{p}_{k}d{E}_k.$$

From Equation (17) in Section 3, we observe that the second sum gives the negative of the SI-macrowork $dW$

$$dW\doteq {\sum }_k{p}_{k}d{W}_k$$

(29a)

as the ensemble average of SI-microwork $d{W}_k$. We identify the first sum, which is $dE+dW$, with SI-macroheat $dQ$ as the ensemble average of SI-microheat $d{Q}_k$

$$dQ\doteq {\sum }_k{p}_{k}d{Q}_k\doteq {\sum }_k{p}_k{E}_{k}d{\eta }_k,$$

(29b)

where ${\eta }_k\doteq ln{p}_k$ is the Gibbs probability index [8]. It should be stressed that the above identification of $dQ$ does not require any size restriction on $\mathsf{\Sigma }$, see Remark 9, and any notion of temperature or entropy; see below. We thus obtain the statement of the first law

$$dE=dQ-dW$$

(30)

for $\overline{\mathsf{\Sigma }}$ in terms of $dQ$ and $dW$, both defined for any N. This completes the step (S3). It should be clear that $dQ$ and $dW$ are the primary concepts in the first law and Gen-Th, making them equivalent as both are independent of SL ($dS\ge 0$).

6.2. Irreversibility Principle for $\overline{\mathsf{\Sigma }}$

We observe that $dE=0$ in $\overline{\mathsf{\Sigma }}$, see Equation (5), so that

$$E={\sum }_k{p}_k\left(t\right)E_k\left(t\right)\equiv {\sum }_k{p}_k^{\mathrm{eq}}{E}_k^{\mathrm{eq}}=E_{\mathrm{eq}},$$

(31)

where $p_k^{\mathrm{eq}}$ is the probability of ${\mathfrak{m}}_{k\mathrm{eq}}$ of microenergy $E_k^{\mathrm{eq}}$, and $E_{\mathrm{eq}}$ is the macroenergy of ${\mathfrak{M}}_{\mathrm{eq}}$. We use this fact in Equation (30) to obtain a simple but very remarkable and extremely profound result, called the irreversibility principle (Irr-P), in Equation (11) during any infinitesimal process $\delta \mathcal{P}$ of $\mathcal{P}$. It provides the mechanical formulationby $dW$ of the stochasticity inherent in the thermodynamic process through $dQ$. We conclude that the stochasticity Definition 4, i.e., the change $\left\{d{p}_k\right\}$ in $dQ$ is strongly constrained by the mechanical work $dW$ and its signature for any $\mathcal{P}$. It is this constraint that allows Equation (11) to be identified as Irr-P. It is derived under steps (S1–S3) and is valid for an isolated $\overline{\mathsf{\Sigma }}$ of any size, and is a direct consequence of the first law. Thus, it is not an independent principle (law) in Gen-Th.

We see from Equation (14) that $d{E}_k\equiv 0$ for uniform microstates so $E_k$cannot change for the uniform $\overline{\mathsf{\Sigma }}$. Accordingly,$dW\equiv 0$. The first law then enforces that $dQ\equiv 0$. However, it is easy to see that it is not possible for the macroenergy E in Equation (31) to remain invariant in general for all possible choices of time-dependent $p_k\left(t\right)$ and constant $\left\{E_k\right\}$ that result in uniform but time-dependent macrostate $\mathfrak{M}$ that was introduced earlier as uniform NEQ macrostate after Remark 1. Thus,

Remark 12.

Uniform NEQ macrostates that must have time-dependent probabilities cannot satisfy energy conservation for $\overline{\mathsf{\Sigma }}$ so they are irrelevant in thermodynamics. The situation can only be salvaged by allowing uncontrollable weak interactions with the outside for even for the isolated system as discussed elsewhere [4] as no system is really isolated in Nature.

The following simple example clarifies the above Remark. Consider a $\overline{\mathsf{\Sigma }}$ containing two microstates with distinct microenergies $E_1,E_2$ and probabilities $p_1,p_2$, respectively. The change in the energy due to probability changes at fixed microenergies is

$$dE=(E_1-E_2)d{p}_1.$$

It is easy to see that, for nonzero $d{p}_1$ for time-dependent probabilities, $dE$ cannot vanish.

The entropy can be determined using Equation (1) to obtain $dS$ during $\delta \mathcal{P}$. In terms of T from the use of Equation (7), we relate $dQ$ with $dS$ by the following simple relation

$$dQ=TdS.$$

(32)

valid for a system $\overline{\mathsf{\Sigma }}$ of any size. The use of this equation in Equation (11) immediately proves the top equation in Equation (13) noted earlier in Gen-Th.

This completes the discussion of Gen-Th and the first law in it.

As $p_k$ remains constant in $dW$, it represents an isentropic macroquantity to justify it as a mechanical quantity, the average of the change $d{W}_k=-d{E}_k$; see Equation (29a). On the other hand, $p_k$ does not remain constant in $dQ$ so it justifies $dQ$ as a stochastic quantity Definition 4 undergoing entropy change $dS$ but the two are simply related by Irr-P in Equation (11).

We are interested in the microwork done during the ${\mathfrak{m}}_{k\mathrm{eq}}$-controlled evolution of ${\mathfrak{m}}_k\left(t\right)$ along its spontaneous trajectory ${\gamma }_k$ in a thermodynamic process $\mathcal{P}$ in ${\mathfrak{S}}_{\mathbf{X}}$. The accumulated microwork along ${\gamma }_k$ in step (S1) follows from Equation (15), and is given by

$$\Delta W_k=-\Delta E_k$$

(33)

by the microenergy change $\Delta E_k$ along ${\gamma }_k$; recall Equation (15). In step (S2), its ensemble average over $\mathcal{P}$ yields the SI-macrowork

$$\Delta W={\int }_{\mathcal{P}}dW\equiv {\sum }_k{\int }_{{\gamma }_k}{p}_{k}d{W}_k\doteq <\Delta W>$$

(34)

in Gen-Th; it remains valid even if $p_k$ remains constant over ${\gamma }_k$. The cumulative formulation of Equation (11) is the cumulative form of Irr-P

$$\Delta Q=\Delta W,$$

(35a)

where $\Delta Q$ is given by

$$\Delta Q={\sum }_k{\int }_{{\gamma }_k}{p}_k{E}_{k}d{\eta }_k,$$

(35b)

and exists if and only if $p_k$ does not remain constant during $\mathcal{P}$. This is consistent with the conventional wisdom that the concept of heat does not apply to mechanical bodies for which $d{p}_k\equiv 0$ and $dQ\equiv 0$. While $\Delta W$ is determined by the instantaneous value of $p_k$ along ${\gamma }_k$, it does not control how it changes along it. The latter is controlled by the macroheat $\Delta Q$.

The macroworks $dW$ and $\Delta W$ play an important role in understanding the function $E_{\mathrm{w}}$ shown in Figure 2. We first introduce the cumulative macrowork

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

(36a)

along a process ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$ starting from EQ point ${\mathfrak{M}}_{\mathrm{eq}}$ to some point $\mathfrak{M}$ on solid curves in Figure 2. We immediately see that ${\mathcal{P}}_{\mathrm{eq}}^{\prime }$ is in the direction opposite to the blue arrow for ${\mathfrak{M}}^{\mathrm{s}}$ but in the direction of the red arrow for ${\mathfrak{M}}^{\mathrm{u}}$. We use it to introduce a macrowork function

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

(36b)

of t or $\mathsf{\xi }$. It is this average $E_{\mathrm{w}}$ that is shown in Figure 2, and differs from $E=E_{\mathrm{eq}}$ by the cumulative macrowork $W_{\mathrm{eq}}^{\prime }$. We see that

$$d{E}_{\mathrm{w}}=-dW,$$

(36c)

a result that will be used in deriving GSL below.

6.3. Generalized Second Law (GSL)

We impose (S4) on Gen-Th above to formulate a new thermodynamics that we have identified as Gen-GSL-Th for $\overline{\mathsf{\Sigma }}$, see Table 2, for the simple reason that (S4) leads to GSL as we now discuss. Due to (S4), we can focus on the extrema of $\left\{E_k\right\}$ and the directional ${\mathfrak{m}}_k$-evolution. The extrema and the forms of $\left\{E_k\right\}$ determine the extremum and the form of $E_{\mathrm{w}}$ for $\mathfrak{M}$ as shown by continuous curves in Figure 2. To see this, we consider the ensemble average $\mathbf{F}=〈\mathbf{F}〉$. By definition,

$$\mathbf{F}={\sum }_k{p}_k{\mathbf{F}}_k=0$$

at the extremum. As this must hold for any $\left\{p_k\right\}$, this requires that ${\mathbf{F}}_k=0$ for all microstates ${\mathfrak{m}}_k$. This thus shows that the averaging also generalizes microstate Mec-EQ-P to the thermodynamical principle of (stability and instability) equilibrium (Th-EQ-P) for $\mathfrak{M}$ in Gen-GSL-Th, see also Theorem 1, according to which the extremum $E_{\mathrm{eq}}$ of E controls $\mathfrak{M}$-evolution. We must not confuse this evolution with that produced by intervention required to prepare nonuniform $\mathfrak{M}$ out of uniform ${\mathfrak{M}}_{\mathrm{eq}}$ by internal constraints as discussed below; see also [16,92]. As ${\mathfrak{m}}_{k\mathrm{ueq}}$is, in effect, physically equivalent to a “nonexistent” microstate because of its instability, ${\mathfrak{M}}_{\mathrm{ueq}}$ is also physically nonexistent [23,24]. Even a realistic isolated system has a boundary separating it from its exterior so it is not truly isolated from the exterior. Because of this, Cl-Th only deals with ${\mathfrak{M}}^{\mathrm{s}}$ having the sink ${\mathfrak{M}}_{\mathrm{seq}}$ to which it asymptotically converges. For example, the behavior for the stable equilibrium is at the heart of Cahn-Hilliard equation for the energy in which the square of the gradient term is included to account for the interfacial energy due to nonuniformity, or a similar term in the Ginzburg-Landau equation. Accordingly, any ${\mathfrak{m}}_k^{\mathrm{s}}\in {\mathfrak{M}}^{\mathrm{s}}$ converges to ${\mathfrak{m}}_{k\mathrm{seq}}\in {\mathfrak{M}}_{\mathrm{seq}}$. Despite this, we also consider ${\mathfrak{M}}^{\mathrm{u}}$ to obtain additional and surprising information at the microstate level that is not available in Cl-Th, and allows for a reformulation of SL for negative T as discussed earlier.

Without (S4), Gen-Th also describes GSL-violation ($dQ=dW<0$), see Equation (11), and SL-violation ($dS<0$), resulting in violation thermodynamics (Viol-GSL-Th) of GSL and (Viol-Th) of SL, respectively, for $\overline{\mathsf{\Sigma }}$.

We justify the emergence of GSL within Gen-Th of $\overline{\mathsf{\Sigma }}$; accordingly, we do not impose any restriction on possible $\left\{p_k\right\}$. We focus on ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ and prove the following Lemma 1 for an equilibrium point of a mechanical ${\mathfrak{m}}_k$ and a thermodynamic $\mathfrak{M}$, which establishes that the inequality

$$dW\ge 0$$

is satisfied in any spontaneous infinitesimal process controlled by Mec-EQ-P due to (S4). This proves the first half of Equation (12). As a consequence, $dW<0$ must only happen in any nonspontaneous infinitesimal process that violates Mec-EQ-P.

Lemma 1.

During ${\mathfrak{m}}_{k\mathrm{eq}}$-controlled spontaneous evolution of ${\mathfrak{m}}_k$ of ${\overline{\mathsf{\Sigma }}}^D$ in Gen-GSL-Th, ${\mathfrak{m}}_k$ performsnonnegativemicrowork $\Delta W_k$ as ${\mathfrak{m}}_k\to {\mathfrak{m}}_k^{\prime }$. Performing ensemble average with arbitrary $\left\{p_k\right\}$ then determines thermodynamic stability (instability) of the resulting macrostate $\mathfrak{M}=({\mathfrak{M}}^s,{\mathfrak{M}}^u)$ in Σ, which spontaneously performs nonnegative macrowork $\Delta W$ in Gen-GSL-Th as $\mathfrak{M}\to {\mathfrak{M}}^{\prime }$ during some spontaneous process $\mathcal{P}$. Dissipation is the nonnegative macroheat $\Delta Q=\Delta W$.

Proof. 

(a) Microstate Evolution: The extremum of $E_k$ at $E_{k\mathrm{eq}}$ represents ${\mathfrak{m}}_{k\mathrm{eq}}=({\mathfrak{m}}_{k\mathrm{seq}},{\mathfrak{m}}_{k\mathrm{ueq}})$ that is uniform in ${\mathfrak{S}}_{\mathbf{X}}$; see Uniformity Theorem 1. We now consider a nonuniform ${\mathfrak{m}}_k=({\mathfrak{m}}_k^{\mathrm{s}},{\mathfrak{m}}_k^{\mathrm{u}})$ away from ${\mathfrak{m}}_{k\mathrm{eq}}$ as it spontaneously evolves to ${\mathfrak{m}}_k^{\prime }=({\mathfrak{m}}_k^{\prime \mathrm{s}},{\mathfrak{m}}_k^{\prime \mathrm{u}})$, see red and blue arrows in Figure 2. During this evolution, $E_k$ spontaneously decreasesto $E_k^{\prime }$ so that $\Delta E_k=-\Delta W_k\doteq E_k^{\prime }-E_k\le 0$. Thus, ${\mathfrak{m}}_k$performs spontaneous nonnegative microwork during its evolution towards ${\mathfrak{m}}_k^{\prime }$ given by

$$\Delta W_k=-\Delta E_k\ge 0.$$

(37a)

This proves the second half of Equation (12).

(b) Macrostate Evolution: We now average over ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{m}}_k^{\mathrm{u}}$ using arbitrary $p_k$ to obtain macroquantities of ${\mathfrak{M}}^{\mathrm{s}}$ and ${\mathfrak{M}}^{\mathrm{u}}$, respectively, in Gen-GSL-Th. The mechanical equilibrium microstate ${\mathfrak{m}}_{k\mathrm{eq}}$ determines the thermodynamic EQ macrostate ${\mathfrak{M}}_{\mathrm{eq}}=({\mathfrak{M}}_{\mathrm{seq}},{\mathfrak{M}}_{\mathrm{ueq}})$. The macroenergy $E\doteq <E>$ gives $E_{\mathrm{eq}}\doteq <E_{\mathrm{eq}}>$ for thermodynamic EQ and uniform macrostate ${\mathfrak{M}}_{\mathrm{eq}}$ in ${\mathfrak{S}}_{\mathbf{X}}$. The spontaneous process $\mathfrak{M}\to {\mathfrak{M}}^{\prime }$ results in the nonnegative spontaneous macrowork

$$\Delta W\doteq <\Delta W>\equiv -<\Delta E>\ge 0.$$

(37b)

Thus, dissipated macroheat $\Delta Q=\Delta W\ge 0$.

This completes the proof. □

It should be remarked that $<\Delta E>\ne \Delta E=0$.

The above proof only uses the behavior of $E_k$ in Figure 2. The same conclusion is also obtained by considering the macrowork function $E_{\mathrm{w}}$ shown there. It is easily seen from the definition of $E_{\mathrm{w}}$ in Equation (36c) that $<\Delta E>$ is nothing but the change $\Delta E_{\mathrm{w}}=-<\Delta W>$ during the spontaneous process $\mathcal{P}$.

We recall that there is no sign restriction on $\Delta W_k$ and $\Delta W$ in Gen-Th of $\mathsf{\Sigma }$. The fixed sign of $dW\ge 0$ in any process $\mathcal{P}$ is due to spontaneous irreversible processes [26], which according to Irr-P is dissipated or wasted as macroheat $dQ$ to ensure unchanging E.

We now prove the main Theorem.

Theorem 2.

We consider Σ in ${\mathfrak{S}}_{\mathbf{X}}$. The spontaneous ${\mathfrak{M}}_{\mathit{eq}}$-evolution of any $\mathfrak{M}$ in Gen-GSL-Th during which it either converges to the sink ${\mathfrak{M}}_{\mathit{seq}}$ for ${\mathfrak{M}}^s$ or runs away from the source ${\mathfrak{M}}_{\mathit{ueq}}$ for ${\mathfrak{M}}^u$ directly leads to GSL in Equation (12), making it internally consistent with analytical mechanics. A violation of GSL requires $\Delta W<0$ that cannot occur in Σ. Emergence of $\Delta S<0$ due to instability in the spontaneous evolution in ${\mathfrak{M}}^u$ is not a violation of SL because of its negative T. This makes Viol-GSL-Th and Viol-Th inconsistent with analytical mechanics and the BCM proposal.

Proof. 

In Gen-GSL-Th, the isolated $\mathsf{\Sigma }$ in $\mathfrak{M}\ne {\mathfrak{M}}_{\mathrm{eq}}$ either spontaneously relaxes as t increases to ${\mathfrak{M}}_{\mathrm{seq}}$ during which $dS\ge 0$ or runs off from ${\mathfrak{M}}_{\mathrm{ueq}}$ during which $dS\le 0$. As $\mathfrak{M}\to {\mathfrak{M}}^{\prime }$, we have $\Delta W\ge 0$ from Lemma 1. It then follows from Equation (12) that the corresponding spontaneous macroheat $\Delta Q\doteq {\int }_{\mathfrak{M}}^{{\mathfrak{M}}^{\prime }}TdS$ is nonnegative, which proves GSL in Equation (13) for any $\mathfrak{M}$ in $\mathsf{\Sigma }$, and establishes its consistency with Mech-EQ-P of analytical mechanics. A GSL-violation requires $dQ<0$ that can only happen if $dW<0$ such as along the double green arrow in Figure 2 near ${\mathfrak{M}}_{\mathrm{seq}}$. As this evolution violates Lemma 1, it is nonspontaneous and cannot occur in $\mathsf{\Sigma }$. This makes any GSL violation impossible. The requirement $dQ\ge 0$ is consistent with $T\ge 0$ and $dS\ge 0$ for ${\mathfrak{M}}^{\mathrm{s}}$, and $T<0$ and $dS<0$ for ${\mathfrak{M}}^{\mathrm{u}}$. The instability in ${\mathfrak{M}}^{\mathrm{u}}$ and $\Delta S<0$ during its spontaneous evolution does not imply violating SL. It is also clear that Viol-GSL-Th and Viol-Th can only hold for nonspontaneous processes. As they cannot occur in $\mathsf{\Sigma }$, they are inconsistent with analytical mechanics and the BCM proposal. This completes the proof. □

The claims also apply to ${\mathsf{\Sigma }}_{\mathrm{E}}$ and ${\mathsf{\Sigma }}_{\mathrm{M}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ without any change.

The following calculation clarifies the claims for a one-particle system in ${\mathfrak{S}}_{\mathbf{X}}$.

6.4. A Simple Example

It is instructive to consider, as an example, a two-level particle as our system $\mathsf{\Sigma }$. The entropy is $S=-p_{1}ln{p}_1-p_{2}ln{p}_2$, where $p_1$and $p_2$ are the probabilities of the two levels. Then, $dS=ln[(1-p_1)/p_1]d{p}_1$. The maximum of S occurs at $p_1=p_2=1/2$, which represents the EQ point. For any other value of $p_1$, the behavior is different for $dS>0$ and $dS<0$. In the former case, $dS$ brings $p_1$ and $p_2$ closer to the EQ point making it a sink, while in the latter case, $p_1$and $p_2$ move away from the EQ point making it a source. This is consistent with our discussion above and brings out the difference of ${\mathfrak{M}}^{\mathrm{s}}$ two ${\mathfrak{M}}^{\mathrm{u}}$ vividly. To make the example more interesting, we need to associate energies to the two level. We capture nonuniformity of the system by adding small contributions ${ϵ}_1$ and ${ϵ}_2$ to the equilibrium energies $e_1$ and $e_2>e_1$, respectively, so that $E_1=e_1+{ϵ}_1$ and $E_2=e_2+{ϵ}_2$. We also take $p_1=1/2-\delta$ and $p_2=1/2+\delta$, with $\delta$ a small quantity. We keep only leading order terms. For ${ϵ}_1>0$ and ${ϵ}_2>0$, we are considering ${\mathfrak{M}}^{\mathrm{s}}$; see the blue solid curve. Along the blue arrow, we have $dW\approx ({ϵ}_1+{ϵ}_2)/2$, and $dQ\approx (e_1-e_2)\delta$. From Irr-P, we have $(e_2-e_1)\delta +({ϵ}_1+{ϵ}_2)/2=0$, which is the condition for $dE=0$, as is easily seen from equating $E_{\mathrm{eq}}=(e_1+e_2)/2$, and $E\approx E_{\mathrm{eq}}+(e_2-e_1)\delta +({ϵ}_1+{ϵ}_2)/2$, and $\delta <0,dS>0$, making the system more uniform and $T>0$. For ${ϵ}_1<0$ and ${ϵ}_2<0$, we are considering ${\mathfrak{M}}^{\mathrm{u}}$; see the red solid curve. Along the red arrow, we again have $dW>0$ but $dS<0$ as system becomes more nonuniform. In both cases, $dQ>0$ so GSL remains intact, but the behavior of $dS<0$ shows that $T<0$ as expected.

7. Restriction Thermodynamics (Rest-Th)

7.1. Formulation

To make S a state function for ${\mathsf{\Sigma }}_{\mathrm{M}}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ requires an important restriction on $\left\{p_k\right\}$, but not on $\left\{d{p}_k\right\}$. We now put a particular form of restriction on $\left\{p_k\right\}$ for $\mathfrak{M}$ in ${\mathfrak{S}}_{\mathbf{Z}}$ of ${\mathsf{\Sigma }}_{\mathrm{M}}$ that ensures that its E becomes a state function of its S and ${\mathsf{\xi }}_k$, which in turn means that S is a state function of E and ${\mathsf{\xi }}_k$ in ${\mathfrak{S}}_{\mathbf{Z}}$, whereas it is not a state function for ${\mathsf{\Sigma }}_{\mathrm{E}}$. Using E as a state function $E(S,{\xi }_{\mathrm{E}},{\xi }_{\mathrm{V}}),N$ and V fixed, we obtain the Gibbs fundamental relation

$$dE=TdS-dW,$$

(38)

where $T\doteq {\left(\partial E/\partial S\right)}_{\mathsf{\xi }}$ is the thermodynamic temperature of ${\mathsf{\Sigma }}_{\mathrm{M}}$, and $dW=\mathbf{F}·d\mathsf{\xi }$, see Equation (18), is the macrowork. It is the first law now in terms of $dS$ and $dW$ and yields the restriction thermodynamics (Rest-Th), which is nothing but Gen-Th for ${\mathsf{\Sigma }}_{\mathrm{M}}$. The first term is nothing but $dQ$ in Equation (30), which immediately establishes

$$dQ=TdS,T\doteq {\left(\partial E/\partial S\right)}_{\mathsf{\xi }}.$$

(39)

It should be noted that the relationship [47] is simply a the mathematical consequence of the state function $E(S,{\mathsf{\xi }}_k)$. The resulting form of Gen-Th with S and E as state functions is called restriction thermodynamics (Rest-Th) for ${\mathsf{\Sigma }}_{\mathrm{M}}$, which still allows $dS$ of either sign so

$$dQ=TdS⪌0$$

in Rest-Th, unless supplemented by (S4). Thus, the functional dependence $E(S,{\mathsf{\xi }}_k)$ still does not enforce SL ($TdS\ge 0$) by itself as was also the case for Gen-Th, so Rest-Th allows us to make direct connection with Cl-Th and Viol-Th.

The justification for keeping T in the new formulation $TdS\ge 0$ will be justified within the context of generalized SL (GSL). We thus note that $dQ$ and $dS$ have the same signature for positive T, but opposite signature for negative T.

From Equation (39) follows a simple but very remarkable and extremely profound relation

$${\left.dQ\right|}_E=TdS={\left.dW\right|}_E$$

(40)

for ${\mathsf{\Sigma }}_{\mathrm{M}}$ in Gen-Th (we add an extra suffix E as a reminder that E is fixed, and should not be confused the suffix in ${\xi }_{\mathrm{E}}$) so it remains valid both in Cl-Th and Viol-Th for ${\mathsf{\Sigma }}_{\mathrm{M}}$:

$$\begin{array}{c}TdS\ge 0⟹{\left.dW\right|}_E\ge 0,\\ TdS<0⟹{\left.dW\right|}_E<0.\end{array}$$

(41)

But what is most remarkable about Equation (40) is that it provides a purely mechanical definition of stochastic entropy change by using the first law that holds in all kinds of thermodynamics discussed here.

We now justify that T for $\mathsf{\Sigma }$ in Equation (7) is no difference than the T defined in Equation (39) for ${\mathsf{\Sigma }}_{\mathrm{M}}$, when $\mathsf{\Sigma }$ becomes ${\mathsf{\Sigma }}_{\mathrm{M}}$. All we need to do is to set $dE=0$ in Equation (38) or use Equation (40). We immediately see that $T\doteq {\left(\partial E/\partial S\right)}_{\mathsf{\xi }}$ for ${\mathsf{\Sigma }}_{\mathrm{M}}$ is no different than T in Equation (7) by letting $\mathsf{\Sigma }$ become equal to ${\mathsf{\Sigma }}_{\mathrm{M}}$. This justifies that

Conclusion 1.

T in Equation (7) is a genuine NEQ definition of temperature for a system of any size.

This completes the introduction of the Rest-Th.

7.2. Consequences of Rest-Th for $dS>0$ and $dS<0$

We apply Rest-Th to a macrostate $\mathfrak{M}$ of ${\mathsf{\Sigma }}_{\mathrm{M}}$ used for Equation (25), in which ${\xi }_{k\mathrm{E}}$ must be replaced by its ensemble average ${\xi }_{\mathrm{E}}\doteq E_1/n_1-E_2/n_2$. A simple calculation using the state function $S(E,{\mathsf{\xi }}_k)$ and $dE=0$ yields

$$dS=n_1{n}_2[({\beta }_1-{\beta }_2)d{\xi }_{\mathrm{E}}+({\beta }_1{P}_1-{\beta }_2{P}_2)d{\xi }_{\mathrm{V}}],$$

(42)

where ${\beta }_1,P_1$, and ${\beta }_2,P_2$ are the inverse temperature and pressure of ${\mathsf{\Sigma }}_{1\mathrm{M}}$ and ${\mathsf{\Sigma }}_{2\mathrm{M}}$, respectively, and $\beta \doteq 1/T$$=n_1{\beta }_1+n_2{\beta }_2$ is the inverse temperature of $\mathsf{\Sigma }$. The two terms on the right side in Equation (42) represent entropic contributions due to the two internal variables ${\xi }_{\mathrm{E}}$ and ${\xi }_{\mathrm{V}}$ in Rest-Th; each must be nonnegative for SL or negative for its violation. In terms of $d{E}_1=-d{E}_2$ as the macroenergy change and $d{V}_1=-d{V}_2$ as the volume change of ${\mathsf{\Sigma }}_{1\mathrm{M}}$, we have

$$d{\xi }_{\mathrm{E}}={\displaystyle \frac{1}{n_1{n}_2}}d{E}_1,d{\xi }_{\mathrm{V}}={\displaystyle \frac{1}{n_1{n}_2}}d{V}_1.$$

(43)

Using $\gamma \doteq -1/T$ introduced by Ramsey [69], whose numerical values define the “hotness” of ${\mathsf{\Sigma }}_{\mathrm{M}}$ as it increases from $-\infty$ to $+\infty$ covering positive and negative temperatures, Equation (42) becomes

$$dQ=TdS=-(\Delta \gamma /\gamma )d{E}_1-(\Delta \left(\gamma P\right)/\gamma )d{V}_1$$

(44)

for ${\mathsf{\Sigma }}_{\mathrm{M}}$ in Rest-Th, where $\Delta \gamma \doteq ({\gamma }_2-{\gamma }_1)$, and $\Delta \left(\gamma P\right)\doteq ({\gamma }_2{P}_2-{\gamma }_1{P}_1)$; $\Delta \gamma >0$ means that ${\mathsf{\Sigma }}_{2\mathrm{M}}$ is hotter than ${\mathsf{\Sigma }}_{1\mathrm{M}}$, and vice-versa. For GSL to hold, we require $dS$ in $dQ=TdS$ to be $\ge 0$ for $T>0$$\left(\gamma <0\right)$ and $<0$ for $T<0$$\left(\gamma >0\right)$. For GSL violation, we require $dS<0$ for $T>0$ and $dS>0$ for $T<0$. Both situations are considered above. Let us consider just the first term above to be specific by setting $d{V}_1=0$. We consider various scenarios; (a) and (b) refer to ${\mathfrak{M}}^{\mathrm{s}}$ having $T>0$, and (c) and (d) refer to ${\mathfrak{M}}^{\mathrm{u}}$ having $T<0$. The analysis here is more extensive compared to an earlier preliminary and incomplete investigation [53], where the issue of catastrophic evaluation was first discussed.

(a)

For $dQ=dW>0$ and $T>0$, we must have $d{E}_1>0$ for ${\mathfrak{M}}^{\mathrm{s}}$ so that macroheat flows from hot to cold as expected in which it converges to ${\mathfrak{M}}_{\mathrm{seq}}$ due to an attractive SI-macroforce ${\mathbf{F}}^{\mathrm{seq}}$ pointing towards SEQ. The SI-evolution of ${\mathfrak{M}}^{\mathrm{s}}$ is spontaneous due to ${\mathbf{F}}^{\mathrm{seq}}$ pointing towards its sink ${\mathfrak{M}}_{\mathrm{seq}}$ as seen from the blue arrows. Therefore, as expected in this case, $dS>0$ so Cl-Th and Gen–GSL-Th remain valid. This is the most common situation.

(b)

For $dQ=dW<0$ and $T>0$, $d{E}_1<0$ so that macroheat flows from cold to hot, and ${\mathfrak{M}}^{\mathrm{s}}$ runs away from ${\mathfrak{M}}_{\mathrm{seq}}$ due to some repulsive macroforce ${\mathbf{F}}_{\mathrm{repu}}^{\mathrm{s}}$ along green arrows, distinct from the SI-macroforce ${\mathbf{F}}^{\mathrm{seq}}$, to eventually converge to ${\mathfrak{M}}_{\mathrm{cata}}^{\mathrm{s}}$ by becoming more and more nonuniform. The evolution of ${\mathfrak{M}}^{\mathrm{s}}$ is not spontaneous as is in (a) and ${\mathfrak{M}}_{\mathrm{seq}}$ is no longer the sink. In this case, $dS<0$ (Viol-Th), but we also violate GSL (Gen–GSL-Th), but the violation is because of nonspontaneous processes.

(c)

For $dQ=dW>0$ but $T<0$ for ${\mathfrak{M}}^{\mathrm{u}}$, $d{E}_1<0$ so that macroheat flows from cold to hot and ${\mathsf{\Sigma }}_{\mathrm{M}}$ becomes more and more nonuniform because of the instability in it as discussed above. The spontaneous evolution of ${\mathfrak{M}}^{\mathrm{u}}$ from its source ${\mathfrak{M}}_{\mathrm{ueq}}$ under the repulsive SI-macroforce ${\mathbf{F}}^{\mathrm{ueq}}$ along the red arrows is catastrophic in that it converges to a catastrophic macrostate ${\mathfrak{M}}_{\mathrm{cata}}^{\mathrm{u}}$. In this case, $dS<0$ so it appears that SL is violated, but GSL (Gen–GSL-Th) remains valid. However, as the process remains spontaneous, $dS<0$ must not be considered as violating SL; GSL remains satisfied.

(d)

For $dQ=dW<0$ and $T<0$, we must have $d{E}_1>0$ so that macroheat flows from hot to cold in ${\mathfrak{M}}^{\mathrm{u}}$. In this case, ${\mathfrak{M}}^{\mathrm{u}}$ nonspontaneously converges to ${\mathfrak{M}}_{\mathrm{ueq}}$ due to an attractive macroforce ${\mathbf{F}}_{\mathrm{attr}}^{\mathrm{u}}$, which is distinct from the repulsive SI-macroforce ${\mathbf{F}}^{\mathrm{ueq}}$, with$dS>0$ so SL seems to remain valid but not Gen–GSL-Th. As ${\mathfrak{M}}_{\mathrm{ueq}}$ is no longer the source for ${\mathfrak{M}}^{\mathrm{u}}$-evolution in the nonspontaneous process that is not covered by SL, $dS>0$ must not be taken as validating SL; GSL fails as expected.

We analyze nonspontaneous processes (b) and (d) further. As these processes are not controlled by SL, they cannot and must not be taken as examples of violating SL. We first consider (b). As ${\mathfrak{M}}^{\mathrm{s}}$ runs away from its sink ${\mathfrak{M}}_{\mathrm{seq}}$ to a new macrostate ${\mathfrak{M}}^{\prime \mathrm{s}}$, the latter further runs away from ${\mathfrak{M}}^{\mathrm{s}}$ by its ${\xi }_{\mathrm{E}}$ deviating further from its value in ${\mathfrak{M}}^{\mathrm{s}}$. Thus, we get successive macrostates ${\mathfrak{M}}^{\left(p\right)\mathrm{s}},p=0,1,2,\cdots$ which run away from the sink farther and farther, during which $d{S}^{\left(p\right)}$ remains non-positive as p increases. Therefore, the evolution to ${\mathfrak{M}}_{\mathrm{cata}}^{\mathrm{s}}$ is catastrophic in that it makes ${\mathsf{\Sigma }}_{\mathrm{M}}$ highly nonuniform due to unexplained nonsystem repelling macroforce ${\mathbf{F}}_{\mathrm{repu}}^{\mathrm{s}}$ that mutilates ${\mathfrak{M}}_{\mathrm{seq}}$. It follows from the stability of ${\mathfrak{M}}^{\mathrm{s}}$ considered here that ${\mathsf{\Sigma }}_{1\mathrm{M}}$ and ${\mathsf{\Sigma }}_{2\mathrm{M}}$ are also stable so their specific heats at constant volume are nonnegative. As $d{E}_1=-d{E}_2<0$, the disparity $\Delta \gamma$ continues to increase with ${\mathsf{\Sigma }}_{2\mathrm{M}}$ getting more hot and ${\mathsf{\Sigma }}_{1\mathrm{M}}$ getting more cold, until $\Delta \gamma$takes some maximum value $\Delta {\gamma }_{\mathrm{cata}}$ in ${\mathfrak{M}}_{\mathrm{cata}}^{\mathrm{s}}$. We now consider (d), where a similar discussion can also be carried out for ${\mathfrak{M}}^{\mathrm{u}}$ but with different conclusions. Here, ${\mathfrak{M}}^{\mathrm{u}}$ gets more uniform but the uniformity is not due to any SI-macroforce so the evolution is not governed by SL.

It should be clear from the discussion that GSL seems to capture spontaneous processes at both positive and negative temperatures. In this sense, GSL subsumes SL.

The discussion is easily extended to include the second term in Equation (44) with same conclusions that remain consistent with Lemma 1.

We now summarize our conclusions from Rest-Th.

C1

$dS>0$ is not always a consequence of spontaneous processes. In (d), it is a consequence not only of negative T but also of negative $dW$ performed by nonsystem forces that result in a nonspontaneous process. This process is not controlled by SL so $dS>0$ has no significance of spontaneity.

C2

$dS<0$ in (b) and (c) shows that it is not always a consequence of nonspontaneous processes. In (b), it is a consequence only of negative $dW$ performed by nonsystem forces that result in a nonspontaneous process such as during the creation of internal constraints as explained by Callen [16] and below. Again, this process is not controlled by SL, while SI-macrowork $dW>0$ or the removal of the internal constraint is controlled by SL so $dS<0$ in (c) has no significance for SL-violation.

C3

From (a) and (c), we observe that GSL is always a consequence of spontaneous processes, but fails for nonspontaneous processes in (b) and (c).

7.3. Internal Constraint (IC)

We now discuss the relevance of internal constraints (IC’s) [16] for GSL/SL by focusing on Figure 4, which describes the creation of an internal constraint in an isolated system $\mathsf{\Sigma }$ that is initially in a stable EQ macrostate. It is shown by ${\mathsf{\Sigma }}_{\mathrm{seq}}$ in Figure 4a. The system is in contact with a work medium $\tilde{\mathsf{\Sigma }}$ that works on the red partition to move it in to and out of ${\mathsf{\Sigma }}_{\mathrm{seq}}$ that divides $\mathsf{\Sigma }$ into two parts of equal volume. The partition is completely noninteracting (inert) in that it does not allow for flow of any kind across it. The blue partition between ${\mathsf{\Sigma }}_{\mathrm{seq}}$ and $\tilde{\mathsf{\Sigma }}$ provides thermal insulation between them so $\tilde{\mathsf{\Sigma }}$ manipulates its macrostate ${\mathfrak{M}}_{\mathrm{seq}}$ by inserting the noninteracting partition instantly into ${\mathsf{\Sigma }}_{\mathrm{seq}}$. The insertion, being instantaneous, picks out a particular microstate ${\mathfrak{m}}_{k\mathrm{seq}}$ in each member of the Gibbs ensemble by giving rise to two microstates of the two parts of $\mathsf{\Sigma }$, which we denote by ${\mathfrak{m}}_{k_{\mathrm{L}}}$ and ${\mathfrak{m}}_{k_{\mathrm{R}}}$. Just as in the Szilard problem [37], in which the insertion leaves the number of particles in the two parts different, there is no guarantee that the insertion divides $\mathbf{X}$ evenly so the density $\mathbf{X}/N$ remains the same in the two parts. Thus, insertion will not divide $\mathbf{X}$ into equal halves for the two parts. This means that the two parts will not be in equilibrium or reach equilibrium with each other because of the noninteracting partition, even though each will come to its own SEQ macrostate in time. This situation is an example of IC discussed by Callen. We denote this state of $\mathsf{\Sigma }$ by ${\mathsf{\Sigma }}_{\mathrm{iceq}}$ as shown in Figure 4b. The insertion results in reducing the entropy of $\mathsf{\Sigma }$ and has been discussed recently in [92]. This reduction is not a violation of SL as the insertion is done by an outside agent so the process is not spontaneous. The same is also true of Maxwell’s demon who reduces the entropy by acting as an outside agent.

In contrast, the process (b)⟶(c) in which IC is removed results in spontaneous mixing in the two parts. It is this part that is controlled by SL.

From what we have discussed above for negative T and ${\mathfrak{M}}_{\mathrm{ueq}}$ for a system ${\mathsf{\Sigma }}_{\mathrm{ueq}}$ in unstable equilibrium, the manipulation, i.e., creation of IC will suppress spontaneous evolution going “downhill” and result in ${\mathfrak{M}}^{\mathrm{u}}$ run “uphill” towards ${\mathfrak{M}}_{\mathrm{ueq}}$. During this “uphill” climb, which is a nonspontaneous reaction to the creation of IC in ${\mathfrak{M}}^{\mathrm{u}}$, entropy will increase so $dS>0$ during (a)⟶(b), and $dQ<0$. Thus, $dQ<0$ happens during a nonspontaneous process, and must not be treated as a violation of GSL/SL. For that, we need to find a violation in a spontaneous process.

8. Summary and Discussion

We are ready to summarize our novel approach to understand irreversibility, conditions required for the validity of SL, and its violation by following the BCM proposal and focusing on an isolated system. As a consequence of this approach, we obtain a generalization GSL of the second law, whch is valid for a system of any size. We also highlight our findings and discuss their tantalizing consequences.

8.1. Reverse Approach, Mec-EQ-P, and Irreversibility

Th-EQ-P for ${\mathfrak{M}}^{\mathrm{s}}$ is a consequence of SL in Cl-Th for ${\mathsf{\Sigma }}_{\mathrm{M}}$ as is well known. Instead, by taking the reverse approach, see Equation (8), we prove not only Th-EQ-P for ${\mathfrak{M}}^{\mathrm{s}}$ and ${\mathfrak{M}}^{\mathrm{u}}$ but also GSL ($dQ=dW\ge 0$) for $\overline{\mathsf{\Sigma }}$ by using (S4), i.e., Mec-EQ-P, of analytic mechanics [instead of SL as is done in Cl-Th] in Gen-GSL-Th. GSL turns out to be a consequence of analytical mechanics and the first law. It is not a different law as is commonly thought of SL in Cl-Th. This shows the strength of our reverse approach. As ${\mathfrak{m}}_k^{\mathrm{s}}$ and ${\mathfrak{M}}^{\mathrm{s}}$ asymptotically converge to ${\mathfrak{m}}_{k\mathrm{seq}}$ and ${\mathfrak{M}}_{\mathrm{seq}}$, respectively, by removing nonuniformity gradually, $S\left(t\right)$ continuously increases and reaches its maximum [44]. As ${\mathfrak{m}}_k^{\mathrm{u}}$ and ${\mathfrak{M}}^{\mathrm{u}}$ run away from ${\mathfrak{m}}_{k\mathrm{ueq}}$ and ${\mathfrak{M}}_{\mathrm{ueq}}$, respectively, by increasing nonuniformity gradually, $S\left(t\right)$ continuously decreases and reaches some minimum $S_{\mathrm{cats}}$, even though GSL remains valid. With (S4) in Gen-GSL-Th, BCM proposal imposes a very strong constraint of nonnegativity on $d{W}_k$ for $\overline{\mathsf{\Sigma }}$. The nonnegative average $dW$ through Irr-P determines $dQ$, which imposes a restriction on $\left\{d{p}_k\right\}$ to have a particular form $\left\{d{p}_k^{\mathrm{GSL}}\right\}$ to ensure $dQ\ge 0$ Definition 4. As a side remark, we observe another important consequence of the impossibility of GSL violation in Theorem 2. It has been used to construct a No-Go theorem for GSL/SL-violation earlier [93], which requires rejecting the BCM proposal by discarding Mec-EQ-P of analytical mechanics used in (S4). This consequence is a disaster for the foundation of thermodynamics, and has not been recognized so far. Indeed, the violation cannot be taken as a viable possibility as it defies analytic mechanics. The conclusion provides not only a tantalizing insight into SL-violation in ${\mathsf{\Sigma }}_{\mathrm{M}}$ for the first time by recognizing that Viol-Th requires rejecting analytical mechanics, but is also a testament to the robustness of the BCM proposal. This elevates Mec-EQ-P to be of primary relevance for thermodynamic foundation of GSL/SL in Gen-GSL-Th Definition 4, and clarifies the significance of “asymptotic approach to stable equilibrium” for a thermodynamic system in ${\mathfrak{M}}^{\mathrm{s}}$ in time after being momentarily disturbed [16,23,24,94] from ${\mathfrak{M}}_{\mathrm{seq}}$, which forms the cornerstone of Cl-Th. We also extend GSL/SL now to cover ${\mathfrak{M}}^{\mathrm{u}}$, where negative T plays an important role. Thus, SL must always be stated along with T, a fact that has not been appreciated so far. Indeed, it is $dQ$ and not $dS$ that controls irreversibility and dissipation.

Irreversibility in $\overline{\mathsf{\Sigma }}$ makes its relaxation to ${\mathfrak{M}}_{\mathrm{eq}}$ acyclic. Even for a stable $\mathsf{\Sigma }$, the relaxation cannot give rise to a cyclic process. The only way to perform a cyclic process is to drive $\overline{\mathsf{\Sigma }}$ that generates a nonspontaneous process in $\overline{\mathsf{\Sigma }}$ by an external work agent, a work medium $\tilde{\mathsf{\Sigma }}$ as in Figure 4. We postpone a discussion of it to a future publication.

8.2. Macroheat, Entropy and NEQ Temperature

For a thermodynamic description and to develop Gen-Th, we follow BCM proposal in which ${\overline{\mathsf{\Sigma }}}^{\mathrm{D}}$ is endowed with stochasticity Definition 4, entropy, macrowork and macroheat. The first law in Gen-Th is formulated in terms of the last two macroquantities in Equation (30), which is valid for $\overline{\mathsf{\Sigma }}$ of any size in our approach. However, SL is introduced in terms of $dS$ in Cl-Th as a law independent of the first law, with temperature playing no role. Therefore, SL is erroneously assumed to hold for both positive and negative temperatures. We avoid this limitation by formulating GSL in terms of $dQ$ and not $dS$. The NEQ temperature is identified by Equation (7), which allows GSL to be expressed in terms of $dS$, but T remains an integral part of GSL. Our earlier work [44] is used to show that stable macrostates ${\mathfrak{M}}^{\mathrm{s}}$ correspond to $T\ge 0,dS\ge 0$, and unstable macrostates ${\mathfrak{M}}^{\mathrm{u}}$ correspond to $T<0,dS<0$. In both cases, $dQ\ge 0$.

8.3. GSL/SL

Gen-Th for $\overline{\mathsf{\Sigma }}$ always satisfies the first law $dE=0$, see Setion Section 6, from which follows Irr-P in (S2), for any $\left\{p_k\right\}$ that determines $dW$. Irr-P intertwines $dW$ and $dQ$, but without imposing restrictions on $d{p}_k$ Definition 4 so $dQ$ has either sign but always determined by $dW$. It is here the relevance of microscopic Mec-EQ-P for $\left\{{\mathfrak{m}}_k\right\}$, which has not been properly recognized so far, becomes central in our reverse approach.

Conclusion 2.

Mec-EQ-P is finally identified as the root cause of the stochastic principle of GSL/SL that is surprisingly and completely determined by mechanical works $\left\{d{W}_k\ge 0\right\}$ and $dW$ alone in Gen-GSL-Th for $\overline{\mathsf{\Sigma }}$.

Thus,

Conclusion 3.

GSL is a consequence of analytical mechanics, the foundation of the BCM proposal, with $dQ\ge 0$ directly related to $dS⪌0$ depending on $T⪌0$, respectively. This conclusion provides a direct proof of SL for ${\mathfrak{M}}^s$ for $T\ge 0$and $dS\ge 0$, and extends it to ${\mathfrak{M}}^u$ for $T<0$and $dS\le 0$, a very tantalizing extension, which corrects a common misconception of SL about negative T [69].

As negative temperatures are not physically impossible [11,69,70,71], it is quite surprising to realize that $dS<0$ is not a violation of SL, if $T<0$. This has not been recognized before; however, see [95]. Thus,

Conclusion 4.

Gen-GSL-Th of $\overline{\mathsf{\Sigma }}$ supersedes Cl-Th for ${\mathsf{\Sigma }}_M$. In addition, Viol-Th and Viol-GSL-Th cannot be taken as a viable possibility within the BCM proposal.

It is clear from Theorem 2 that $dS⪌0$ alone without any reference to T cannot be used to describe SL or its violation, a point that has not been appreciated so far and highlights the role of T for SL.

8.4. Violation of GSL/SL and IC

Let us consider a nonspontaneous evolution requiring $T\ge 0$ and $dS<0$ for ${\mathfrak{M}}^{\mathrm{s}}$, and $T<0$ and $dS>0$ for ${\mathfrak{M}}^{\mathrm{u}}$. In both cases, $dQ<0$. During this evolution, ${\mathfrak{M}}^{\mathrm{s}}$ becomes less and less uniform ($dS<0$) and ${\mathfrak{M}}^{\mathrm{u}}$ becomes more and more uniform ($dS>0$).Thus, ${\mathfrak{M}}^{\mathrm{s}}$ runs off from the uniform sink ${\mathfrak{M}}_{\mathrm{seq}}$ catastrophically to asymptotically approach an extremely nonuniform macrostate ${\mathfrak{M}}_{\mathrm{cats}}^{\mathrm{s}}$ so that $S_{\mathrm{cats}}^{\mathrm{s}}<<S_{\mathrm{seq}}$ as if ${\mathfrak{M}}_{\mathrm{seq}}$ is unstable. Similarly, ${\mathfrak{M}}^{\mathrm{u}}$ runs towards and terminates in the uniform source ${\mathfrak{M}}_{\mathrm{ueq}}$ along the direction opposite to red arrows with S increasing to $S_{\mathrm{ueq}}$ as if ${\mathfrak{M}}_{\mathrm{ueq}}$ is stable. Both possibilities require some external agent as discussed above in Section 7.3, see Figure 4, for $T>0$ ($T<0$) to obtain $dS<(>)0$ by “manipulating ${\mathfrak{M}}_{\mathrm{seq}}$ (${\mathfrak{M}}_{\mathrm{ueq}}$)” to drive ${\mathfrak{M}}^{\mathrm{s}}$ (${\mathfrak{M}}^{\mathrm{u}}$) run away from (towards) the sink ${\mathfrak{M}}_{\mathrm{seq}}$ (${\mathfrak{M}}_{\mathrm{ueq}}$) in a nonspontaneous manner; see Section 7, where various possibilities are discussed. If $\overline{\mathsf{\Sigma }}$ in Figure 4b is now detached from $\tilde{\mathsf{\Sigma }}$, the result will be an internal constraint discussed by Callen [16], removal of which in (b)→(c) will initiate spontaneous processes to increase the entropy in accordance with GSL/SL as it must. A similar situation occurs in the demon paradox [42,96,97], in which the demon starts manipulating ${\mathfrak{M}}_{\mathrm{seq}}$ in a nonspontaneous manner as discussed recently [92,98,99]. Thus, there cannot be a genuine GSL/SL violation in $\overline{\mathsf{\Sigma }}$, which is consistent with the No-Go theorem [93].

8.5. Interacting System

To verify some of the predictions of GSL/SL in the labratory, one has to treat an interacting system, and not an isolated system; however, also see Remark 6. It is then important to consider a macroscopic system ${\mathsf{\Sigma }}_{\mathrm{M}}$ in contact with a medium $\tilde{\mathsf{\Sigma }}$ as shown in Figure 1b. This is so that not only their energies but also entropies satisfy additivity.

We will only focus on positive temperatures, where most of the experiments will be performed that deal with stable macrostates ${\mathfrak{M}}^{\mathrm{s}}$. Extension to negative temperatures is easily done. It is easy to show [40,44,47,53] that the Irr-P in Equation (11) remains valid except that it is now expressed in terms of irreversible macroquantities:

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

(45)

GSL is now expressed as

$$d_{\mathrm{i}}Q\equiv d_{\mathrm{i}}W\ge 0;d_{\mathrm{i}}{W}_k\ge 0,\forall k,$$

(46)

where the internal microwork

$$d_{\mathrm{i}}{W}_k\doteq d{W}_k-d_{\mathrm{e}}{W}_k$$

follows from the dashed-dot curves in Figure 2, which now express the change $d_{\mathrm{i}}{E}_k$ due to explicit time dependence in ${\mathfrak{S}}_{\mathbf{X}}$ or due to internal variables in ${\mathfrak{S}}_{\mathbf{Z}}$. The changes due to $\mathbf{X}$ govern the exchange contribution $d_{\mathrm{e}}{E}_k$ that is subtracted from $d{E}_k$ to determine $d_{\mathrm{i}}{E}_k$.

Consider $d_{\mathrm{i}}Q\equiv dQ-d_{\mathrm{e}}Q=TdS-T_0{d}_{\mathrm{e}}S$, $T_0$ being the temperature of the medium, in which the generalized macroheat $dQ$ must be replaced by $TdS$ in terms of the temperature $T\doteq \partial E/\partial S$ of ${\mathsf{\Sigma }}_{\mathrm{M}}$ [40,47,53]. We use this to recast Irr-P as

$$d_{\mathrm{i}}Q=T{d}_{\mathrm{i}}S+(T-T_0)d_{\mathrm{e}}S=d_{\mathrm{i}}W.$$

(47)

We now point out an important aspect of dissipation in the interacting case, which is either measure by the lost macrowork $d_{\mathrm{i}}W$ or equivalently by dissipated macroheat $d_{\mathrm{i}}Q$. We see that

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

(48)

What we observe is that the dissipated macroheat or lost macrowork is not identical to $T{d}_{\mathrm{i}}S$ as was the case for an isolated ${\mathsf{\Sigma }}_{\mathrm{M}}$ for which the top equation in Equation (13) holds. Indeed, their difference

$$T{d}_{\mathrm{i}}S-d_{\mathrm{i}}Q=(T_0-T)d_{\mathrm{e}}S\doteq T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}\ge 0$$

is the irreversible macroheat $T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}$ generated due to exchange macroheat between $\mathsf{\Sigma }$ and $\tilde{\mathsf{\Sigma }}$. To understand the significance of $T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}$, we turn to the discussion in Section 7.2 in which we treat ${\mathsf{\Sigma }}_1$ and ${\tilde{\mathsf{\Sigma }}}_2$ as $\mathsf{\Sigma }$ and $\tilde{\mathsf{\Sigma }}$, respectively, and take the limit $n_1\to 0$ and $n_2\to 1$. We consider the first term in Equation (42) by not allowing any change in the volume V of $\mathsf{\Sigma }$ so $d_{\mathrm{i}}W=0$. We also express $d{\xi }_{\mathrm{E}}$ in terms of $d{E}_1=d_{\mathrm{e}}Q$ from Equation (43). It can be easily verified that the second term on the right side above is nothing but the irreversible macroheat $T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}$ due to macroheat exchange. Our new approach has thus reproduced the well-known result in classical thermodynamic [26].

The important observation, however, is that

$$T{d}_{\mathrm{i}}S\ge d_{\mathrm{i}}Q,$$

(49)

the difference being $T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}$; the equality occurs only when there is no macroheat exchange between $\mathsf{\Sigma }$ and $\tilde{\mathsf{\Sigma }}$. This inequality is not always appreciated in the literature. Thus,

Claim 2.

Dissipation measured by dissipated macroheat or lost macrowork is not always proportional to the irreversible entropy $d_{\mathrm{i}}S$. The difference is the irreversible macroheat $T{d}_i{S}^Q$ generated due to macroheat exchange.

The observation that the irreversible macroheat $T{d}_{\mathrm{i}}{S}^{\mathrm{Q}}$ must be subtracted from $T{d}_{\mathrm{i}}S$ is consistent with our previous observation in Equation (24) and the discussion of it. However, Equation (24) has a much deeper consequence, which states that variation of ${\xi }_{k\mathrm{E}}$ does not generate any SI-microwork $d_{\mathrm{i}}{W}_{k\mathrm{E}}$ so it does not generate any macroscopic dissipation, although it creates irreversible entropy generation $d_{\mathrm{i}}{S}^{\mathrm{Q}}$.