# Foundations of Nonequilibrium Statistical Mechanics in Extended State Space

## Abstract

**:**

`q`during a process, and the concept of reduction. The mechanical process quantities (no stochasticity) like macrowork are given by $\widehat{A}{d}_{\alpha}\mathsf{q}$, but the stochastic quantities ${\widehat{C}}_{\alpha}\mathsf{q}$ like macroheat emerge from the commutator ${\widehat{C}}_{\alpha}$ of ${d}_{\alpha}$ and $\widehat{A}$. Under the very common assumptions of quasi-additivity and quasi-independence, exchange microquantities ${d}_{\mathrm{e}}$

`q`${}_{k}$ such as exchange microwork and microheat become nonfluctuating over $\left\{{\mathfrak{m}}_{k}\right\}$ as will be explained, a fact that does not seem to have been appreciated so far in diverse branches of modern statistical thermodynamics (fluctuation theorems, quantum thermodynamics, stochastic thermodynamics, etc.) that all use exchange quantities. In contrast, dq${}_{k}$ and ${d}_{\mathrm{i}}$

`q`${}_{k}$ are always fluctuating. There is no analog of the first law for a microstate as the latter is a purely mechanical construct. The second law emerges as a consequence of the stability of the system, and cannot be violated unless stability is abandoned. There is also an important thermodynamic identity ${d}_{\mathrm{i}}Q\equiv {d}_{\mathrm{i}}W$$\ge 0$ with important physical implications as it generalizes the well-known result of Count Rumford and the Gouy-Stodola theorem of classical thermodynamics. The $\mu $NEQT has far-reaching consequences with new results, and presents a new understanding of thermodynamics even of an isolated system at the microstate level, which has been an unsolved problem. We end the review by applying it to three different problems of fundamental interest.

## 1. Introduction

**${x}^{\u2034}$**$,\cdots $ for different $n=1,2,3,\cdots $

#### 1.1. Scope of the Review

#### 1.2. System-Intrinsic and Medium-Intrinsic Thermodynamics

- A.
- The internal microwork ${\Delta}_{\mathrm{i}}{W}_{k}$ has no analog in the former because it uses the following questionable conjecture:$${\Delta}_{\mathrm{e}}{W}_{k}\stackrel{?}{=}-\Delta {E}_{k},$$$${d}_{e}W\stackrel{?}{=}-{\textstyle {\sum}_{k}}{p}_{k}d{E}_{k},{d}_{e}Q\stackrel{?}{=}{\textstyle {\sum}_{k}}{E}_{k}d{p}_{k}.$$$$d{W}_{k}\equiv -d{E}_{k},$$$${d}_{e}{W}_{k}={d}_{e}W,\forall k;$$
- B.
- Consequently, the microforce imbalance ($\mu $FI) that results in fluctuating ${\Delta}_{\mathrm{i}}{W}_{k}=-{\Delta}_{\mathrm{i}}{E}_{k}$, a ubiquitous quantity, is absent in the former in that ${\Delta}_{\mathrm{i}}{W}_{k}=\Delta {W}_{k}-{\Delta}_{\mathrm{e}}{W}_{k}\equiv 0$ but is always present (${\Delta}_{\mathrm{i}}{W}_{k}\ne 0$) in the latter.
- C.
- The former results in a first law of thermodynamics ($\Delta {E}_{k}={\Delta}_{\mathrm{e}}{Q}_{k}-{\Delta}_{\mathrm{e}}{W}_{k}$) for each ${\mathfrak{m}}_{k}$, while the latter has it hold ($\Delta E={\Delta}_{\mathrm{e}}Q-{\Delta}_{\mathrm{e}}W$) only for a $\mathfrak{M}$; however, see Equation (243).
- D.
- The lost or dissipated macrowork ${\Delta}_{\mathrm{lost}}W$ measured by the average ${\Delta}_{\mathrm{i}}{W}_{k}$ should be absent in the former due to its above conjecture, but is always present in the latter.
- E.
- The exchange microwork ${\Delta}_{\mathrm{e}}{W}_{k}$ depends on the entire trajectory ${\gamma}_{k}$ in the former to make it fluctuating over ${\gamma}_{k}$, while in the latter, $\Delta {W}_{k}$ depends only on the terminal microstates of ${\gamma}_{k}$, and ${\Delta}_{\mathrm{e}}{W}_{k}\equiv {\Delta}_{\mathrm{e}}W$ is nonfluctuating (it is the same for all ${\gamma}_{k}$’s).

#### 1.3. Main Results

**Remark 1.**

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

#### 1.4. Layout

## 2. Notation, Definitions and New Concepts

#### 2.1. Systems and State Variables

**Definition 1.**

**Remark 2.**

**Definition 2.**

**Definition 3.**

**Remark 3.**

**Definition 4.**

#### 2.2. Microstates and Macrostates

**Remark 4.**

**Remark 5.**

**Claim 1.**

**Definition 5.**

**Definition 6.**

**Remark 6.**

#### 2.3. Micro–Macro Variables

**Notation 1.**

**Notation 2.**

#### 2.4. Random Variable and Average

**Remark 7.**

**Definition 7.**

`q`is defined by

`q`for which ${\mathsf{q}}_{k}=\mathsf{q},\forall k$. We have used $\widehat{A}$ to denote the above averaging operator in Equation (12).

**Remark 8.**

**Notation 3.**

`q`$\left(t\right)$ (see Equation (11b)) during an arbitrary infinitesimal process $d\mathcal{P}$ can be partitioned as

`q`$\left(t\right)$ is the change caused by exchange (“e”) with the surroundings such as the medium and ${d}_{\mathrm{i}}$

`q`$\left(t\right)$ is its change due to internal or irreversible (“i”) processes going on within ${\Sigma}_{b}$. As mentioned earlier, the term external quantity will also be used for an exchange quantity to emphasize its external nature in this review. The partition also applies to the outcome $d{\mathsf{q}}_{k}$ as follows:

**Claim 2.**

#### 2.5. Different States in NEQT

**Definition 8.**

**Definition 9.**

- (a)
- Internal-equilibrium macrostate (IEQ): The nonequilibrium entropy $S(\mathbf{X},\phantom{\rule{3.33333pt}{0ex}}t)$ for such a macrostate is a state function $S\left(\mathbf{Z}\right)$ in the larger nonequilibrium state space ${\mathfrak{S}}_{\mathbf{Z}}$ spanned by $\mathbf{Z}$; ${\mathfrak{S}}_{\mathbf{X}}$ is a proper subspace of ${\mathfrak{S}}_{\mathbf{Z}}$: ${\mathfrak{S}}_{\mathbf{X}}\subset {\mathfrak{S}}_{\mathbf{Z}}$. As there is no explicit time dependence, there is no memory of the initial macrostate in IEQ macrostates.
- (b)
- Non-internal-equilibrium macrostate (NIEQ): The nonequilibrium entropy for such a macrostate is not a state function of the state variable $\mathbf{Z}$. Accordingly, we denote it by $S(\mathbf{Z},t)$ with an explicit time dependence. The explicit time dependence gives rise to memory effects in these NEQ macrostates that lie outside the nonequilibrium state space ${\mathfrak{S}}_{\mathbf{Z}}$. An NIEQ macrostate in ${\mathfrak{S}}_{\mathbf{Z}}$ becomes an IEQ macrostate in a larger state space ${\mathfrak{S}}_{{\mathbf{Z}}^{\prime}},{\mathbf{Z}}^{\prime}\supset \mathbf{Z}$, with a proper choice of ${\mathbf{Z}}^{\prime}$.

**Definition 10.**

#### 2.6. Mechanical Description

**Claim 3.**

- Nonfluctuating (NFl) approach: It can be treated as a nonfluctuating (fixed) parameter in the Hamiltonian of ${\Sigma}_{b}$ so that it is the same for all of its microstates. If we alter $\mathbf{W}$, it changes the same way for all ${\mathfrak{m}}_{k}$’s. We say that $\mathbf{W}$ is a NFl-parameter over ${\mathfrak{m}}_{k}$’s. This results in fluctuating generalized microforce$${\mathbf{F}}_{\mathrm{w}k}\doteq -\partial {E}_{k}/\partial \mathbf{W}$$$${\mathbf{F}}_{\mathrm{w}}\doteq {\textstyle {\sum}_{k}}{p}_{k}{\mathbf{F}}_{\mathrm{w}k}=-\partial E/\partial \mathbf{W}.$$
- Fluctuating (Fl) approach: Alternatively, we let $\mathbf{W}$ fluctuate over ${\mathfrak{m}}_{k}$’s and think of it conceptually as a random variable $\mathbb{W}$ with outcomes $\left\{{\mathbf{W}}_{k}\right\}$, even though ${\mathbf{W}}_{k}$ is a microvariable. To be consistent with the NFl-approach (see below), we require that ${\mathbf{F}}_{w}$ becomes nonfluctuating (fixed) defined by$$\forall k,\phantom{\rule{4pt}{0ex}}-\partial {E}_{k}/\partial {\mathbf{W}}_{k}=-\partial E/\partial \mathbf{W}={\mathbf{F}}_{\mathrm{w}}.$$In this view, the macroforce ${\mathbf{F}}_{w}$ is fixed (so it is the same for all macrostates) with the result that ${\mathfrak{m}}_{k}\left({\mathbf{W}}_{k}\right)$ is determined by the fluctuating random variable $\mathbb{W}$ over ${\mathfrak{m}}_{k}$’s, with its average (see Equation (112)) given by$$\mathbf{W}\doteq {\textstyle {\sum}_{k}}{p}_{k}{\mathbf{W}}_{k}.$$We use the notation $\left\{{\mathbf{W}}_{k},{\mathbf{F}}_{w}\right\}$ to compactly refer to this case.

**Claim 4.**

**Claim 5.**

**Remark 9.**

**Claim 6.**

**Remark 10.**

**Remark 11.**

**Remark 12.**

**Remark 13.**

**Definition 11.**

**Definition 12.**

#### 2.7. Entropy and Stochastic Description

**Definition 13.**

**Definition 14.**

**Remark 14.**

**Remark 15.**

#### 2.8. Reduction

**Definition 15.**

**Definition 16.**

**Claim 7.**

**Remark 16.**

**Remark 17.**

**Theorem 1.**

**Proof.**

#### 2.9. Process Quantities

**Remark 18.**

**Definition 17.**

**Definition 18.**

**Remark 19.**

**Conclusion 1.**

**Remark 20.**

**Definition 19.**

**Definition 20.**

#### 2.10. ${\Sigma}_{0}$ (Isolated Body) and $\tilde{\Sigma}\phantom{\rule{3.33333pt}{0ex}}$(Medium)

**Remark 21.**

**Remark 22.**

**Remark 23.**

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

#### 3.1. Generalizing $d\equiv {d}_{e}+{d}_{i}$

**Remark 24.**

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

**Remark 25.**

**Remark 26.**

**Remark 27.**

**Theorem 2.**

**Proof.**

**Claim 8.**

`q`${}_{k}$ that do not require such averaging. To make this clear distinction, we call these microquantities simply internal.

**Summary 1.**

`q`${}_{k}=(d{S}_{k},d{E}_{k},d{W}_{k},d{Q}_{k})$ and ${d}_{i}$

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

#### 3.2. Consequences of Theorem 60

**Remark 28.**

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

**Claim 9.**

#### 3.3. Some Simple Examples

**Conclusion 2.**

#### 3.4. Manipulations with ${d}_{\alpha}$

**Definition 21.**

**Definition 22.**

**Definition 23.**

`q`${}_{\mathrm{m}}$ for brevity. Infinitesimal macroquantities

`q`${}_{\mathrm{s}}$ for brevity. Together, they determine the change ${d}_{\alpha}$q:

**Remark 29.**

**Remark 30.**

**Remark 31.**

**Theorem 3.**

**Proof.**

**Definition 24.**

## 4. Internal Variables

**Claim 10.**

**Claim 11.**

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

**Remark 32.**

**Axiom 1.**

**Axiom 2.**

**Axiom 3.**

**Axiom 4.**

**Axiom 5.**

**Axiom 6.**

**Axiom 7.**

#### 5.1. Fundamental Axiom

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

see also the last paragraph on p. 106 in Jaynes [174].“...that the averages obtained on successive trials of the same experiment will agree with the ensemble average, thus permitting any particular individual system to exhibit a behavior in time very different from the average;”

#### 5.2. Parameter Description

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

`E`taking Fl-values $\left\{{E}_{k}\right\}$; their average value is determined by a fixed ${f}_{\mathrm{s}}=-T$; see Equation (24) and Claim 5. In this ensemble, $T,V$ and $\xi $ are fixed so we can also call it a $(T$-V-$\xi )$-ensemble. In this case, ${E}_{k},{P}_{k}$, and ${A}_{k}$ are fluctuating over $\left\{{\mathfrak{m}}_{k}\right\}$. If we take ${\mathbf{W}}^{\mathrm{NF}}=\left(\xi \right)$ and ${\mathbf{W}}^{\mathrm{F}}=\left(V\right)$, then ${E}_{k},{V}_{k}$, and ${A}_{k}$ are fluctuating over $\left\{{\mathfrak{m}}_{k}\right\}$ with $T,P$ and $\xi $ kept fixed in this ensemble, which we can call a $(T$-P-$\xi )$-ensemble. We can also consider an ensemble with ${\mathbf{W}}^{\mathrm{NF}}=\left(V\right)$ and ${\mathbf{W}}^{\mathrm{F}}=\left(\xi \right)$. In this ensemble, ${E}_{k},{P}_{k}$, and ${\xi}_{k}$ are fluctuating over $\left\{{\mathfrak{m}}_{k}\right\}$; $T,V$ and A are kept fixed so we can call it a $(T$-V-$A)$-ensemble. For these ensembles to represent the same physical system thermodynamically, we must have $V=\u2329V\u232a,P=\u2329P\u232a,\xi =\u2329\xi \u232a$, and $A=\u2329A\u232a$ in accordance with Equation (109).

**Remark 33.**

#### 5.3. Ensemble of Replicas

#### 5.4. Concept of Probability

**Claim 12.**

**Definition 25.**

**Remark 34.**

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

**Claim 13.**

**Remark 35.**

**Remark 36.**

**Remark 37.**

#### 5.6. Principle of Additivity

#### 5.6.1. Additivity

#### 5.6.2. Quasi-Additivity

**Remark 38.**

**Remark 39.**

**Remark 40.**

**Remark 41.**

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

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

**Remark 42.**

**Remark 43.**

**Conclusion 3.**

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

**Definition 26.**

**Remark 44.**

**Remark 45.**

**Proposition 1.**

**Remark 46.**

**Remark 47.**

**Remark 48.**

**Remark 49.**

#### 5.10. Consequences of the Second Law

**Theorem 4.**

**Proof.**

**Corollary 1.**

**Proof.**

**Corollary 2.**

**Proof.**

**Corollary 3.**

**Proof.**

#### 5.11. Assumptions

#### 5.11.1. N Fixed for $\Sigma $

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

## 6. Mechanical Aspects

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

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

#### 6.3. SI-Legendre Transform

**Remark 50.**