A Novel Trick to Overcome the Phase Space Volume Change and the Use of Hamiltonian Trajectories with an emphasis on the Free Expansion
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b UATP/2002
A Novel Trick to Overcome the Phase Space Volume Change and the Use ofHamiltonian Trajectories with an emphasis on the Free Expansion
P.D. Gujrati ∗ Department of Physics, Department of Polymer Science, The University of Akron, Akron, OH 44325 (Dated: January 31, 2020)
Abstract
We extend and successfully apply a recently proposed microstate nonequilibrium thermodynamics ( µ NEQT) to study expan-sion/contraction processes. Here, the numbers of initial and final microstates { m k } are different so they cannot be connected byunique Hamiltonian trajectories. This commonly happens when the phase space volume changes, and has not been studied sofar using Hamiltonian trajectories that can be inverted to yield an identity mapping T : m k ( Z E in ) ⇄ m k ( Z E fin ) as the parameter Z E in the Hamiltonian is changed. We propose a trick to overcome this hurdle with a focus on free expansion ( P vacuum = 0) inan isolated system, where the concept of dissipated work is not clear. The trick is shown to be thermodynamically consistentand can be extremely useful in simulation. We justify that it is the thermodynamic average ∆ i W ≥ i W k done by m k that is dissipated; this microwork is different from the exchange microwork ∆ e W with the vacuum, whichvanishes. We also establish that ∆ i W k ≥ Keywords : Microstate irreversible thermodynamics, Dissipated work in free expansion, Mising microstates, Internal vari-ables, Modern fluctuation theorems.
1. INTRODUCTION1.1. Background
Free (unrestricted) expansion is an undergraduateparadigm of irreversibility, in which the exchangemacrowork ∆ e W and macroheat ∆ e Q , see Fig. 1, areidentically zero. It is also accompanied by an increasein the volume | Γ | of the phase space Γ of the system Σ.One can study it as an irreversible process P going onwithin an isolated system from an initial (in) macrostate M in to a final (fin) macrostate M fin . This ensures thatits energy remains constant even if the system remainsout of equilibrium (EQ) during the entire process includ-ing the initial and final macrostates. The expansion issudden at t = 0, but it takes a while ( t = τ eq >
0) forEQ to emerge. As is known, a macrostate M of Σ refersto a collection { m k , E k , p k } of its microstates m k of ener-gies E k that appear with probabilities p k in M ; the p k ’sgive rise to stochasticity required for a proper thermody-namics. We use macro - and micro - in this study to referto quantities pertaining to macrostates and microstates,respectively, with a macroquantity referring to the ther-modynamic average of related microquantities. A micro-quantity will always carry a subscript k as a reminderthat it is associated with m k . We will use Z for a statevariable (see Sec. 2.2 for explanation), a macrovariable,and Z k for its value, a microvariable, for m k .The study of a particular form of the free expansion iswell known at the undergraduate level in the traditionalmacroscopic nonequilibrium (NEQ) thermodynamics [1–6] based on the above exchange quantities; see also [7–10] for modern treatment. We denote this traditionalNEQ thermodynamics by ˚MNEQT in the following; hereM stands for macroscopic and the small circle refers tothe use of the exchange quantities. The study worksif and only if the initial and final macrostates M in,eq and M fin,eq , respectively, are in EQ so that the entropychange ∆ S = S fin − S in and, therefore, the net irre-versible entropy change ∆ i S = ∆ S over the process canbe evaluated without knowing the entire history. Butthe ˚MNEQT does not provide any information duringthe relaxation ( t < τ eq ) towards M fin,eq such as the ir-reversible entropy generation d i S ( t ) associated with anysegment δ P of the process between intermediate NEQmacrostates M ( t ). Thus, the use of the ˚MNEQT is lim-ited in its scope.A NEQ process P undergoes dissipation at all times t < τ eq , and is usually described by the dissipated work∆ i W >
0, which in turn is directly related to ∆ i S over P under suitable conditions; see later. Here, free expan-sion poses another hurdle as the common understandingis that any internal work done by the ”vacuum” (ab-sence of matter and radiation) into which the gas ex-pands must be zero; see Fig. 2. This makes it hardto understand what work is being dissipated as the gasmost certainly generates irreversible entropy ∆ i S >
0. Acentral aspect of this investigation is to obtain a betterunderstanding of dissipated work d i W and the sourceof d i S over δ P in an interacting and an isolated sys-tem; see Corollary 2. This is achieved by focusing on system-intrinsic (SI) quantities dZ, dZ k (which we nowallow to also include dW, dW k and dQ, dQ k , which shouldnot be confused with their exchange analogs d e W, d e W k and d e Q, d e Q k ; see below) that are uniquely determinedby the system itself. They contain all the information in-cluding the one about internal processes that we wish tounderstand. The exchange quantities d e Z, d e Z k (whichalso include d e W, d e W k and d e Q, d e Q k ) are primarily de-termined by the macrostate of the medium e Σ; we willrefer to them as medium-intrinsic (MI) quantities here.They are easily determined by focusing on the medium,which is always taken to be in EQ. Thus, we can deter-mine d i Z . = dZ − d e Z, d i Z k . = dZ k − d e Z k that directlyTypeset by REVTEX 1 solated Macroscopic System S T ( t ), P ( t ),……, A ( t ) d i Z Macroscopic System S T ( t ), P ( t ),……, A ( t ) d i Zd e Z Surrounding Environment (Medium) T , P ,……, A =0 (a) (b) S FIG. 1: (a) An isolated nonequilibrium system Σ with in-ternally generated d i Z driving it towards equilibrium, dur-ing which its SI-fields T ( t ) , P ( t ) , · · · , A ( t ) associated with SI-variables S ( t ) , V ( t ) , · · · , ξ ( t ) continue to change towards theirequilibrium values; d i Z k denote the microanalog of d i Z . Thesign of d i Z is determined by the second law. (b) A nonequilib-rium systen Σ in a surrounding medium e Σ, both forming theisolated system Σ . The macrostates of the medium and thesystem are characterized by their fields T , P , ..., A = 0 and T ( t ) , P ( t ) , ..., A ( t ), respectively, which are different when thetwo are out of equilibrium. Exchange quantities ( d e Z ) carrya suffix ”e” and irreversibly generated quantities ( d i Z ) withinthe system by a suffix ”i” by extending the Prigogine notation.Their sum d e Z + d i Z is denoted by dZ , which is a system-intrinsic quantity (see text). In a nonequilibrium system, thenonzero differences F ht = T − T and F wt = ( P − P , · · · , A )denote the set of thermodynamic forces, where we have alsoincluded the affinity A for internal variables ξ ; see text. Amicrostate m k of Σ is specified by appending a subscript k to F wt so that F wt, k = ( P k − P , · · · , A k ). describe the irreversibility in the system.We have recently developed a version of nonequilib-rium thermodynamics (NEQT) that is expressed in termsof only SI quantities so we have a direct access to d i Z and d i Z k . It has appeared in a series of papers [11–15] cover-ing separate aspects, and reviewed in [16, 17]. We havelabeled it MNEQT to distinguish it from the ˚MNEQT. Itis briefly introduced in Sec. 3. The theory is applicable tosystems that are either isolated or in a medium within thesame framework. The corresponding microstate versionof the MNEQT is called the µ NEQT, with µ - referringto the use of SI microquantities. It is capable of studyingexpansion/contraction at the microstate level in interact-ing and isolated system that has not been possible so far as we will discuss shortly. Another reason to focus on theproblem of expansion/contraction in the µ NEQT is dueto its close connection with Maxwell’s demon and Lan-dauer’s eraser. Both versions of the theory also involve internal variables [4, 6, 7, 18, 19] that are required, seeSec. 2.1, to explain nonequilibrium internal processes.Thus, they provide a very general framework of NEQTto understand a majority of nonequilibrium processes aswe will explain.We know that the classical thermodynamics is basedon the concept of work and heat so we need to identifythem in a NEQ process to make any progress. The cen-tral concept in the MNEQT is that of the generalized SImacrowork, see Fig. 1, dW = P ( t ) dV ( t )+ · · · + A ( t ) · dξ ( t )and SI macroheat dQ = T ( t ) dS ( t ), see Eq. (14), thatare different from the (exchange) MI macrowork d e W = P dV ( t ) + · · · and MI macroheat d e Q = T d e S , respec-tively, see Eq. (15), by irreversible contributions: d i W = dW − d e W ≥ , (1a) d i Q = dQ − d e Q ≥ . (1b)The ability to directly deal with d i W and d i Q makesthe MNEQT not only perfectly suited to study isolatedsystems as we will do, but also ensures that the general-ized macrowork dW is isentropic and that the generalizedmacroheat dQ satisfies the Clausius identity dQ = T dS ,see Eq. (14b), in all processes that we are interested inhere; S is always the Gibbs statistical entropy [1, 5] S . = − P k p k ln p k . (2)The µ NEQT was first introduced a while back [15] andapplied to a few simple examples including a brief appli-cation to the Brownian motion with a goal to compareits predictions with those from the work fluctuation the-orem (WFT) due to Jarzynski [20]; see Eq. (57) for itsprecise formulation. The importance of microforce im-balance F wt, k , see Fig. 1 caption and later, between ex-ternally applied macroforce and internally generated mi-croforce was pointed out there for the first time. It is ubiquitous in nature [15, 21] as it is always present inall (EQ and NEQ) macrostates. The macroforce imbal-ance F t = ( F ht , F wt ) between the fields of the system andthe medium, see Fig. 1 caption, determines irreversiblecontribution ( d i Q, d i W ) and is well defined even for anisolated system. It vanishes only in EQ. This makes F t and F wt, k central in the µ NEQT, which has recently beenapplied [21] to study the Brownian motion in full detail,where the relative motion of the Brownian particle withrespect to the medium generates F wt, k . Thus, the µ NEQTis also capable of tackling small systems like Brownianparticles under NEQ conditions.
Our goal here is to use the µ NEQT and the MNEQT tostudy general irreversible processes in interacting and iso-2ated systems with emphasis on those undergoing phasespace volume change and the resulting irreversibility ata deeper, microscopic level in terms of microstates { m k } .The forthcoming demonstration of the success of our ap-proach for free expansion, which has not been studiedso far, shows its usefulness as a general theory for bothinteracting and isolated systems. As a general setup,we consider an interacting NEQ system Σ in a verylarge medium e Σ, see Fig. 1(b). The two form an iso-lated system Σ = Σ ∪ e Σ. Quantities pertaining to Σ carry a suffix 0, those pertaining to e Σ carry a tilde, andthose pertaining to Σ carry no suffix. For example, themacroworks are dW , d f W , and dW , respectively. Themedium, being in EQ at all times, has no irreversibilityin it so that d i e Q = d i f W = 0. Because of its large size,its temperature, pressure, etc. are the same as for Σ sothey are denoted by T , P , etc. as seen in Fig. 1(b).The thermodynamic macroforce [6, 19] F t = ( T − T , P − P , · · · ) must be nonzero in a NEQ macrostateand vanish only in an EQ macrostate, i.e. , when Σ is inEQ by itself or with e Σ, if the latter is present. But themicroforce F wt, k = ( P k − P , · · · ) is ubiquitous as notedabove in all macrostates but independent of them, i.e.,of { p k } . Unfortunately, as we will see, this is not alwaysenforced in many current microscopic approaches to NEQthermodynamics.Care must be exercised if the medium is not extremelylarge such as in Fig. 2.Our methodology in the µ NEQT will ensure that themicroforces are always accounted for. Given F t, k , thechoice of { p k } determines whether F t = 0 or not sothe methodology will describe thermodynamics correctly.The temporal development of M in any P can also bestudied by following the deterministic Hamiltonian evo-lution along Hamiltonian trajectories { γ k } of microstates { m k } described by its Hamiltonian H . The trajectories,therefore, describe deterministic evolution during which { p k } does not change. As dW is isentropic, the evolutioninvolves the performance of microworks dW k at fixed p k ;see later. The stochasticity is due to microheat dQ k thatmodifies p k . Thus, dW k and dQ k control different aspectsof the evolution in M so their combined effect completesthe stochastic evolution in the µ NEQT.The trajectories have been recently popularized bymodern fluctuation theorems (MFTs) [22, 23]; see also[24, 25]. Among these is the Jarzynski’s WFT [20], whichis the most celebrated one for the simple reason that theother MFTs are related to it; see for example, Ref. [26].Thus, we will comment mostly on the WFT, commonlyknown as the JE, in the following, but the comments areequally valid for other MFTs.There are four important and independent aspects thatrequire careful consideration here.(i)
Internal variables.
The importance of internalvariables [4, 6, 7] and their affinities to describe NEQmacrostates has been well documented and is an integralpart of the MNEQT and µ NEQT used in this study; see also [15, 21]. We will give a simple argument for theirrelevance and the significance of affinities in Sec. 2.1.(ii)
Nonequilibrium Entropy . The MI d e Z aloneprovides no insight into d i Z during relaxation unless SI dZ is also identified. This creates a problem as M ( t )’sdenote NEQ macrostates in general so the SI dS is notknown if S is defined only for EQ macrostates. Thus, weneed to identify S for NEQ macrostates. We have shownthat for a NEQ system that is in internal equilibrium , thestatistical entropy given in Eq. (2) is a state functionin an enlarged state space involving internal variables[11, 12]; see Eq. (47). It is then used in the MNEQT todetermine the irreversible contributions directly. We seefrom Eq. (17b) that F t is an integral part of the MNEQTas promised. We then use the MNEQT to derive the µ NEQT.(iii)
Phase space volume change ∆ | Γ | 6 = 0. Asthe number of microstates depends on | Γ | , there can-not be a one-to-one mapping between the sets of mi-crostates in the two phase spaces in a process of expan-sion/contraction. The same problem arises if d | Γ | /dt =0 even if at the end ∆ | Γ | = 0 such as in a cyclic process.(iv) Dissipated work.
We need to provide a physicalexplanation of the macrowork that is being dissipated inthe free expansion (see Corollary 2) and the correspond-ing microworks.As interacting systems are also included in our analy-sis, we make a few comments in passing about the MFTs,with special attention to the WFT, that are derived forinteracting systems and where trajectories are also ex-ploited. The formulation invariably uses exchange quan-tities ∆ e W and ∆ e Q directly but ∆ i W and ∆ i Q are notpart of the formulation. Our comments basically sum-marize the results already available in the literature.The MFTs are claimed to describe NEQ processes, be-cause of which they have attracted a lot of attention.However, despite being part of an highly active field, wefind that they do not provide a useful methodology forour NEQ consideration here. There is no direct proof offor their NEQ nature that we are aware. The only indi-rect proof for the WFT is through the application of theJensen’s inequality to demonstrate its compliance withthe second law since the inequality leads to∆ e W J ≤ − ∆ F , (3)where ∆ e W J (”J” for Jarzynski’s formulation) is a par-ticular ”average exchange” work (properly defined in Eq.(58) later) that is obtained by using the initial prob-ability over the entire process. It turns out to be a non-thermodynamic average [26], and ∆ F is the differ-ence of the equilibrium (and, therefore, thermodynamic)Helmholtz free energies in the process; see also comment(f) below in the subsection. The above inequality looksvery similar to the following thermodynamic inequalityinvolving thermodynamic average (exchange macrowork) R = − ∆ e W , where R is the exchange work ∆ e f W done3y e Σ on Σ, R ≥ ∆ F , (4)a well-known consequence of the second law, but only if T remains a constant in the process [5]. In the lattercase, the dissipated work defined as∆ diss W . = R − ∆ F = T ∆ i S ≥ . (5)To provide an ”indirect proof” that the JE is a nonequi-librium result, Jarzynski sets without any proof that∆ e W J conjecture = − R (6)to turn Eq. (3) into Eq. (4). However, as shown recently[26], Jensen’s inequality applied to the MFTs does notprove compliance with the second law inequality so ∆ e W J in Eq. (3) cannot be equated with ∆ e W even when T = const .There are other concerns about the MFTs, which raisedoubts about their usefulness for our investigation. (a)They do not include any internal variables, necessary forirreversibility; see Sec. 2. (b) The external macroforce(such as the pressure P ) is always assumed to be equalto the macroforce (such as the pressure P ) in the system;hence, they implicitly assume that d e W = dW , which re-sults in d i W ≡
0; see Eq. (16a). This was first pointedout in Ref. [15]. Thus, they do not include any thermo-dynamic macroforce F wt necessary for d i W and for irre-versibility [6]. (c) From d i W ≡ d i Q ≡
0, see Eq.(11). If the temperature of the system is always equal to T , i.e. , F ht = T − T = 0, then it follows from Eqs. (17a,17b) that d i S ≡
0. Cohen and Mauzerall [27, 28] werethe first to raise concern that T may not even exist for aNEQ process; see also [29] for counter-arguments, someof which we will discuss later. The concern was justifiedas correct later by Muschik [30] so T = T will make theMFTs unsuitable for a NEQ process. (d) MFTs are basedon a fixed set of classical microstates { m k } or trajectories { γ k } as the use of Hamiltonian dynamics is consistentlyprevalent. Thus, their applicability is limited to the sit-uation d | Γ | = ∆ | Γ | = 0; see Sec. 2.4. This was firstpointed out by Sung [31]. Unfortunately, this limitationis not well recognized in the field. (e) The WFT shouldalso apply to an isolated system [32]. Because d e W = 0in this case, they do not. (f) The free expansion in Fig. 2refers to an isolated system so the WFT should be appli-cable in this case [33], but does not as first pointed out bySung [31]; see also [34, 35] for the ensuing debate. We willcome back to this issue later when we discuss free expan-sion. (g) In addition, the averaging in the WFT is not athermodynamic averaging over the process as first hintedby Cohen and Mauzerall [27], and established rigorouslyrecently by us [26], whereas we require a thermodynamicaveraging in our investigation.Because of all these limitations, the MFTs are not ofcentral interest to us in this study except to draw atten-tion to the differences with our approach. Therefore, we will discuss and substantiate the above points again laterin Sec. 5.4 within our theoretical framework; we focus onthe WFT for simplicity.There have been several numerical attempts to studyrestricted expansion in an interacting system [36–41] butwith a goal only to verify the WFT. Because of this,these numerical studies are also not helpful to us for thereasons stated above.In conclusion, it is not a surprise that we are left toexclusively use the MNEQT and µ NEQT in this studyof interacting and isolated systems. We have already ap-plied the MNEQT to briefly study free expansion [42, 43].Here, we wish to go beyond the earlier study to demon-strate how the µ NEQT can be used to study expan-sion/contraction with special attention to free expansionby including internal variables also. The µ NEQT has alsobeen recently applied successfully to provide a thermo-dynamic alternative to study Brownian motion withoutusing the mechanical approach involving the Langevinequation [21]. The macroscopic friction emerges as a con-sequence of the relative motion of the Brownian particle,an internal variable, with respect to the medium. We donot need to postulate the Langevin noise term; it emergesas a consequence of thermodynamic averaging mentionedabove.Our methodology and theory will be formulated forany arbitrary process in both interacting and isolatedsystems. The theory is derived from the MNEQT so itis always consistent with classical thermodynamics. Theprocess will also include expansion and contraction asspecial cases but the main focus will be mostly on thespontaneous process of unrestricted, i.e., free expansionfor the reason explained above. Whenever we study freeexpansion, we will consider the gas as a closed system Σ,which is in a medium e Σ that happens to be the vacuum;see Fig. 2. Their combination forms the isolated systemshown by Σ in Fig. 1. For the set up for free expansion,we follow Kestin [33] as we want to make the system(Σ) and the vacuum ( e Σ) independent. As e Σ is devoidof matter and radiation, Σ is nothing but Σ. We canreplace the partition by a piston exerting an externalpressure P from e Σ for a general expansion/contractionprocess. For P < P , the gas will expand; for P > P , thegas will contract. As the piston is an insulator, e Σ onlyacts as a working medium e Σ w . We need to bring in athermal medium e Σ h to bring about thermal equilibrium.Such an expansion/contraction process is covered by theWFT [20]. As P →
0, and
P >
0, we obtain the limitingcase of free expansion. Thus, free expansion is merely alimiting case of expansion/contraction in our approach,and does not require a separate approach. In all thesecases, we require the gas particles initially to be alwaysconfined in the left chamber.4 b)(a)Gas VacuumVacuum
FIG. 2: Free expansion of a gas. The gas is confined to the leftchamber, which is separated by a hard partition (shown by asolid black vertical line) from the vacuum in the right chamberas shown in (a). At time t = 0, the partition is removedabruptly as shown by the broken line in its original place in(b). The gas expands in the empty space, devoid of matterand radiation, on the right but the expansion is gradual asshown by the solid front, which separates it from the vacuumon its right. We can also think of the hard partition in (a) as apiston, which maintains the volume of the gas on its left. Thepiston can be moved slowly or rapidly to the right within theright chamber with a pressure P < P to change this volume.The free expansion occurs when the piston moves extremely(infinitely) fast by letting P → We briefly review some useful new concepts in the nextsection. We begin the discussion with the need for in-ternal variables in a NEQ process. The nature of theparameters in the Hamiltonian of a NEQ system is dis-cussed after that, which is then followed by the definitionof Hamiltonian trajectories. We close this section by in-troducing the central concept of internal equilibrium. Wefollow this section by a brief introduction to the MNEQTin Sec. 3. Here, we show the importance of internal vari-ables for an isolated system to determine the irreversiblemacrowork; see Theorem 1. This partially answers oneof the motivating questions. This is then followed by anintroduction to the µ NEQT in Sec. 4, where we intro-duce the concepts of various microworks and microheats.We introduce the moment generating function in Sec. 5,which gives all the moments including fluctuations of mi-croworks from this single function. We then turn to thefree expansion of a classical gas and study it by restrict-ing to use only two internal variables in the MNEQT inSec. 6. We study the free expansion in the µ NEQT in thenext section. We first study it in a quantum case in Sec.7.1 and then in the classical case in Sec. 7.2, where weintroduce the important trick that allows us to considerfree expansion for any arbitrary expansion. The moment generating function is used to directly demonstrate thatthe trick does not affect thermodynamics. The trick canbe extended to a cyclic process during which the phasespace volume changes nonmonotonically or to restrictedexpansion and contraction. A brief discussion of our re-sults is presented in the last section.
2. BASIC CONCEPTS2.1. Need for an Internal Variable
Consider a simple example of a NEQ system of N par-ticles, each of which can be in two levels, forming anisolated system Σ of volume V . Let ρ l and e l ( V ) , l = 1 , ρ , ρ keep chang-ing. We have e = ρ e + ρ e for the average energy perparticle, which is a constant, and dρ + dρ = 0 as aconsequence of ρ + ρ = 1. Using de = 0, we get dρ + dρ e /e = 0 , which, for e = e , is inconsistent with the second equa-tion (unless dρ = 0 = dρ , which corresponds to EQ).Thus, e l ( V ) cannot be treated as constant in determining de . In other words, there must be an extra dependencein e l so that e dρ + dρ e + ρ de + ρ de = 0 , and the inconsistency is removed. This extra dependencemust be due to independent internal variables that arenot controlled from the outside (isolated system) so theycontinue to relax in Σ as it approaches EQ. Let us imag-ine that there is a single internal variable ξ so that wecan express e l as e l ( V, ξ ) in which ξ continues to changeas the system comes to equilibrium. The above equationthen relates dρ and dξ ; they both vanish simultaneouslyas EQ is reached. We also see that without any ξ , theisolated system cannot equilibrate.The above discussion is easily extended to a Σ withmany energy levels of a particle with the same conclu-sion that at least a single internal variable is required toexpress e l = e l ( V, ξ ) for each level l . We can also visualizethe above system in terms of microstates. A microstate m k refers to a particular distribution of the N particles inany of different levels with energy E k = P l N l e l , where N l is the number of particles in the l th level, and isobviously a function of N, V, ξ so we will express it as E k ( N, V, ξ ). This makes the average energy of the systemalso a function of
N, V, ξ , which we express as E ( N, V, ξ ).An EQ system is uniform. Thus, the presence of ξ suggests some sort of nonuniformity in the system. Toappreciate its physics, we consider a slightly different sit-uation below as a possible example of nonuniformity.We consider as a simple NEQ example a compositeisolated system Σ consisting of two identical subsystems5 and Σ of identical volumes and numbers of parti-cles but at different temperatures T and T at any time t ≤ τ eq before EQ is reached at t = τ eq so the sub-systems have different time-dependent energies E and E , respectively. We assume a diathermal wall sepa-rating Σ and Σ . Treating each subsystem in EQ ateach t , we write their entropies as S ( E , V / , N/ S ( E , V / , N/ S of Σ is a func-tion of E , E , V , and N . Obviously, Σ is in a NEQmacrostate at each t < τ eq . From E and E , we formtwo independent combinations E = E + E =constantand ξ = E − E so that we can express the entropyas S ( E, V, N, ξ ). Here, ξ plays the role of an internalvariable, which continues to relax towards zero as Σ ap-proaches EQ. For given E and ξ , S ( E, V, N, ξ ) has themaximum possible values since both S and S havetheir maximum value. As we will see below, this is theidea behind the concept of internal equilibrium in which S ( E, V, N, ξ ) is a state function of state variables andcontinues to increase as ξ decreases and vanishes in EQ.We assume Σ to be in IEQ at each t in this simpleexample. From 1 /T = ∂S/∂E and A/T = ∂S/∂ξ , A being the activity associated with ξ , we find that T = 2 T T / ( T + T ) , A = ( T − T ) / ( T + T ) . As EQ is attained, T → T eq , the EQ temperature of bothsubsystems and A → A = 0 as expected. We see thatin this simple example Adξ/T is the contribution due toirreversiblity in dS , which also shows that ( − Adξ ) is thecontribution due to irreversiblity in dE .In general, the activity A controls the behavior of ξ in a NEQ macrostate and vanishes when EQ is reached.Here, we will take a more general view of A , and extendits definition to X also [11]. Comparing A with F ht , weclearly see that F ht also plays the role of an activity. Thesame reasoning also shows that F wt plays the role of anactivity.The example can be easily extended to the case of ex-pansion and contraction by replacing E, E , and E by N, N L , and N R , see Fig. 2, to describe the diffusion ofparticles. The role of β and E , etc. are played by βµ and N , etc. It is clear that in order to capture a NEQ process,internal variables are necessary . Another way to appre-ciate this fact is to realize that for an isolated system,all the observables in X = ( E, V, N, · · · ) are fixed so ifthe entropy is a function of X only, it cannot change[11, 12, 14, 16, 17]. Thus, we need additional indepen-dent variables to ensure the law of increase of entropyfor a NEQ system. An EQ macrostate is representedby a point in the state space S spanned by X , but aNEQ macrostate by a point in an enlarged state space S spanned by Z . =( X , ξ ), where ξ is the set formed by in-ternal variables. Internal variables cannot be controlled from the outside of the system; they are only controlledby the processes within the system. On the other hand,the observables in X are controlled from the outside. Wewill call X a set of observables and Z a set of state vari-ables. In EQ, internal variables are no longer indepen-dent of the observables. Consequently, their affinities (seelater) vanish in EQ. It is common to define the internalvariables so their EQ values vanish.As we will be dealing with the Hamiltonian of the sys-tem, it is useful to introduce the sets X E . = X \ E =( V, N, · · · ), and Z E . = Z \ E = ( V, N, · · · , ξ ) = ( X E , ξ ).Then, E and E k become a function of Z E as we saw inSec. 2.1. Here, Z E appears as a parameter in the Hamil-tonian, which we will write as H ( z | Z E ), where z is apoint (collection of coordinates and momenta of the par-ticles) in the phase space Γ( Z E ) specified by Z E . As anexample, N, V , and ξ are the parameters in the previ-ous section. When the system moves about in the phasespace Γ( Z E ), z changes but Z E as a parameter remainsfixed in a state subspace S E . = S \ E . Traditional formulation of statistical thermodynamics[1, 5, 44] takes a mechanical approach in which m k followsits classical or quantum mechanical evolution dictated byits SI Hamiltonian H ( z | Z E ). The quantum microstatesare specified by a set of good quantum numbers, which wehave denoted by k above as a single quantum number forsimplicity; we take k ∈ N , N denoting the set of naturalnumbers. We will see below that k does not change as Z E changes. In the classical case, we can use a small cell δ z k of size (2 π ~ ) N around z k = z as the microstate m k .In the rest of the work, we will keep N fixed to fix thesize of the system. Therefore, from now on, X and Z willnot contain it. The Hamiltonian gives rise to a purelymechanical evolution of individual m k ’s, which we willcall the Hamiltonian evolution , and suffices to providetheir mechanical description. The change in H ( z | Z E ) ina process is d H = ∂ H ∂ z · d z + ∂ H ∂ Z E · d Z E . (7a)The first term on the right vanishes identically due toHamilton’s equations of motion for any m k . Thus, forfixed Z E , the energy E k = H k . = H ( z k | Z E ) remainsconstant as m k moves about in Γ( Z E ). Only the variation d Z E in S generates any change in E k . Consequently, wedo not worry about how z k changes in H ( z | Z E ) in thephase space, and focus, instead, on the state space S , inwhich can write dE k = ∂E k ∂ Z E · d Z E = − dW k , (7b)where dW k denotes the generalized microwork producedby the generalized microforce F Ek : dW k = F Ek · d Z E , F Ek . = − ∂E k /∂ Z E . (7c)6e can now identify Z E as the work parameter , whosevariation d Z ( t ) . = ( dE ( t ) , d Z E ( t )) in S defines not onlythe microworks { dW k } , but also a thermodynamic pro-cess P . The trajectory γ k in S followed by m k as afunction of time will be called the Hamiltonian trajectory during which Z E varies from its initial (in) value Z E in toits final (fin) value Z E fin during P . The variation producesthe generalized microwork dW k ; p k plays no role so dW k is purely mechanical, which simplifies its determinationin our theory. The microwork dW k also does not changethe index k of m k as said above.Being purely mechanical in nature, a trajectory is com-pletely deterministic and cannot describe the evolution ofa macrostate M during P unless supplemented by ther-modynamic stochasticity, which requires p k ( M ) as dis-cussed above [5], and is related to dQ k as shown later;see Eq. (38). Thermodynamics emerges when quanti-ties pertaining to the trajectories are averaged over thetrajectory ensemble { γ k } with appropriate probabilitiesthat will usually change during the process. In this sense,our approach is different from approaches using stochas-tic trajectories [22, 23], where dE k is identified with theexchange microwork dR k . = − d e W k ; see Remark 6.The development of the µ NEQT requires pursuing in-dividual trajectory γ k of m k ( Z E ). In the following, m k will usually stand for classical microstates unless speci-fied otherwise, and follows its deterministic trajectory γ k as m k ,in . = m k ( Z E in ) evolves into m k ,fin . = m k ( Z E fin ). Byreversing the change in Z E from Z E fin to Z E in , m k ( Z E fin )comes back to m k ( Z E in ). Thus, γ k defines an identity map T : T : m k ( Z E in ) ⇄ m k ( Z E fin ) (8)without altering the index k so it is a one-to-one (1-to-1)or identity mapping of microstates. The two arrows meanthat the mapping can be inverted without altering theindex k so it is a one-to-one (1-to-1) or identity mappingof microstates. However, Hamiltonian trajectories for classical mi-crostates are not suitable for processes that involve ex-pansion and contraction in the volume V and/or otherparameters in Z E of the system with a correspondingchange in the phase space volume | Γ | . In the follow-ing, we will think of V as the varying work parameterfor simplicity. Then, during expansion, the initial vol-ume | Γ | in . = (cid:12)(cid:12) Γ( Z E in ) (cid:12)(cid:12) is smaller than the final volume | Γ ′ | fin . = (cid:12)(cid:12) Γ( Z E fin ) (cid:12)(cid:12) as shown in Fig. 3. This means thatthere are microstates such as δζ ′ ( = δ z ′ ) in Γ ′ fin . = Γ( Z E fin )that cannot be reached from any of the microstate δ z inΓ in . = Γ( Z E in ) along Hamiltonian trajectories; the lattertake δ z into δ z ′ inside the broken horizontal ellipse. Theconverse is true for contraction.Note also that we are not interested in the cardinal-ity of the initial and final sets of microstates { m in } and G¢ in G¢ fin G fin G in d z γ in d z ¢d Q ¢ γ fin d Q FIG. 3: The evolution of a microstate δ z ∈ Γ in into δ z ′ ∈ Γ ′ fin following the variation in Z E (green arrows). The initial andfinal phase spaces are Γ in and Γ ′ fin shown by the interiors ofthe red solid ellipses. By changing Z E in the reverse ordermaps δ z ′ into δ z as implied by the reverse green arrows. Themicrostates δζ ′ and δζ , and other quantities are explained inthe text. { m fin } ,respectively. We are interested in how they mapunder Hamiltonian evolution; see Sec. 8 for more clarifi-cation.It is clear that in the classical case, we require a newapproach to overcome the loss of the 1-to-1 mapping if weconfine ourselves to only Hamiltonian trajectories. To thebest of our knowledge, the problem of how to overcomethis hurdle of phase space volume change using Hamilto-nian trajectories has not been solved.We overcome the hurdle by introducing a novel butsimple trick. The clue for the new approach comesfrom considering trajectories in quantum mechanics. Asquantum microstates form a denumerable set with index k ∈ N as Z E changes, there is a 1-to-1 mapping as inEq. (8) between m k ,in and m k ,fin during expansion andcontraction, which helps us remedy the lack of one-to-one correspondence due to volume change in the classicalcase. The trick is to enlarge the smaller phase space tobecome equal to the larger phase space by adding miss-ing microstates that appear with nonzero but vanishinglysmall probabilities. As the work along deterministic tra-jectories in Eq. (7b) is oblivious to their probabilities(even though they continue to change in thermodynam-ics), we can add trajectories initiating at the missing mi-crostates to obtain a enlarged trajectory ensemble { γ } . At the end of the computation of ensemble aver-ages, a formal limit of vanishing probabilities of missinginitial microstates is taken. The central concept of the NEQT exploited here is thatof the internal equilibrium (IEQ) according to which theentropy S of a NEQ macrostate is a state function ofthe state variables in the enlarged state space S [11–13]7elative to the EQ state space S due to independentinternal variables [6, 7, 11, 12, 18, 19] that are requiredto describe a NEQ macrostate as explained above. InEQ, the internal variables are no longer independent ofthe observables forming the space S . As a consequence,their affinities vanish in EQ. In general, the temperature T of the system in IEQ is identified in the standard man-ner by the relation 1 /T = ∂S/∂E (9)using the fact that S is state function in S .An important property of IEQ macrostates is the fol-lowing: It is possible in an IEQ macrostate to have dif-ferent degrees of freedom or different parts of a systemto have different temperatures than T . For example, ina glass, it is well known that the vibrational degrees offreedom have a different temperature than the configu-rational degrees of freedom [45, 46]. In the viscous dragproblem, the CM-motion of the Brownian particle canhave a different temperature than T of the rest of theparticles in the fluid [21]. This observation is easily veri-fied in MNEQT based on the concept of IEQ as done else-where [46, see Sec. 8.1 and Eq. (58)]. By taking a largerand larger set of internal variables, we can treat almostall NEQ macrostates as if in IEQ. Thus, the MNEQT isan extremely useful thermodynamics for NEQ systems.
3. THE MNEQT
An EQ macrostate is described by X = ( E, V, · · · ), andits entropy is a state function S ( X ). Away from EQ, S ( Z ) becomes a state function for NEQ macrostates inIEQ. In the following, we will focus on V and ξ as mem-bers of Z E . = Z \ E for simplicity but the discussion is gen-eral and applies to any Z E . Indeed, we use two internalvariables when we study free expansion. The microstate m follows its evolution dictated by its (SI) Hamiltonian H ( z | V, ξ ); the interaction with e Σ is usually treated asa very weak stochastic perturbation, which immediatelysuggests adopting a SI description.
There are two kinds of macrowork,the SI and MImacroworks dW and d e W , respectively, in NEQT; seeEq. (1a). The irreversible macrowork d i W ≥ dQ and the MIexchange macroheat d e Q ; see Eq. (1b). The irreversibleheat d i Q ≥ dE = dQ − dW, (10a) dE = d e Q − d e W ; (10b) the first equation follows from Eq. (31a) and the discus-sion following it. From the above two equations followsthe important identity d i Q ≡ d i W, (11)which establishes that internal processes ensure that ir-reversible macroheat and macrowork within Σ are iden-tically equal in magnitude. This equality is very generaland will be used extensively in this study. It is a con-sequence of a very general result of NEQT that no irre-versible process can generate any internal change in the(average) energy of the system, i.e., d i E ≡ , (12)so that d e E = dE . From now on, we will refer to dW and dQ ( d e W and d e Q ) simply as macrowork and macroheat(exchange macrowork and exchange macroheat), respec-tively, so no confusion can arise.The discussion above is valid for any arbitrary pro-cess, but, from now on, we restrict to the case when S = S ( Z ) is a state function in S for simplicity, i.e. ,for the macrostate to be an IEQ macrostate [11, 12] todefine T . The discussion to an arbitrary process, whichcan be done, will be avoided here.For a process requiring pressure-volume macroworkonly, we have dW = P dV done by Σ (SI) or d f W = − P dV done by e Σ (MI) in terms of the instantaneouspressure P = − ∂E/∂V of Σ or P of e Σ, and their vol-ume change dV or − dV , respectively. The exchangemacrowork is d e W = − d f W = P dV . For irreversibility, P = P , with P − P playing the role of an activity [11].The irreversible or dissipated work is d i W = ( P − P ) dV [3, 4, 6].Comparing Eq. (10a) for an IEQ macrostate with dS = ( ∂S/∂ X ) · d X +( ∂S/∂ ξ ) · d ξ , (13)allows us to identify the macroheat and macrowork as dW = T ( ∂S/∂ Z E ) · d Z E = P dV + · · · + A · d ξ , (14a) dQ = T dS ; (14b)where T . = 1 / ( ∂S/∂E ) = 1 /β, β A . = ( ∂S/∂ ξ ) identifiesthe affinity A , and · · · refers to other elements in X .Recalling that for e Σ, T = T , P = P , · · · , A = 0, wehave in general, d e W = − d f W = P dV + · · · , (15a) d e Q = − d e Q = T d e S. (15b)We can identify the irreversible macrowork due to F wt : d i W = ( P − P ) dV + · · · + A · d ξ = F wt · d Z E ≥ , (16a)where F wt = { P − P , · · · , A = A − A } (16b)8s the thermodynamic force , which also include the affin-ity A , driving the system towards EQ when d i W → F wt →
0. For irreversibility, d i W >
0, which requires F wt to be non-zero as asserted earlier in Sec. 1.1. Eachcomponent ( P − P ) dV, · · · , A dξ , A dξ , · · · of F wt · d Z E must be positive separately for irreversibility.Using d e Q = T d e S in Eq. (14b), we find d i Q = (cid:26) ( T − T ) d e S + T d i S ( T − T ) dS + T d i S ≥ . (17a)Using d i Q = d i W and Eq. (16a), we also obtain d i S = (cid:26) (cid:8) ( T − T ) d e S + F wt · d Z E (cid:9) /T (cid:8) ( T − T ) dS + F wt · d Z E (cid:9) /T ≥ . (17b)We see that F ht = T − T can also be thought of as a”thermodynamic force” due to thermal imbalance drivingthe system towards EQ via heat transfer; it ensures that F ht d e S ≥
0, in accordance with the second law. Thus,both contributions to d i S are always nonnegative as ex-pected. In the absence of any heat exchange ( d e S = 0)or for an isothermal system ( T = T ), we have d i Q = T d i S = d i W, (18)where d i W is given by Eq. (16a).We can use F t . = ( F ht , F wt ) (19)as the generalized thermodynamic force , which includesthe thermal imbalance F ht and the work imbalance F wt . Itshould be obvious that F ht is meaningless for an isolatedor an isothermal system, while F wt is meaningful for allNEQ systems, interacting or not. The above formulation of MNEQT is perfectly suitedfor considering an isolated system Σ ( d e W = d e Q ≡ d X = 0 sothat d i W = A · d ξ as seen from Eq. (16a). Theorem 1
The irreversible entropy generated withinan isolated system is still related to the dissipated workperformed by the internal variables.
Proof. As E remains fixed for an isolated system ( dQ = T d i S ), we have from Eq. (10a) T d i S = d i W = A · d ξ ≥ d i W here is sim-ply A · d ξ and not the full expression in Eq. (16a). Same conclusion is also obtained when we apply Eq. (17b) toan isolated system.The above theorem thus clarifies the unsettling factabout the significance of dissipated macrowork that mo-tivated this study; see also Eq. (26). The dissipatedmacrowork d i W in an isolated system is performed bythe internal variable ξ , and can be identified with d i S asnoted in Sec. 1.1. Corollary 2
Neither the entropy can increase nor willthere be any dissipated work unless some internal vari-ables are present in an isolated system. If no internalvariables are used to describe an isolated system, thenthermodynamics requires it to be in EQ.
Proof.
The proof follows trivially from Eq. (20).
Let us consider a thermodynamic process P be-tween two macrostates M in . = M ( T in , Z E in ) and M fin . = M ( T fin , Z E fin ) at temperatures T in and T fin , respectively.The system may be isolated and in IEQ so its temper-ature is well defined. It may be very different from thetemperature T of the medium, if Σ is not isolated. Ineach case, the cumulative macroquantities of Σ are ob-tained by simple integration along the process:∆ α E = R P d α E, ∆ α S = R P d α S, (21a)∆ α W = R P d α W, ∆ α Q = R P d α Q. (21b)Similar definitions also apply to e Σ and Σ . Above wehave used the compact notation d α = d, d e , and d i toindicate various infinitesimal forms, which we can treatas linear operators. They will be useful in the rest of thework.We now consider an interacting system and determine∆ W between two EQ macrostates M in,eq . = M ( T , Z E in )and M fin,eq . = M ( T , Z E fin ) at the same temperature. Wedenote the corresponding process by P , which may pos-sibly be irreversible. We recall that the Helmholtz freeenergy F = E − T S (22a)(conventionally written as F but we will use that for a SIfree energy here) is also the Helmholtz free energy of aNEQ system [11] in a canonical ensemble in a medium attemperature T ; the temperature T of the system doesnot appear in F explicitly. It is this free energy thatfollows the second law [11] and not F , which is the SIfree energy F = E − T S + A · ξ. (22b)It depends only on the system and is very different from F . It will be useful later. In terms of the difference ∆ F W = ∆ Q − ∆ E =∆ Q − ∆ F − ∆( T S ). Thus,∆ W = R P [( T − T ) dS − SdT ] − ∆ F , (23a)∆ e W = − R P [ T d i S + SdT ] − ∆ F , (23b)∆ i W = R P [ T dS − T d e S ] (23c)The corresponding infinitesimal form is dW = ( T − T ) dS − SdT − dF , (24a) d e W = − [ T d i S + SdT ] − dF , (24b) d i W = T dS − T d e S. (24c)If the medium is maintained at a fixed temperature dur-ing P , we must remove the dT term above.We see that if and only if T = T = const over theentire process P so that it is isothermal , we have∆ W isoth = − ∆ F , ∆ e W isoth = − T ∆ i S − ∆ F , (25a)in terms of the Helmholtz free energy difference so that∆ i W isoth = T ∆ i S. (25b)This is a very strong requirement when e Σ remains in con-tinuous contact with Σ, since it requires complete ther-mal equilibrium at all times. In this case, d i Q = T d i S and ∆ i Q = T ∆ i S ; see also Eq. (18).We now consider an isolated system for which ∆ E = 0so that∆ i W = R P T d i S = T ∆ i S + R P [( T − T ) d i S ≥ , (26)which is in accordance with Theorem 1 and fi-nally explains the physical meaning of the irreversiblemacrowork, which was one of the questions that hadprompted this investigation. It should be clear that thederivation is not restricted to a process between two EQmacrostates. As remarked earlier, we use a single internal variable ξ in addition to V for Z E for simplicity so that we have dW = P dV + Adξ, d e W = P dV. (27a)The dissipated work is d i W = ( P − P ) dV + Adξ ≥
0; (27b)in the absence of ξ . We also have T d i S = ( T − T ) d e S + ( P − P ) dV + Adξ ≥ , (28)where we have used d i Q = d i W . If no exchangemacrowork is done, dV = 0 and T d i S = ( T − T ) d e S + Adξ ≥
0. In the absence of any exchange macroheat,we have
T d i S = ( P − P ) dV + Adξ ≥
0. In an isolatedsystem, we have
T d i S = Adξ ≥ , (29)which is a special case of the general result in Eq. (20).
4. THE µ NEQT
We will closely follow Refs. [15, 21] to provide a briefpedagogical review of the µ NEQT for the sake of con-tinuity and demonstrate its successful application to thefree expansion, which has not been attacked by any otherapproach so far. The main idea is to cast any macro-quantity in the MNEQT as a thermodynamic average,see Eq. (30a), over microstates. Then, we can identifycorresponding microquantities. Some care must be ex-ercised to ensure their uniqueness as we will see. Thesemicrostate microquantities can be used to identify thecontribution along a trajectory by simple integration.
The theory was first presented in a very condensedform in Ref. [15]. It was successfully applied [21] to pro-vide an alternative, but a much simpler, approach (usingdeterministic microforces F Ek ) to study Brownian motionwithout the use of Langevin’s stochastic noise term so itdoes not require the use of the stochastic theory. Themicroforce responsible for the Brownian motion is asso-ciated with the relative motion of the center of mass ofthe Brownian particle with respect to the medium. Agood description of the salient features of the µ NEQT isavailable there.It follows from H ( z | Z E ) that E k also depend on Z E .A macroquantity O (except the temperature) in theMNEQT appear as a thermodynamic average h O i over { m k } : O = h O i . = P k p k O k , (30a)where O k is the value of O associated with m k . Thus, E = P k p k E k , P = P k p k P k , S = P k p k S k , (30b)etc. Here, E k , P k . = − ∂E k /∂V, S k . = − ln p k , etc. are thevalues that are only determined by m k . However, p k or S k , although associated with m k is not determined by italone because of the constraint P k p k = 1, which makesit depend on the macrostate also.From E , we have the change in the macroenergy dE = P k E k dp k + P k p k dE k (31a)between two neighboring macrostates. We will use thecompact notation dE h . = P k E k dp k = h Edη i ,dE w . = P k p k dE k = h dE i , (32)where we have introduced η k . = ln p k so that dη k = dp k /p k . As { E k } does not change but { p k } changes in dE h , it must depend on dS ; see Eq. (14b). As { p k } is not changed in dE w , it is evaluated at fixed entropy [16, 42]. Comparing dE with the first law in Eq. (10a),we identify dQ = dE h . = h Edη i , (33a) dW = − dE w . = − h dE i (33b)Thus, the generalized macroheat dQ is a contributionproportional to dS and the generalized macrowork dW is an isentropic contribution as noted in Sec. 1.1. Theidentification of macroheat and macroworks above alsoexplains the choice of h for heat and w for work as suffixabove and superfix in F ht and F wt .We can now identify the (generalized) microheat andmicrowork along γ k ; they are given by dQ ′ k . = E k dη k , dW k = − dE k, (34)respectively. The reason for the prime in d α Q ′ k will be-come clear below. The second equation above is simplythe previously derived mechanical identity in Eq. (7b).We summarize this important result, which is not prop-erly appreciated in the field [15], in the form of a Theorem 3
The mechanical microwork dW k done bythe system in the k th microstate is the negative of thechange dE k = ( ∂E k /∂ Z E ) · d Z E : dW k = − dE k (35) Proof.
See the derivation of Eq. (7b).As shown in Eq. (7b), the temporal evolution of m k isdue to F Ek , which changes its microenergy E k but doesnot change m k . The average over all { m k } of dW k due to F Ek , see Eq. (7c), gives the generalized macrowork dW ,see Eq. (33b), due to the macroforce F E . = (cid:10) F E (cid:11) = T ( ∂S/∂ Z E ) , see Eq. (14a), so that dW = F E · d Z E as expected. We can summarize the above result as thefollowing theorem because it plays a central role in the µ NEQT.In general, macroworks d α W are thermodynamic av-erages of microworks d α W k : d α W = h d α W i . = P k p k d α W k . (36)As it is easy to determine the mechanical microworks d α W k , we can extend the identity dE k = − dW k to intro-duce microenergies d e E k = − d e W k , d i E k = − d i W k ; (37)they define what is meant by d e E k and d e E k .Shifting E k by a constant does not affect dW and dQ ,showing their unique nature. While dW k is also not af-fected by the shift, it does affect dQ ′ k . Therefore, insteadof using dQ = dE h to identify dQ k , we instead use the identity dQ = T dS in Eq. (14b) to identify dQ k as thereis no ambiguity in the definition of the statistical entropy S in Eq. (2). We thus find that dQ k . = T dS | k . = − T ( η k + 1) dη k , (38)where d S | k = − ( η k + 1) dη k , not to be confused with dS k = − dη k , is the microquantity corresponding to themacroquantity dSdS = − P k ( η k + 1) dp k . = P k p k dS | k . Similarly, we use Eqs. (15b) and (17a) to identify d e Q k and d i Q k , respectively. We are not going to be directlyinvolved with microheats in our investigation here so wewill not spend time with them further; we will treat themin a separate publication. However, we need various microworks in this investi-gation as our focus is to understand the dissipated work.Therefore, we give the results for them. For the simplecase Z E = ( V, ξ ), compare with Sec. 3.4, we have dW k = P k dV + A k dξ. (39a)From d e W = P dV , we obtain d e W k = P dV , whichdefines d e E k = − P dV . We also find d i W k = ( P k − P ) dV + A k dξ (39b)which identifies d i E k = − ( P k − P ) dV − A k dξ and ex-plains how it becomes nonzero due to internal processes.For an isolated NEQ system, we must d i W k = A k dξ = dW k . For free expansion, we must set P = 0 so d i W free k = P k dV + A k dξ = − d i E k . (39c) There have been several attempts to formulate mi-crostate trajectory thermodynamics [47, 48] based on uti-lizing the work fluctuation theorem so they are not di-rectly applicable to an isolated system. Our own attemptthat includes an isolated system was briefly outline inRef. [15] and elaborated recently in Ref. [26]. Here,we briefly summarize it for continuity. We will assumethe medium e Σ to consist of two noninteracting media e Σ h that controls macroheat exchange and e Σ w that controlsmacrowork exchange. We are interested in a NEQ workprocess P as the system evolves from one EQ macrostateto another by changing X E from X E in to X E fin by manip-ulating the medium e Σ w . It is usually the case that when X E = X E fin , the system is not yet in EQ so the internalvariables have not come to their EQ values. We denotethis part of P by P . It takes a while during ¯ P ( P =11 ∪ ¯ P ) for the system to reach the final EQ macrostate.We may allow the temperature T of e Σ h to change during P , or disconnect e Σ h from Σ during P . In both cases,the temperature T of the system in the final macrostate M fin . = M ( T fin , Z E fin ) may be different from that of the ini-tial macrostate M in . = M ( T in , Z E in ). While the microstatemaintains its identity ( k does not change) as shown in Eq.(8), the microenergy E k changes during the entire evolu-tion over P in accordance with Eq. (7b). Let us focuson dW k = − dE k during t and t + dt along γ k . Its integralalong γ k determines the accumulated microwork ∆ α W k ∆ W k . = R γ k dW k = − R γ k dE k . (40)The integral is not affected by how p k changes during P so it is the same for all processes between Z E in and Z E fin . Thus, we can evaluate { ∆ W k } for a single processsuch as an EQ process P but can us it for every otherpossible process P ( Z E in → Z E fin ). On the other hand, theaccumulated macrowork ∆ W over P , see Eq. (21b), isaffected by p k , see Eq. (36), so it is different for differentprocesses P ( Z E in → Z E fin ). Theorem 4
Let ∆ Z E denote the change in Z E in theprocess P ( Z E in → Z E fin ) . The cumulative microwork ∆ W k = − ∆ E k = E k ( Z E in ) − E k ( Z E in + ∆ Z E ) is thesame for all processes, including the reversible one, thatundergo the same net change ∆ Z E : Z E in → Z E fin = Z E in + ∆ Z E . However, the cumulative macrowork ∆ W depends on the process. Proof. As E k is specific to the microstate m k , the inte-gral in Eq. (40) is the required difference:∆ W k = − ∆ E k = E k ( Z E in ) − E k ( Z E in + ∆ Z E ) . (41)Let us consider various processes that occur when chang-ing Z E in by ∆ Z E to Z E in + ∆ Z E , regardless of the process.As E k is a microproperty, the net difference ∆ E k is thesame for all these processes. As { p k } is different for dif-ferent processes, the macrowork ∆ W , see Eq. (21b), isusually different for different processes as expected.The theorem has far-reaching consequences. Accord-ing to this, we can evaluate { ∆ W k } for a single processsuch as an EQ process P but can use it for every otherpossible process P ( Z E in → Z E fin ). On the other hand,the accumulated macrowork ∆ W over P is affected by p k , see Eq. (36), so it is different for different processes P ( Z E in → Z E fin ).The identification in Eq. (35) or in (41) is themost important feature of the µ NEQT that distinguishesit from current stochastic thermodynamic approaches,which invariably identifies d e W k with − dE k or ∆ e W k with − ∆ E k [15]; see also Remark 6.As the set { ∆ W k } is the same in all possible pro-cesses with Z E in → Z E in + ∆ Z E , we can introduce a ran-dom variable W such that it takes the value (outcome) W k = − E k , or a random variable d W that takes the value dW k = − dE k in m k with probability p k . Corollary 5
For a spontaneous process in an isolatedsystem, ∆ i W k ≥ , ∆ i W ≥ . (42) Proof.
We see from the definition of generalized forcesin Eq. (7c) that E k acts like the potential energy of a me-chanical system. According to the principle of (potential)energy minimization, these forces spontaneously causethe isolated mechanical system to decrease E k . There-fore, for a spontaneous process, E k ( Z E in ) ≥ E k ( Z E in +∆ Z E ) in Eq. (41). As ∆ W k = ∆ i W k for an isolated sys-tem, we have ∆ i W k ≥
0; hence ∆ i W ≥
0. This provesthe corollary.
Remark 6 As ∆ e W k is associated with the MImacrowork ∆ e W , it cannot be related to the difference dE k of microstate energies that are SI quantities. Thisclearly shows that the µ NEQT is very different fromcurrent stochastic thermodynamic approaches as notedabove.
The most important aspect of the Hamiltonian trajec-tory is the identity nature of T , which ensures that everyinitial microstate m k , k ∈ N , is mapped onto itself at theend of P , although its energy will have changed. Thus,each γ k is unique and a sum over its ensemble { γ k } is thesame as the sum over k ∈ N . This is a major simplifica-tion of our approach, and plays a major role in the restof the study.
5. MOMENT GENERATING FUNCTION5.1. Trajectory Probability
Let us consider an arbitrary process P between M in and M fin . We consider two terminal microstates m k ,in and m k ,fin along P and introduce the following tra-jectory probability [26] between them for any system,interacting or isolated, p γ k ≡ R γ k p k dW k R γ k dW k = R γ k p k dw k ; (43)here, dw k . = dW k / ∆ W k and is independent of p k . Wecan generalize the above definition to introduce p γ k ,α byreplacing dW k by d α W k ; see Ref. [26]. We see that P k p γ k ∆ W k = R γ k dW = ∆ W, (44)the macrowork as introduced in Eq. (21b). It is clearthat P k p γ k = 1 as expected, and that p γ k is nothing butthe (thermodynamic) probability of having a particularvalue ∆ W k = E k, in − E k, fin , determined only by ∆ Z E asseen from Theorem 4; here E k, in = E k ( Z E in ) and E k, fin = E k ( Z E in +∆ Z E ). It is clear that p γ k is the joint probabilityof m k ,in and m k ,fin , which can be expressed in terms ofthe conditional probability p ( m k ,fin | m k ,in ): p γ k = p ( m k ,in ) p ( m k ,fin | m k ,in ) , (45a)12here p k ,in . = p ( m k ,in ) (45b)is the probability of m k ,in . If M in is an EQ one at T in = T , then the EQ probability p (0) k ,in in the canoni-cal ensemble is p (0) k ,in . = e β ( F in − E k, in ) , (45c)where F in is the initial EQ free energy and β = 1 /T ;for an isolated system, T denotes its EQ temperature.For a system in IEQ, with Z E = ( V, ξ ), the microstateprobability looks very similar to the above Boltzmannprobability p k = exp { β [Φ − ( E k + P k V + A k · ξ )] } , (46)as given in Ref. [21, Eq. (20) with F k ,BP · R replacedby A k · ξ ]; here Φ is the SI thermodynamic potentialobtained by ensuring P k p k = 1.Alternatively, we first determine S using Eq. (2). Thisgives S = β ( E + P V + A · ξ − Φ) . (47)Using p k in a process involving IEQ macrostates, we de-termine p γ k above, which will be used in the rest of thepaper. To obtain the NEQ version of the canonical en-semble, the contribution from V is conventionally notincluded. In EQ, we must also remove the contributionfrom ξ as it is not an independent variable anymore. Thisthen gives Eq. (45c).We now determine Φ from Eq. (47) to obtainΦ = E − T S + P V + A · ξ. This is the SI potential in terms of system’s macro-quantities, and should not be confused with the cor-responding conventional thermodynamic potential Φ = E − T S + P V in terms of the fields of the medium. Ina NEQ canonical ensemble ( V fixed), Φ reduces to theSI free energy F in Eq. (22b), whereas the Helmholtzfree energy is F = E − T S . In EQ, Φ = Φ and F = F ;otherwise, they are different macroquantities. We introduce the moment generating function (MGF)for the random variable ∆ W , with outcomes ∆ W k over m k , along P : W ( ˙ β (cid:12)(cid:12)(cid:12) ∆ W ) = D e ˙ β ∆ W E . = P k p γ k e ˙ β ∆ W k , (48)where ˙ β is some independent parameter that p γ k and∆ W k do not depend upon, and the sum is over the en-semble of all trajectories γ k originating at m k ,in . Thedefinition is valid for any system, interacting or isolated, and for any terminal macrostates M in and M fin , neitherof which has to be an EQ macrostate. Implicit in the def-inition is the Hamiltonian characteristic of the process inEq. (8) and the thermodynamic nature of the trajec-tory probability p γ k in Eq. (44). The latter makes W athermodynamic function. The independent parameter ˙ β should not be confused with any inverse temperature ofΣ.Various moments of ∆ W are obtained by differentiat-ing W with respect to the parameter ˙ β and then setting˙ β = 0. For the first two moments, we have d W /d ˙ β (cid:12)(cid:12)(cid:12) ˙ β =0 = P k p γ k ∆ W k = ∆ W, (49a) d W /d ˙ β (cid:12)(cid:12)(cid:12) ˙ β =0 = P k p γ k (∆ W k ) = ∆ W >
0; (49b)∆ W introduced above should not be confused with(∆ W ) . Recalling Eq. (44), we see that ∆ W is thecumulative macrowork along the arbitrary process P forwhich there is no sign restriction. Indeed, all momentsderived from W are thermodynamic in nature so theyremain unchanged under any transformation such as inEq. (67) later that leaves W invariant.The MGF, apart from yielding all the moments, is alsoquite useful in establishing the following theorem for P between two EQ macrostates M in,eq and M fin,eq . Theorem 7
For ˙ β = β = 1 /T in , the initial value of β in M in,eq , the MGF becomes independence of the initialvalues of the probabilities n p (0) k ,in o . Proof.
Recalling Eq. (45a) and setting p ( m k ,in ) = p (0) k ,in ,we see that the MGF reduces to a function W given by W ( β | ∆ W ) = e β F in P k p eq ( m k ,fin | m k ,in ) e − β E k, fin , (50)which clearly shows its independence from the initialprobabilities n p (0) k ,in o .The theorem proves extremely useful in the trick inSec. 7.2 that is needed to overcome phase space volumechange. The bar on W is a reminder that we are consid-ering a NEQ process P between two EQ macrostates.We also note that since β is no longer an independentparameter, W is no longer a MGF. The sum in W turns into an EQ partition function ifwe set p ( m k ,fin | m k ,in ) = 1 , ∀ k , a poor approximation, inwhich case p k ,fin = p (0) k ,in , p γ k ( P ) = p (0) k ,in , ∀ k. (51)This leads to a new function c W (the caret on top is areminder of the choice in Eq. (51)), defined by c W ( β | ∆ W ) = \ h e β ∆ W i . = P k p (0) k ,in e β ∆ W k , (52)13here the ”average” denoted by b hi is with respect to n p (0) k ,in o as trajectory probabilities. The new functionsimplifies to c W ( β | { ∆ W } ) = e − β ∆ F , (53)where ∆ F . = F fin − F in . Comparing the ”average”macrowork \ h ∆ W i . = P k p (0) k ,in ∆ W k . (54)with Eq. (49a), we conclude that∆ W = \ h ∆ W i . In general, the non-thermodynamic probability choice in c W will not result in a thermodynamic macrowork ∆ W .It is clear that the assumption p γ k = p (0) k ,in does not resultin thermodynamic average; see [26] for more details. As c W is a non-thermodynamic function, we will refer to Eq.(53) as a mathematical identity to differentiate it from athermodynamic identity. Despite this, it is an interest-ing function in that it can be evaluated in a closed formas seen above. The assumption of a constant p γ k , ∀ k inEq. (51) and the choice ˙ β = β has been popularized byJarzynski through his WFT to be discussed below.The issue of non-thermodynamic averaging was firstraised by Cohen and Mauzerall [27, 28] in the contextof MFTs. But its most significant consequence is aboutjustifying the second law as discussed earlier with respectto Eqs. (3-6). The issue has been discussed and settledonly recently in Ref. [26]. We will establish below thatthe non-thermodynamic identity in Eq. (53) is satisfiedeven for free expansion as expected as the µ NEQT isperfectly capable of describing isolated systems. For freeexpansion, ∆ W k does not identically vanish for all k as weclearly see from Eq. (39c). In this regard, our approachis very different from the one used in the WFT, to whichwe now turn. The non-thermodynamic function c W is closely relatedto the well-known Jarzynski’s WFT [20] given below inEq. (57). We first introduce the function used by Jarzyn-ski W J ( β | ∆ e W ) . = P k p (0) k ,in e β ∆ e W k , (55a)which is obtained by replacing ∆ W by ∆ e W in c W ( β | { ∆ W } ). Here, ∆ e W is assumed to be a randomvariable with outcomes ∆ e W k over m k with some proba-bility, and the suffix ”J” is a reminder for Jarzynski’s non-thermodynamic average in which the probability is re-placed by p (0) k ,in . Jarzynski further assume, without proof,that ∆ e W k assump = ∆ W k = − ∆ E k , ∀ k, (56) as has been discussed recently for its validity [15].This work-energy assumption is in addition to the non-thermodynamic average conjecture in defining c W in Eq.(52), and violates Theorem 4. With the two assumptions, W J becomes W J ( β | ∆ e W ) = e − β ∆ F , (57)which is the well-known WFT. Let us consider the fol-lowing average∆ e W J . = \ h ∆ e W i = P k p (0) k ,in ∆ e W k . (58)From ∆ e W in Eq. (36), we see that in general∆ e W J = ∆ e W = − R. This should be contrasted with the fundamental assump-tion of the WFT in Eq. (6), which has already been ques-tioned earlier [26, 27]. Thus, the conjecture in Eq. (6)cannot be justified. Indeed, by considering an EQ processin an interacting system, for which ∆ e W = ∆ W = − ∆ F ,it has been shown [26, see Eq.(20) there] that∆ e W J < ∆ e W, violating Eq. (6). For an isolated system undergoingfree expansion for which ∆ e W k ≡
0, the ”mathematicalidentity” in Eq. (57) obviously fails [31, 34]. Jarzynski[35] argues that such a system does not start in EQ sothe WFT should not apply there; however, see [33] forcounter-argument.The ”mathematical identity” in Eq. (57) is evidentlydifferent from the previous mathematical identity in Eq(53) unless the above work-energy assumption in Eq.(56) is taken to be valid ; then the two are the same.However, the WFT is considered a mathematical identitysatisfied for a class of NEQ processes by most workers inthe field who have not appreciated the implications ofEq (53). Let us determine these implications within the µ NEQT. It follows from Eq. (56) that ∆ i W k = 0 , ∀ k ;see Theorem 4. Consequently, we must have ∆ i W = 0as noted in Sec. 1.2 so no irreversibility is captured bythe WFT as was first concluded a while back in Ref.[15]. Using thermodynamic probabilities p γ k with theassumption in Eq. (56), we find ∆ W = ∆ e W as a ther-modynamic consequence. This says nothing about \ h ∆ W i or ∆ e W J that are not thermodynamic quantities. How-ever, it is commonly believed that ∆ e W J = ∆ e W , whichcannot be justified [26].
6. THE FREE EXPANSION IN THE MNEQT
Our derivation of the identity in Eq. (53) is exact soit should be valid for all processes including free expan-sion P as we will now show. In free expansion, there isno exchange of any kind so d = d i . This simplifies ournotation as we do not need to use d i when referring to14. The gas Σ expands freely in a vacuum ( e Σ) from V in ,the volume of the left chamber, to V = V fin , the volumeof Σ ; the volume of the right chamber is V fin − V in . Thevacuum exerts no pressure ( e P = P vacuum = 0). The left(L) and right (R) chambers are initially separated by animpenetrable partition, shown by the solid partition inFig. 2(a), to ensure that they are thermodynamically in-dependent regions, with all the N particles of Σ in theleft chamber, which are initially in an EQ macrostate M in,eq with entropy S in . For ideal gas, we have S in = N ln( eV in /N );here, we are not including a temperature-dependent func-tion [5], which does not play any role as we will be consid-ering an isothermal free expansion. The initial pressureand temperature of the gas prior to expansion at time t = 0 are P in and T in = T , respectively, that are relatedto E = E in and V in by its EQ equation of state. As Σ is isolated, the expansion occurs at constant energy E ,which is also the energy of Σ.It should be stated, which is also evident from Fig.2(b), that while the removal of the partition is instanta-neous, the actual process of gas expanding in the rightchamber is continuous and gradually fills it. This isobviously a very complex internal process in a highlyinhomogeneous macrostate. As thus, it will requiremany internal variables to describe different number ofparticles, different energies, different pressures, differentflow pattern which may be even chaotic, etc. in eachof the chambers. For example, we can divide thevolume V fin into many layers of volume parallel tothe partition, each layer in equilibrium with itself butneed not be with others; see the example in Sec. 2.1.Here, we will simplify and take a single internal variable ξξ . = N L − N R (59)by considering only two layers to describe different num-bers N L = ( N + ξ ) / N R = ( N − ξ ) / V = V ( t ) to the rightof which exists a vacuum. This means that at each in-stant when there is a vacuum to the right of this front, thegas is expanding against zero pressure so that d e W = 0.Since we have a NEQ expansion, dW >
0. As V ( t ) can-not be controlled externally, it also represents an internalvariable. The two internal variables ξ ( t ) and V ( t ) allowus to distinguish between P and ¯ P as we will see below.We assume that the expansion is isothermal so there is noadditional internal variable associated with temperaturevariation. As dQ = dW = 0, the expansion is irreversibleso the entropy continues to change (increase).At t = 0, the partition is suddenly removed, shown bythe broken partition in Fig. 2(b) and the gas expandsfreely to the final volume V ( t ′ ) = V fin at time t ′ < τ eq during P . At t ′ , the free expansion stops but there is noreason a priori for ξ = 0 so the gas is still inhomogeneous( ξ = 0). This is in a NEQ macrostate until ξ achieves itsEQ value ξ = 0 during ¯ P , at the end of which at t = τ eq the gas eventually comes into M fin,eq isoenergetically. Webriefly review this expansion in the MNEQT [43].We work in the state space S . Using Eq. (14b), wehave dS ( t ) = dW ( t ) /T ( t ) . (60a)Setting P = 0 in Eq. (16a), we have dW ( t ) = (cid:26) P ( t ) dV ( t ) + A ( t ) dξ ( t ) for t < t ′ < τ eq ,A ( t ) dξ ( t ) for t ′ < t ≤ τ eq ; (60b)here, we have used the fact that V ( t ) does not change for τ ′ < t ≤ τ eq . Thus,∆ S = Z P dW ( t ) T ( t ) > , ∆ Q = Z P dW ( t ) = ∆ W > S from EQ macrostatefrom M in,eq to M fin,eq during P is the EQ entropychange ∆ i S is ∆ S ≡ S fin − S in , (61)and can be directly obtained if the EQ entropy S ( E, V )is known. The above analysis is also valid for any arbi-trary free expansion process P ; we must carry out theintegration over P above. We can evaluate ∆ i S by using S from Eq. (47).The above exercise allows us to identify ∆ W as thedissipated work over the entire process even if the pro-cess. We have ∆ W = ∆ i W = T ∆ i S for an isothermalprocess T ( t ) = T ; see also Eq. (25b). However, withthe inclusion of the internal variables above, we are alsoable to determine d i S = dS = dW/T using Eq. (60b)for any infinitesimal segment δ P of the process in theMNEQT that was one of our goals. Thus, d i W = T d i S is the infinitesimal ”dissipated work” over δ P but the re-lationship contains T and not T as it must; see Theorem1. Let us consider an ideal gas for which V fin = 2 V in sothat ∆ i S = N ln 2, a well-known result [6]. Here, weprovide a more general result for the entropy for t ≤ t ′ ,which can be trivially determined: S ( t, ξ ) = N L ln( eV in /N L ) + N R ln( eV ′ /N R ) , with ξ >
0; here V ′ = V − V in . Thus, for arbitrary ξ , we have ∆ i S ( t, ξ ) = S ( t, ξ ) − S in . At EQ, not only V ′ fin = V fin − V in , but also ξ = 0 so the EQ entropy isgiven by S fin = ( N/
2) ln(2 eV in /N ) + ( N/
2) ln(2 eV ′ fin /N ) , i S = N ln 2, as expected. Wecan also take the initial macrostate to be not an EQ onein P by using one or more additional internal variables.Thus, the approach is very general.
7. THE FREE EXPANSION IN THE µ NEQT7.1. Quantum Free Expansion
The expansion/contraction of a one-dimensional quan-tum ideal gas with moving walls has been treated in manydifferent ways [49–52] but none deal with sudden expan-sion. The latter, however, has been studied [26, 53, 54]quantum mechanically (without any ξ ) as a particle inan isolated box Σ of length L fin , which we restrict to2 L in here, with rigid, insulating walls. We briefly revisitthis study and expand on it by introducing a ξ to set thestage for the classical expansion using the µ NEQT in thefollowing section. We will follow Ref. [26] closely.We make the very simplifying assumptions in the pre-vious section to introduce ξ . At time t = 0, all the N particles (or their wavefunctions) are confined in EQ inthe left chamber of length L in so that N L = N initially.We can think of an intermediate length L fin ≥ L ( t ) > L in ,in analogy with V ( t ) in the previous section, so that N R = N − N L particles are simultaneously confined inthe intermediate chamber of size L ( t ), while N L particlesare still confined in the left chamber for all t > t = τ eq , allthe N R = N particles are confined in the larger cham-ber of size L fin so that there are no particles in the initialchamber. We let ξ = N L , which gradually decreases from ξ = N to ξ = 0. Note that this definition is different fromthe previous section but we make this choice for the sakeof simplicity. At some intermediate time τ ′ < τ eq thatidentifies P , L ( t ) = L fin , but N R is still not equal to N ( ξ = 0). We then follow its equilibration during ¯ P as thegas come to EQ in the larger chamber at the end of P when ξ = 0. Again, there are two internal variables L and ξ . The expansion is isoenergetic at each instant. Aswe will see below, this means that it is also isothermal.However, dQ = dW = 0 ensuring a irreversible processso the microstate probabilities continue to change.Since we are dealing with an ideal gas, we can focus ona single particle whose energy levels are in appropriateunits E k = k /l , where l is the length of the chamberconfining it. The single-particle partition function forarbitrary l and inverse temperature β = 1 /T is given by Z ( β, l ) = P k e − βE k ( l ) , from which we find that the single particle free energy is F = − ( T /
2) ln( πT l /
4) and the average single particleenergy is E = 1 / β , which depends only on β but not on l . Assuming that the gas is in IEQ so that the particlesin each of the two chambers are in EQ (see the secondexample in Sec. 2.1) at inverse temperatures β L and β , we find that the N -particle partition function is given by Z N ( β L , β ) = [ Z ( β L , L in )] ξ [ Z ( β, L )] N − ξ so that the average energy is E N ( β L , β, L in , L, ξ ) = ξ/ β L +( N − ξ ) / β . As this must equal N/ β for all val-ues of L and ξ , it is clear that β L = β = β , which provesthe above assertion of an isothermal free expansion at T .To determine ∆ W k , we merely have to determine themicroenergy change ∆ E k = E k ,fin − E k ,in . It is triviallyseen that Eq. (53) is satisfied as was reported earlier[15, 26].Below we will show that the quantum calculation heredeals with an irreversible P . The single-particle energychange ∆ E k is∆ E k = k (1 /L − /L ) < , L > L in . The micropressure P k = − ∂E k /∂L = 2 E k /L = 0 (62)determines the microwork∆ W k = Z L fin L in P k dL > . (63)It is easy to see that this microwork is precisely equalto ( − ∆ E k ) as expected; see Eq. (40). It is also evidentfrom Eq. (62) that for each L between L in and L fin , P = P k p k P k = 2 E/L = 0 , We can use this average pressure to calculate the ther-modynamic macrowork∆ W = Z L fin L in P dL = 2 P k Z L fin L in p k E k dL/L = 0 . as expected. As ∆ E = 0, this means that the irreversiblemacroheat and macrowork are ∆ Q = ∆ W >
0. Thisestablishes that the expansion we are studying is irre-versible .We now turn to the entire system in which the work isdone by N R particles. We need to think of the microstateindex k as an N -component vector k = { k i } denoting theindices for the single-particle microstates. For a given ξ ,we have ∆ W k ( L, ξ ) = − P i ∆ E k i , where i runs over the N R particles. We can compute the macrowork, whichturns out to be ∆ W N ( ξ ) = ( N − ξ )∆ W >
0. The corre-sponding change in the free energy is∆ F N ( L, ξ ) = ( N − ξ )[ F ( β , L ) − F ( β , L in )]= − ∆ W N ( ξ ) , which is consistent with Eq. (25a) for an isolated systemfor any ξ .At the end of P , ∆ W N (0) = N ∆ W >
0, and∆ F N (0) = N [ F ( β , L fin ) − F ( β , L in )]. We can now setup the MGF W ( β | { ∆ W } ) for any L and ξ so that we16an compute all the moments. However, we will onlyconsider the entire process P so that ξ = 0 at the end.For the first moment in Eq. (49a) we find that for theisothermal expansion∆ W N = − ∆ F N = T ∆ i S N > , (64)after using Eq. (25b). The same result is also obtainedfrom the classical isothermal expansion; see Eq. (60a).All this is in accordance with Theorem 1 in the MNEQT,as expected.For the discussion below, we suppress N as the sub-script for simplicity. The benefit of using the µ NEQTis that we get a much better perspective of the dissi-pation in the free expansion. In terms of the SI mi-crowork, ∆ i W k = 0 , ∀ k even though the SI microwork∆ e W k ≡ , ∀ k . However, what is more revealing aboutthe free expansion is that ∆ W k > γ k as seen from Eq. (63), which is in accor-dance with Corollary 5. This is not true in a generalprocess. For example, ∆ i W = 0 in a reversible processeven though ∆ i W k = 0 , ∀ k . In fact, ∆ i W k must be ofeither sign to ensure a vanishing average. The change in the phase space volume in classical sta-tistical mechanics destroys the required unique mappingin Eq. (8) as discussed in Sec. 2.4, and causes a prob-lem with the use of the Hamiltonian trajectories thatwere used in deriving the µ NEQT and introducing theMGF W . Their use for classical expansion will requiressome modification, which we describe below. To intro-duce the required modification as simply as possible, wewill first consider V fin = 2 V in that results in doubling thevolume after expansion; see Fig. 2(a). Later, we willgeneralize to any arbitrary expansion/contraction. Wewill use the notation of Sec. 6, and restrict ourselves to Z E = ( V in , V fin , V, ξ ); see Eq. (59) for ξ and Fig. 3.Let δ z ( t ) . = δ z ( V, ξ ) denote a microstate at some time t with work variables V . = V ( t ) and ξ . = ξ ( t ). Let E k ( t ) . = E ( δ z ( t )) denote the instantaneous microenergyof δ z ( t ). Let δ z be some initial microstate at t = 0 with N L = N, N R = 0 , V = V in , and ξ = ξ in = N . We de-note the number of microstates in the initial phase spacethat is denoted by the interior of the solid red ellipse Γ in = Γ in ( Z E in ) on the left by N . The final phase spaceis shown by the solid red ellipse Γ ′ fin = Γ ′ fin ( Z E fin ) on theright, which contains twice as many (2 N ) microstatesas are in Γ in . We will assume that both the initial andthe final macrostates ( M in,eq and M fin,eq ) are in EQ forwhich V and ξ are no longer independent state variables.Therefore, we will not use V and ξ in Z E for the twophase spaces at t = 0 and t = τ eq . We set P = 0 in Eq.(39b) to obtain the microwork for t > dW k = P k dV + A k dξ, which is nonzero, while d e W k = 0. The EQ gas at t = 0has microstate probabilities p in ( δ z ) = e − β E ( δ z ) /Z in ( β , V in ) ≥ , (65a) Z in ( β , V in ) . = P δ z ∈ Γ in e − β E ( δ z ) . (65b)Under T , δ z ( V in ) ∈ Γ in maps onto its image (see thegreen double arrow γ in ) δ z ′ ( V fin ) ∈ Γ ′ in = Γ ′ in ( V fin ), where Γ ′ in is shown schematically by the broken red ellipse onthe right. Thus, Γ ′ in contains N microstates. We thenpick a microstate δζ ′ ( V fin ) ∈ Γ ′ diff ( V fin ) . = Γ ′ fin \ Γ ′ in ; here, Γ ′ diff denotes Γ ′ fin from which Γ ′ in has been removed soit also contains N microstates. Because of the unique-ness of γ fin , δζ ′ ( V fin ) / ∈ Γ ′ in is the image of a microstate δζ ( V in ) ∈ Γ diff . = Γ fin \ Γ in , where Γ fin is shown by thebroken ellipse on the left from which Γ in has been takenout to obtain Γ diff . Again, Γ diff contains N microstates.To find δζ , we follow the inverse of γ fin along which Z E fin → Z E in ( V fin → V in ); this is shown by the left arrowon γ fin in accordance with the reversibility of T . Thesame number of microstates in Γ diff and Γ ′ diff ensures theuniqueness of γ fin , a prerequisite for Hamiltonian trajec-tories. Physically, the N microstates in Γ diff correspondsto as if there are N particles in the right chamber in Fig.2(a), with the partition intact so that each particle isconfined in the volume V of the right chamber only. Butnote that there are no particles in the right chamber at t = 0.The situation was very simple for the uniqueness of γ fin because of the choice V fin = 2 V in : Γ ′ diff and Γ diff havethe same number of N microstates. This will not be thecase if V in < V fin < V in , since the number of microstates δζ ′ ( V fin ) in Γ ′ diff is now N ′ < N , while Γ diff has N mi-crostates δζ ( V in ). To identify the set { δζ ( V in ) } ⊂ Γ diff of theses N ′ microstates, we follow the inverse of γ fin foreach δζ ′ ( V fin ) ∈ Γ ′ diff ( V fin ) to identify the image δζ ( V in ) asdescribed above. From now on, we will use Γ diff ( V in | V fin )to denote the required set of N ′ microstates. The N − N ′ remaining microstates in Γ diff are superfluous. We cansimilarly extend the above discussion to V fin = 3 V in , andthen to 2 V in < V fin < V in , and so on.With the above understanding of Γ diff ( V in | V fin ), therewill be no confusion to simplify the notation and use Γ diff for it. With this understanding, Γ fin as the unionof Γ in and Γ diff only contains the required N + N ′ mi-crostates. This will be understood below.Let us pick two microstates δ z ( V in ) ∈ Γ in and δζ ( V in ) ∈ Γ diff . Their image microstates δ z ′ ( V fin ) ∈ Γ ′ in and δζ ′ ( V fin ) ∈ Γ ′ diff are obtained by the deterministic evo-lution along γ in = γ ( δ z ) and γ fin = γ ( δζ ), respectively.The corresponding microworks are given by∆ W ( δ z ) = − ( E ( δ z ′ ) − E ( δ z )) , ∆ W ( δζ ) = − ( E ( δζ ′ ) − E ( δζ )) . (66)However, the situation is still not fully resolved as δζ ( Z E in ) ∈ Γ diff does not represent a physical initial mi-17rostate in the left chamber; instead, it refers to a mi-crostate in the right chamber so its probability vanishes: p in ( δζ ) = 0 , δζ ∈ Γ diff . We recall that we do not need p k to determine ∆ W k so it does not matter if p in ( δζ ) = 0. During expansion, p ( δζ ) at t > formally assume that the initial probability distribution p in ( δζ ) is infinitesimally small for δζ by shifting its initialenergy by a very large positive contribution E ( δζ ) → E ( δζ ) + e ( δζ ) /ε > , δζ ∈ Γ diff at t = 0 (67)using an infinitesimal quantity ε >
0. At the end of thecalculation, the limit ε → + will be taken to ensure p in ( δζ ) ε → + →
0. This trick of energy shift transforms W as W ( ˙ β, ε (cid:12)(cid:12)(cid:12) ∆ W ) → P δ z ∈ Γ in p ( γ in ) e ˙ β ∆ W ( δ z ) + P δζ ∈ Γ diff p ( γ fin ) e ˙ β ∆ W ( δζ ) ; (68)however, the energy shift leaves ∆ W ( δζ ) invariant. Thus,we see that letting ε → + makes the second term vanishso W ( ˙ β, ε (cid:12)(cid:12)(cid:12) ∆ W ) ε → + → W ( ˙ β (cid:12)(cid:12)(cid:12) ∆ W ). As W and all the mo-ments remain invariant, the trick (energy shift followedby ε → + ) does not affect thermodynamics in anyway.In particular, it leaves the first law unaffected. We willverify this below by direct manipulation.It is clear that the two sums in Eq. (68) can be ex-pressed as a single sum over all microstates δ ¯z ∈ Γ fin ,which refers to microstates in Γ fin . For example, theshifted initial partition function also remains invariantunder the trick: Z in ( β , V in , ε ) . = P δ ¯z ∈ Γ fin e − β E ( δ ¯z ) ε → + → Z in ( β , V in ) . (69)This is consistent with Theorem 7. Thus, we can focuson Γ fin from the start instead of Γ in . This allows us tobasically use the identity mapping T between initial mi-crostates δ ¯z ∈ Γ fin and final microstates δ ¯z ′ ∈ Γ ′ fin usingHamiltonian trajectories γ as ( V in , ξ in ) → ( V fin , ξ fin = 0).In the following, we will use { δ ¯z } and { δ ¯z ′ } to show thesesets. We will use { γ ( t ) } to denote the ensemble of Hamil-tonian trajectories at any time t .We can combine the two equations in Eq. (66) in a sin-gle equation ∆ W ( δ ¯z ) = − ( E ( δ ¯z ′ ) − E ( δ ¯z )), with E ( δ z )playing the role of E k along γ . Let us consider W for P W ( β | { ∆ W } ) . = lim ε → + P δ ¯z ∈ Γ fin p eq ( γ ) e β ∆ W ( δ ¯z ) . Recalling Eqs. (45a) and (65a), the summand becomes p eq ( δ ¯z ′ | δ ¯z ) e − β E ( δ ¯z ′ ) /Z in ( β , V fin , ε ), in which e β E ( δ ¯z ) from p in,eq ( δ ¯z ) exactly cancels with e β E ( δ ¯z ) comingfrom ∆ W ( δ ¯z ). As the conditional probability p ( δ ¯z ′ | δ ¯z ) is not affected by p in ( δ ¯z ), taking the limit is triv-ial as it affects only Z in ( β , V fin , ε ) in accordancewith Eq. (69) so the above summand converges to p ( δ ¯z ′ | δ ¯z ) e − β E ( δ ¯z ′ ) /Z in ( β , V in ). Thus, W ( β | ∆ W ) = P δ ¯z ∈ Γ fin p eq ( δ ¯z ′ | δ ¯z ) e − β E ( δ ¯z ′ ) Z in ( β , V in ) . (70)Before proceeding further, we wish to confirm that theinclusion of microstates in Γ diff does not violate the firstlaw in MNEQT by focusing on h ∆ W i . We consider amicrostate δ ¯z ( t ) along an infinitesimal segment dγ on γ between t > t + dt . Over this segment, Z E ( t ) =( V, ξ ) → Z E ( t + dt ) = ( V + dV, ξ + dξ ). Thus, dW ( t ) = − P { δ ¯z ( t ) } p ( δ ¯z ( t ))[ E ( Z E ( t + dt )) − E ( Z E ( t ))] . Introducing dp ( t ) . = p ( δ ¯z ( t + dt )) − p ( δ ¯z ( t )), and dE ( t ) = E ( t + dt ) − E ( t ), we have dW ( t ) = − dE ( t ) + P { δ ¯z ( t ) } dp ( t ) E ( Z E ( t + dt )) ≃ − dE ( t ) + dQ ( t ) , where we have neglected the second-order term dp ( t ) dE ( Z E ( t )) as is a common practice and used Eq.(33a) to identify dQ . Thus, for any t >
0, we have sat-isfied Eq. (10a). To consider t = 0, we need to takethe limit ε → + , which limits the sum in dW above to δ ¯z ∈ Γ in . This ensures that Eq. (10a) remains satis-fied at t = 0 with our modification in Eq. (67). Thus,we have established that our trick of using ∀ δ ¯z ∈ Γ fin is consistent with the MNEQT. It follows then that thediscussion here in the µ NEQT will finally result in Eq.(61) once we realize that dE = 0 so that dW = T dS asin Eq. (60a).We now consider c W ( β | ∆ W ), which can be exactlyevaluated. For this, we set p eq ( δ ¯z ′ | δ ¯z ) = 1 above anduse the 1-to-1 mapping T : δ ¯z → δ ¯z ′ to replace the sumover δ ¯z to a sum over δ ¯z ′ to obtain c W ( β | ∆ W ) = P δ ¯z ′ ∈ Γ ′ fin e − β E ( δ ¯z ′ ) Z in ( β , V in ) = Z fin ( β , V fin ) Z in ( β , V in ) , (71)which is precisely Eq. (53). Notice that the initial EQmacrostate in proving the above relation corresponds tothe one with all the particles confined in the left chamber;see, however, [33].Above, we have considered the case of free expansion.The same trick will also work if the expansion is gradualand not abrupt. The only difference will be that V ( t )will not be an internal variable as it is controlled exter-nally. We still will need the trick of inserting ”missing”microstates as above. By interchanging the role of theinitial and final phase spaces above, we can also use thetrick to investigate the case of contraction. The only dif-ference is that in the last two cases, the system is not18solated so we must make a distinction between dW (or∆ W ) and d i W (or ∆ i W ).If it happens that the phase space volume | Γ | contin-ues to change during a process but | Γ in | = | Γ fin | , such asa cyclic process, we can treat it as a combination of ex-pansion and contractions processes. To see this, we lookfor the time t m when | Γ | is maximum (minimum). Then,we are dealing with expansion (contraction) over t > t m ,and contraction (expansion) over t > t m . The same ap-proach of a combination of expansion/contraction can betaken when | Γ | does not change monotonically during aprocess, whether | Γ in | = | Γ fin | or not.
8. DISCUSSION AND CONCLUSION
The current investigation was motivated by a desire tounderstand the following two very important aspects ofNEQ thermodynamics at the microstate level1. how to use Hamiltonian trajectories to describephase space volume changes in a process to con-struct a microstate NEQ thermodynamics of a sys-tem, interacting or not , and2. the nature of microworks that give rise to the dis-sipated work in a noninteracting , i.e., an isolatedsystem for which no exchange of macroheat andmacrowork is allowed (∆ e Q = 0 , ∆ e W = 0),as these issues have not been addressed in the liter-ature but lie at the heart of the many common NEQprocesses including free expansion taught to undergrad-uates in macroscopic thermodynamics. Recently, we havedeveloped a macroscopic and microscopic SI NEQ ther-modynamics (the MNEQT and the µ NEQT) that di-rectly include the macroforce imbalance F t or the micro-force imbalance F wt, k , an important concept introducedrecently by us, to ensure describing an interacting anda noninteracting system within the same framework. Asdiscussed in Sec. 1.2, no other current NEQ thermody-namics directly captures the microforce imbalance. Inits absence ( F wt, k = 0 , ∀ k ), a situation common in variousMFTs noted in Sec. 1.2 and discussed in Sec. 5.4, therecannot be any work irreversibility so we cannot overem-phasize its importance for any NEQ processes.The phase space Γ of a classical system with finiteparameter Z has a finite volume | Γ | . In particular, thisrequires { E k } to have only finite number of values. Thisis the case for all numerical simulations so our approachhere provides a useful approach to carry out numericalsimulation of a finite system.Let us consider the following two cases that can arise.1. If the volume | Γ | does not change at all during aprocess ( | Γ in | = | Γ fin | ), the finite number of microstates N . = | m | also does not change, and one can always use theidentity property of the Hamiltonian trajectory in Eq.(8) to follow the temporal evolution of each microstate m k along γ k . 2. When the volume | Γ | changes monotonically dur-ing a process, which is a common situation such as dur-ing expansion or contraction of a classical system, then N in . = | m in | 6 = N fin . = | m fin | . Then these microstates cannotbe connected 1-to-1 by Hamiltonian trajectories. The ex-cess microstates N diff . = | m diff | between { m in } and { m fin } will have no Hamiltonian trajectories associated withthem as seen in Fig. 3 for the case of expansion. This iswhen our trick in Sec. 7.2 will be useful. We introduceexcess microstates in the phase space with the smallernumber of microstates but with extremely high positivemicroenergy shift to ensure that | m | is the same in theinitial and final macrostates. This trick is crucial andenables us to use only Hamiltonian trajectories to con-nect the microstates in Γ in and Γ fin in a 1-to-1 manner.Eventually, the microenergy of the missing microstatesis taken to diverge to + ∞ to ensure their probabilitiesvanish in the phase space of smaller volume. However,if | Γ | changes nonmonotonically during the process evenif | Γ in | = | Γ fin | , then, as discussed in Sec. 7.2, we aredealing with a combination of expansion and contractionso it belongs to this case.The above argument suggests that when phase spacevolume | Γ | is not finite, the Hamiltonian trajectories willnot work, since there is no way to argue that N in = N fin when both are infinitely large. This is not correct. Letus assume that | m | is infinitely large but the set { m k } isdenumerable as is the case with the quantum expansionin Sec. 7.1. We still have 1-to-1 Hamiltonian trajectoriesconnecting m k ,in with m k ,fin , ∀ k ∈ N , where N is the setof natural numbers, i.e. { } . It is this propertythat allows us to introduce the µ NEQT in Sec. 4 andthe MGF in Sec. 5. This is true despite the sets { m k ,in } and { m k ,fin } having ”equal” but infinite large number ofmicrostates so we can say that | m in | ≡ | m fin | .In the classical expansion/contraction, the equality ofthe numbers of microstates is clearly not valid despitethe fact that they both have the same cardinality as thatof N . What is important for the cardinality considera-tion is the idea of association and not the unique 1-to-1mapping. This is the same when we compare N with theset N o of odd natural numbers, i.e. { } or the set N e of even natural numbers, i.e . { } . All thesesets have the same cardinality, but N has twice as manymembers as either of the sets N o or N e . It is the numberof members that is important in the mapping in Eq. (8).To understand it, we proceed as follows. Think of N as an example of the set of microstates in the interior ofthe dotted ellipse Γ fin , N o as an example of the set ofmicrostates in the interior of the solid ellipse Γ in , and N e as an example of the set of microstates in the difference Γ diff of the two ellipses Γ fin and Γ in . We can similarlyintroduce N ′ for the interior of the solid ellipse Γ ′ fin , N ′ o for the interior of the broken ellipse Γ ′ in , and N ′ e for thedifference Γ ′ diff . Under the 1-to-1 mapping, N o goes to N ′ o , with no mapping leading to N ′ e . This clearly showsthat the concept of cardinality is irrelevant as noted inSec. 2.4. Now we can use our trick introducing excess19icrostates to ensure N in = N fin . Using the MGF W ,we have also established that the trick does not destroythe consistency of the µ NEQT with the MNEQT so thetrick is thermodynamically consistent. It is clear that thetrick is very general and will work in all cases of missing microstates.The most important signature of a NEQ process is theexistence of (generalized) thermodynamic force F t , seeEq. (19), which drives the system towards EQ. It is com-mon to study a NEQ isothermal process ( T = T ⇒ F ht =0) so we must look for a nonvanishing F wt in this case,regardless of whether the system is isolated or interact-ing. In the former case, F wt reduces to A as seen in Eq.(20), and contains A in the latter case. The presence of F wt in a NEQ system, including the free expansion thatwe are interested in, is necessary along with the presenceof internal variables. Both of these features are absentin MFTs so they provide no guide in our investigation asdiscussed in Sec. 1.2. We are therefore left to use our re-cently developed NEQ thermodynamics, which containsboth features. For the sake of continuity, we have pro-vided a brief review of the MNEQT and the µ NEQT inSecs. 3 and 4. As F wt, k is ubiquitous (it is present even inEQ), it is clear that its absence in the MFTs and theirinability to treat free expansion make them very differentfrom the µ NEQT.We now discuss some of the important results of ouranalysis .The SI microwork ∆ W k along γ k is shown to be givenby the negative of the microenergy change ∆ E k so it isunaffected by a constant shift proportional to ε − in E k .Moreover, the addition of missing microstates does notchange the MGF W ( β | ∆ W ). This function is importantto justify that the trick leaves our approach consistentwith thermodynamics. The presence of internal variablesmakes the MNEQT and µ NEQT perfectly suited to studyany system, interacting or isolated. The success of our theory to study the behavior of a Brownian particle [21]also shows that they can also be used to study smallsystems.We have shown that the best way to understand theorigin of dissipated work is to focus on dW and dQ , with d i W = d i Q ≥
0, as is done in the MNEQT and µ NEQT.In an isolated system, dW = d i W = dQ = d i Q . There-fore, only d i W k resulting from F wt, k needs to be deter-mined in the µ NEQT, which is easier to do with the useof the Hamiltonian trajectories. The presence of F wt, k andinternal variables for isolated systems finally explains thesource of irreversibility ( d i S ≥ µ NEQT to small-scalesystems (Brownian particles) in Ref. [21], it is possibleto extend the expansion/contraction of the gas investi-gated here to study Maxwell’s demon and the problem ofLandauer’s eraser. We hope to come to these problemsin future.Finally, we mention some of the important predictionsof our approach. We restrict ourselves to free expan-sion here. While in general, dW k = d i W k can have anysign even though d i W ≥
0, we must have dW k ≥ { ∆ W k } . One shouldnot see any negative ∆ W k . By determining ∆ W andcomparing it with ∆ W isoth = − ∆ F = T ∆ S , see Eqs.(25a, 25b), we can determine whether the process isisothermal or not; see Eq. (26).Declarations of interest: none ∗ Electronic address: [email protected][1] J.W. Gibbs,
Elementary Principles in Statistical Me-chanics , Charles Scribner’s Sons, N.Y. (1902).[2] E. Fermi,
Thermodynamics , Dover, New York (1956).[3] L.C. Woods,
The Thermodynamics of Fluids Systems ,Oxford University Press, Oxford (1975).[4] J. Kestin,
A Course in Thermodynamics, Revised Print-ing,
McGraw-Hill, New York (1979).[5] L.D. Landau, E.M. Lifshitz,
Statistical Physics , Vol. 1,Third Edition, Pergamon Press, Oxford (1986).[6] D. Kondepudi and I. Prigogine,
Modern Thermodynam-ics , John Wiley and Sons, West Sussex (1998).[7] S.R. de Groot and P. Mazur, nonequilibrium Thermody-namics , First Edition, Dover, New York (1984).[8] D. Jou, J. Casas-V´azquez, and G. Lebon,
Extended Irre-versible Thermodynamics , Springer, Berlin (1996).[9] H.C. ¨Ottinger,
Beyond Equilibrium Thermodynamics ,Wiley Interscience, New Jersey (2005).[10] B.C. Eu,
Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrody-namics,
Vol. 1, Springer, Switzerland (2016).[11] P.D. Gujrati, Phys. Rev. E , 051130 (2010); P.D. Gu-jrati, arXiv:0910.0026.[12] P.D. Gujrati, Phys. Rev. E , 041128 (2012); P.D. Gu-jrati, arXiv:1101.0438.[13] P.D. Gujrati, Phys. Rev. E , 041129 (2012); P.D. Gu-jrati, arXiv:1101.0431.[14] P.D. Gujrati, arXiv:1304.3768.[15] P.D. Gujrati, arXiv:1702.00455; P.D. Gujrati,arXiv:1803.09725v2.[16] P.D. Gujrati, Entropy, , 710 (2015).[17] P.D. Gujrati, Entropy , 149 (2018).[18] B.D. Coleman, J. Chem. Phys. , 597 (1967).[19] G.A. Maugin, The Thermodynamics of Nonlinear Irre-versible Behaviors: An Introduction , World Scientific,Singapore (1999).[20] C. Jarzynski, Phys. Rev. Lett. , 2690 (1997).[21] P.D. Gujrati, Phys. Rev. E , 012140 (2020).
22] U. Seifert, Eur. Phys. J. B , 423 (2008).[23] C.Van den Broeck and M. Esposito, Physica A , 6(2015).[24] S.K. Blau, Phys. Today , 19 (2002).[25] C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today, , 43 (2005).[26] P.D. Gujrati, Phys. Lett. A , 126460 (2020).[27] E.G.D. Cohen and D. Mauzerall, J. Stat. Mech. P07006(2004).[28] E.G.D. Cohen and D. Mauzerall, Mol. Phys. , 2923(2005).[29] C. Jarzynski, arXiv:cond-mat/0407340v2.[30] W. Muschik, arXiv:1603.02135; Continuum Mech. Ther-modyn., DOI 10.1007/s00161-016-0517-y (online).[31] J. Sung, arXiv:cond-mat/0506214v4.[32] Eq. (17) in [29] gives the WFT for a thermally isolatedsystem, which Jarzynski calls a strong result as it doesnot depend on the specific protocol for how the work pa-rameter is varied. Its derivation does not depend in anyway on the fact that the work parameter is externally controlled; its external nature appears only in interpret-ing the work as external work. Thus, the work parametercan as well be controlled internally . In this case, ther-mally isolated system turns into an isolated system. Forexample, we can consider the partition in Fig. 2 beingconnected to a spring, whose other end is connected tothe far right wall of the right chamber. If the spring con-stant changes in time, this will control the position of thepartition and internal work will be done. Thus, the WFTshould equally hold for an isolated system.[33] There seems to some confusion in the literature aboutfree expansion set up [35]. We take all the particles to beconfined in the left chamber initially. For a careful de-scription of the set up and the process of free expansionthat we consider, see Fig. 4.16 and the discussion on pp.130-31 in [4, vol. 1]. The partition and the walls are com-pletely insulating (adiabatic) and impenetrable (no par-ticle transfer) to make the two chambers independent. Inother words, the presence of the vacuum does not affect the system Σ (the gas in the left chamber) so the lattercan be initially prepared in EQ at a given temperature.Jarzynski [35] assumes that the two chambers are inter-acting so the initial EQ macrostate corresponds to equalnumber of particles in each chamber. This is not what weconsider in this study.[34] D.H.E. Gross, arXiv:cond-mat/0508721v1.[35] C. Jarzynski, arXiv:cond-mat/0509344v1.[36] R.C. Lua, arXiv:cond-mat/0511302v1.[37] R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B ,6805 (2005).[38] C. Jarzynski, Phys. Rev. E , 046105 (2006).[39] A. Baule, R.M.L. Evans, and P.D. Olmsted, Phys. Rev.E , 061117 (2006).[40] G.E. Crooks and C. Jarzynski, Phys. Rev. E , 021116(2007).[41] S.J. Davie, J.C. Reid, and D.J. Searles, J. Chem. Phys. , 174111 (2012).[42] P.D. Gujrati, arXiv:1105.5549.[43] P.D. Gujrati, arXiv:1206.0702.[44] P.D. Gujrati, Symmetry , 1201 (2010).[45] P.G. Debenedetti, Metastable Liquids; Conc epts andPrinciples , Priceton University Press, Princeton (1996).[46] P.D. Gujrati, Entropy, , 149 (2018); see Sec. 8.1 andEq. (58).[47] F. Ritort, J. Stat. Mech.: Theor. Exp. P10016 (2004).[48] C.Vanden Broeck and M. Esposito, Physica A , 6(2015).[49] S.W. Doescher and M.H. Rice, Am. J. Phys. , 1246(1969).[50] O.Foj´on, M.Gadella, and L.P.Lara, Comput. Math. Appl.59, 964 (2010).[51] S. Di Martino, et. al. J. Phys. A , 365302 (2013).[52] K. Cooney, arXiv:1703.05282v1.[53] C.M. Bender, D.C. Brody and B.K. Meister, J. Phys. A , 4427 (2000).[54] P.D. Gujrati, arXiv:1512.08744., 4427 (2000).[54] P.D. Gujrati, arXiv:1512.08744.