# A Novel Trick to Overcome the Phase Space Volume Change and the Use of
Hamiltonian Trajectories with an emphasis on the Free Expansion

P.D. Gujrati<EMAIL_ADDRESS>Department of Physics, Department of Polymer
Science, The University of Akron, Akron, OH 44325

(January 31, 2020)

###### Abstract

We extend and successfully apply a recently proposed microstate nonequilibrium
thermodynamics ($\mu$NEQT) to study expansion/contraction processes. Here, the
numbers of initial and final microstates $\left\\{\mathfrak{m}_{k}\right\\}$
are different so they cannot be connected by unique Hamiltonian trajectories.
This commonly happens when the phase space volume changes, and has not been
studied so far using Hamiltonian trajectories that can be inverted to yield an
identity mapping
$\mathcal{T}:\mathfrak{m}_{k}(\mathbf{Z}_{\text{in}}^{E})\rightleftarrows\mathfrak{m}_{k}(\mathbf{Z}_{\text{fin}}^{E})$
as the parameter $\mathbf{Z}^{E}$ in the Hamiltonian is changed. We propose a
trick to overcome this hurdle with a focus on free expansion
($P_{\text{vacuum}}=0$) in an isolated system, where the concept of dissipated
work is not clear. The trick is shown to be thermodynamically consistent and
can be extremely useful in simulation. We justify that it is the thermodynamic
average $\Delta_{\text{i}}W\geq 0$ of the internal microwork
$\Delta_{\text{i}}W_{k}$ done by $\mathfrak{m}_{k}$ that is dissipated; this
microwork is different from the exchange microwork $\Delta_{\text{e}}W$ with
the vacuum, which vanishes. We also establish that $\Delta_{\text{i}}W_{k}\geq
0$ for free expansion, which is remarkable, since its sign is not fixed in a
general process.

Keywords: Microstate irreversible thermodynamics, Dissipated work in free
expansion, Mising microstates, Internal variables, Modern fluctuation
theorems.

††preprint: UATP/2002

##  I Introduction

###  I.1 Background

Free (unrestricted) expansion is an undergraduate paradigm of irreversibility,
in which the exchange macrowork $\Delta_{\text{e}}W$ and macroheat
$\Delta_{\text{e}}Q$, see Fig. 1, are identically zero. It is also accompanied
by an increase in the volume $\left|\Gamma\right|$ of the phase space $\Gamma$
of the system $\Sigma$. One can study it as an irreversible process
$\mathcal{P}$ going on within an isolated system from an initial (in)
macrostate $\mathfrak{M}_{\text{in}}$ to a final (fin) macrostate
$\mathfrak{M}_{\text{fin}}$. This ensures that its energy remains constant
even if the system remains out of equilibrium (EQ) during the entire process
including the initial and final macrostates. The expansion is sudden at $t=0$,
but it takes a while ($t=\tau_{\text{eq}}>0$) for EQ to emerge. As is known, a
macrostate $\mathfrak{M}$ of $\Sigma$ refers to a collection
$\left\\{\mathfrak{m}_{k},E_{k},p_{k}\right\\}$ of its microstates
$\mathfrak{m}_{k}$ of energies $E_{k}$ that appear with probabilities $p_{k}$
in $\mathfrak{M}$; the $p_{k}$’s give rise to stochasticity required for a
proper thermodynamics. We use _macro_ \- and _micro_ \- in this study to refer
to quantities pertaining to macrostates and microstates, respectively, with a
macroquantity referring to the _thermodynamic_ average of related
microquantities. A microquantity will always carry a subscript $k$ as a
reminder that it is associated with $\mathfrak{m}_{k}$. We will use $Z$ for a
state variable (see Sec. II.2 for explanation), a macrovariable, and $Z_{k}$
for its value, a microvariable, for $\mathfrak{m}_{k}$.

The study of a particular form of the free expansion is well known at the
undergraduate level in the traditional macroscopic nonequilibrium (NEQ)
thermodynamics Gibbs ; Fermi ; Woods ; Landau ; Prigogine ; Kestin  based on
the above exchange quantities; see also Eu ; Jou ; Ottinger ; deGroot  for
modern treatment. We denote this traditional NEQ thermodynamics by M̊NEQT in
the following; here M stands for macroscopic and the small circle refers to
the use of the exchange quantities. The study works if and only if the initial
and final macrostates $\mathfrak{M}_{\text{in,eq}}$ and
$\mathfrak{M}_{\text{fin,eq}}$, respectively, are in EQ so that the entropy
change $\Delta S=S_{\text{fin}}-S_{\text{in}}$ and, therefore, the net
irreversible entropy change $\Delta_{\text{i}}S=\Delta S$ over the process can
be evaluated without knowing the entire history. But the M̊NEQT does not
provide any information during the relaxation ($t<\tau_{\text{eq}}$) towards
$\mathfrak{M}_{\text{fin,eq}}$ such as the irreversible entropy generation
$d_{\text{i}}S(t)$ associated with any segment $\delta\mathcal{P}$ of the
process between intermediate NEQ macrostates $\mathfrak{M}(t)$. Thus, the use
of the M̊NEQT is limited in its scope.

A NEQ process $\mathcal{P}$ undergoes dissipation at all times
$t<\tau_{\text{eq}}$, and is usually described by the dissipated work
$\Delta_{\text{i}}W>0$, which in turn is directly related to
$\Delta_{\text{i}}S$ over $\mathcal{P}$ under suitable conditions; see later.
Here, free expansion poses another hurdle as the common understanding is that
any internal work done by the ”vacuum” (absence of matter and radiation) into
which the gas expands must be zero; see Fig. 2. This makes it hard to
understand what work is being dissipated as the gas most certainly generates
irreversible entropy $\Delta_{\text{i}}S>0$. A central aspect of this
investigation is to obtain a better understanding of dissipated work
$d_{\text{i}}W$ and the source of $d_{\text{i}}S$ over $\delta\mathcal{P}$ in
an interacting and an isolated system; see Corollary 2. This is achieved by
focusing on _system-intrinsic_ (SI) quantities $dZ,dZ_{k}$ (which we now allow
to also include $dW,dW_{k}$ and $dQ,dQ_{k}$, which should not be confused with
their exchange analogs $d_{\text{e}}W,d_{\text{e}}W_{k}$ and
$d_{\text{e}}Q,d_{\text{e}}Q_{k}$; see below) that are _uniquely_ determined
by the system itself. They contain all the information including the one about
internal processes that we wish to understand. The exchange quantities
$d_{\text{e}}Z,d_{\text{e}}Z_{k}$ (which also include
$d_{\text{e}}W,d_{\text{e}}W_{k}$ and $d_{\text{e}}Q,d_{\text{e}}Q_{k}$) are
primarily determined by the macrostate of the medium $\widetilde{\Sigma}$; we
will refer 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 determine $d_{\text{i}}Z\doteq dZ-d_{\text{e}}Z,d_{\text{i}}Z_{k}\doteq
dZ_{k}-d_{\text{e}}Z_{k}$ that directly describe the irreversibility in the
system.

Figure 1: (a) An isolated nonequilibrium system $\Sigma_{0}$ with internally
generated $d_{\text{i}}Z$ driving it towards equilibrium, during which its SI-
fields $T(t),P(t),\cdots,\mathbf{A}(t)$ associated with SI-variables
$S(t),V(t),\cdots,\mathbf{\xi}(t)$ continue to change towards their
equilibrium values; $d_{\text{i}}Z_{k}$ denote the microanalog of
$d_{\text{i}}Z$. The sign of $d_{\text{i}}Z$ is determined by the second law.
(b) A nonequilibrium systen $\Sigma$ in a surrounding medium
$\widetilde{\Sigma}$, both forming the isolated system $\Sigma_{0}$. The
macrostates of the medium and the system are characterized by their fields
$T_{0},P_{0},...,\mathbf{A}_{0}=0$ and $T(t),P(t),...,\mathbf{A}(t)$,
respectively, which are different when the two are out of equilibrium.
Exchange quantities ($d_{\text{e}}Z$) carry a suffix ”e” and irreversibly
generated quantities ($d_{\text{i}}Z$) within the system by a suffix ”i” by
extending the Prigogine notation. Their sum $d_{\text{e}}Z+d_{\text{i}}Z$ is
denoted by $dZ$, which is a system-intrinsic quantity (see text). In a
nonequilibrium system, the nonzero differences
$F_{\text{t}}^{\text{h}}=T-T_{0}$ and
$\mathbf{F}_{\text{t}}^{\text{w}}=(P-P_{0},\cdots,\mathbf{A})$ denote the set
of thermodynamic forces, where we have also included the affinity $\mathbf{A}$
for internal variables $\boldsymbol{\xi}$; see text. A microstate
$\mathfrak{m}_{k}$ of $\Sigma$ is specified by appending a subscript $k$ to
$\mathbf{F}_{\text{t}}^{\text{w}}$ so that
$\mathbf{F}_{\text{t,}k}^{\text{w}}=(P_{k}-P_{0},\cdots,\mathbf{A}_{k})$.

We have recently developed a version of nonequilibrium thermodynamics (NEQT)
that is expressed in terms of only SI quantities so we have a direct access to
$d_{\text{i}}Z$ and $d_{\text{i}}Z_{k}$. It has appeared in a series of papers
Gujrati-I ; Gujrati-II ; Gujrati-III ; Gujrati-Entropy1 ; Gujrati-
GeneralizedWork  covering separate aspects, and reviewed in Gujrati-Entropy2 ;
Guj-entropy-2018 . We have labeled it MNEQT to distinguish it from the M̊NEQT.
It is briefly introduced in Sec. III. The theory is applicable to systems that
are either isolated or in a medium within the same framework. The
corresponding microstate version of the MNEQT is called the $\mu$NEQT, with
$\mu$\- referring to the use of SI microquantities. It is capable of studying
expansion/contraction at the microstate level in interacting and isolated
system that has not been possible so far as we will discuss shortly. Another
reason to focus on the problem of expansion/contraction in the $\mu$NEQT is
due to its close connection with Maxwell’s demon and Landauer’s eraser. Both
versions of the theory also involve _internal variables_ Maugin ; Kestin ;
Prigogine ; deGroot ; Coleman  that are required, see Sec. II.1, to explain
nonequilibrium internal processes. Thus, they provide a very general framework
of NEQT to understand a majority of nonequilibrium processes as we will
explain.

We know that the classical thermodynamics is based on the concept of work and
heat so we need to identify them in a NEQ process to make any progress. The
central concept in the MNEQT is that of the generalized SI macrowork, see Fig.
1, $dW=P(t)dV(t)+\cdots+\mathbf{A(}t)\cdot d\mathbf{\xi(}t)$ and SI macroheat
$dQ=T(t)dS(t)$, see Eq. (14), that are different from the (exchange) MI
macrowork $d_{\text{e}}W=P_{0}dV(t)+\cdots$ and MI macroheat
$d_{\text{e}}Q=T_{0}d_{\text{e}}S$, respectively, see Eq. (15), by
irreversible contributions:

$\displaystyle d_{\text{i}}W$ $\displaystyle=dW-d_{\text{e}}W\geq 0,$ (1a)
$\displaystyle d_{\text{i}}Q$ $\displaystyle=dQ-d_{\text{e}}Q\geq 0.$ (1b) The
ability to directly deal with $d_{\text{i}}W$ and $d_{\text{i}}Q$ makes the
MNEQT not only perfectly suited to study isolated systems as we will do, but
also ensures that the generalized macrowork $dW$ is isentropic and that the
generalized macroheat $dQ$ satisfies the Clausius identity $dQ=TdS$, see Eq.
(14b), in all processes that we are interested in here; $S$ is always the
Gibbs statistical entropy Gibbs ; Landau
$S\doteq-{\textstyle\sum\nolimits_{k}}p_{k}\ln p_{k}.$ (2)

The $\mu$NEQT was first introduced a while back Gujrati-GeneralizedWork  and
applied to a few simple examples including a brief application to the Brownian
motion with a goal to compare its predictions with those from the work
fluctuation theorem (WFT) due to Jarzynski Jarzynski ; see Eq. (57) for its
precise formulation. The importance of _microforce imbalance_
$\mathbf{F}_{\text{t,}k}^{\text{w}}$, see Fig. 1 caption and later, between
externally applied macroforce and internally generated microforce was pointed
out there for the first time. It is _ubiquitous_ in nature Gujrati-
GeneralizedWork ; Gujrati-LangevinEq  as it is always present in all (EQ and
NEQ) macrostates. The macroforce imbalance
$\mathbf{F}_{\text{t}}=(F_{\text{t}}^{\text{h}},\mathbf{F}_{\text{t}}^{\text{w}})$
between the fields of the system and the medium, see Fig. 1 caption,
determines irreversible contribution
$\left(d_{\text{i}}Q,d_{\text{i}}W\right)$ and is well defined even for an
isolated system. It vanishes only in EQ. This makes $\mathbf{F}_{\text{t}}$
and $\mathbf{F}_{\text{t,}k}^{\text{w}}$ central in the $\mu$NEQT, which has
recently been applied Gujrati-LangevinEq  to study the Brownian motion in full
detail, where the relative motion of the Brownian particle with respect to the
medium generates $\mathbf{F}_{\text{t,}k}^{\text{w}}$. Thus, the $\mu$NEQT is
also capable of tackling small systems like Brownian particles under NEQ
conditions.

###  I.2 Why A New Approach?

Our goal here is to use the $\mu$NEQT and the MNEQT to study general
irreversible processes in interacting and isolated systems with emphasis on
those undergoing phase space volume change and the resulting irreversibility
at a deeper, microscopic level in terms of microstates
$\left\\{\mathfrak{m}_{k}\right\\}$. The forthcoming demonstration of the
success of our approach for free expansion, which has not been studied so far,
shows its usefulness as a general theory for both interacting and isolated
systems. As a general setup, we consider an interacting NEQ system $\Sigma$ in
a very large medium $\widetilde{\Sigma}$, see Fig. 1(b). The two form an
isolated system $\Sigma_{0}=\Sigma\cup\widetilde{\Sigma}$. Quantities
pertaining to $\Sigma_{0}$ carry a suffix $0$, those pertaining to
$\widetilde{\Sigma}$ carry a tilde, and those pertaining to $\Sigma$ carry no
suffix. For example, the macroworks are $dW_{0},d\widetilde{W}$, and $dW$,
respectively. The medium, being in EQ at all times, has no irreversibility in
it so that $d_{\text{i}}\widetilde{Q}=d_{\text{i}}\widetilde{W}=0$. Because of
its large size, its temperature, pressure, etc. are the same as for
$\Sigma_{0}$ so they are denoted by $T_{0},P_{0}$, etc. as seen in Fig. 1(b).

The _thermodynamic macroforce_ Prigogine ; Maugin
$\mathbf{F}_{\text{t}}=(T-T_{0},P-P_{0},\cdots\mathbf{)}$ must be nonzero in a
NEQ macrostate __ and vanish only in an EQ macrostate, i.e., when $\Sigma$ is
in EQ by itself or with $\widetilde{\Sigma}$, if the latter is present. But
the microforce
$\mathbf{F}_{\text{t,}k}^{\text{w}}=(P_{k}-P_{0},\cdots\mathbf{)}$ is
_ubiquitous_ as noted above in _all_ macrostates but independent of them,
i.e., of $\left\\{p_{k}\right\\}$. Unfortunately, as we will see, this is not
always enforced in many current microscopic approaches to NEQ thermodynamics.

Care must be exercised if the medium is not extremely large such as in Fig. 2.

Our methodology in the $\mu$NEQT will ensure that the microforces are _always_
accounted for. Given $\mathbf{F}_{\text{t,}k}$, the choice of
$\left\\{p_{k}\right\\}$ determines whether $\mathbf{F}_{\text{t}}=0$ or not
so the methodology will describe thermodynamics correctly. The temporal
development of $\mathfrak{M}$ in any $\mathcal{P}$ can also be studied by
following the deterministic Hamiltonian evolution along _Hamiltonian
trajectories_ $\left\\{\gamma_{k}\right\\}$ of microstates
$\left\\{\mathfrak{m}_{k}\right\\}$ described by its Hamiltonian
$\mathcal{H}$. The trajectories, therefore, describe deterministic evolution
during which $\left\\{p_{k}\right\\}$ does not change. As $dW$ is isentropic,
the evolution involves the performance of microworks $dW_{k}$ at fixed
$p_{k}$; see later. The stochasticity is due to microheat $dQ_{k}$ that
modifies $p_{k}$. Thus, $dW_{k}$ and $dQ_{k}$ control different aspects of the
evolution in $\mathfrak{M}$ so their combined effect completes the stochastic
evolution in the $\mu$NEQT.

The trajectories have been recently popularized by modern fluctuation theorems
(MFTs) Seifert ; Broeck ; see also Blau ; Ritort0 . Among these is the
Jarzynski’s WFT Jarzynski , which is the most celebrated one for the simple
reason that the other MFTs are related to it; see for example, Ref. Gujrati-
JensenInequality . Thus, we will comment mostly on the WFT, commonly known as
the JE, in the following, but the comments are equally valid for other MFTs.

There are four important and independent aspects that require careful
consideration here.

(i) Internal variables. The importance of internal variables Prigogine ;
Kestin ; deGroot  and their affinities to describe NEQ macrostates has been
well documented and is an integral part of the MNEQT and $\mu$NEQT used in
this study; see also Gujrati-GeneralizedWork ; Gujrati-LangevinEq . We will
give a simple argument for their relevance and the significance of affinities
in Sec. II.1.

(ii) Nonequilibrium Entropy. The MI $d_{\text{e}}Z$ alone provides no insight
into $d_{\text{i}}Z$ during relaxation unless SI $dZ$ is also identified. This
creates a problem as $\mathfrak{M}(t)$’s denote NEQ macrostates in general so
the SI $dS$ is not known if $S$ is defined only for EQ macrostates. Thus, we
need to identify $S$ for NEQ macrostates. We have shown that for a NEQ system
that is in _internal equilibrium_ , the statistical entropy given in Eq. (2)
is a state function in an enlarged state space involving internal variables
Gujrati-I ; Gujrati-II ; see Eq. (47). It is then used in the MNEQT to
determine the irreversible contributions directly. We see from Eq. (17b) that
$\mathbf{F}_{\text{t}}$ is an integral part of the MNEQT as promised. We then
use the MNEQT to derive the $\mu$NEQT.

(iii) Phase space volume change $\Delta\left|\Gamma\right|\neq 0$. As the
number of microstates depends on $\left|\Gamma\right|$, there cannot be a one-
to-one mapping between the sets of microstates in the two phase spaces in a
process of expansion/contraction. The same problem arises if
$d\left|\Gamma\right|/dt\neq 0$ even if at the end
$\Delta\left|\Gamma\right|=0$ such as in a cyclic process.

(iv) Dissipated work. We need to provide a physical explanation of the
macrowork that is being dissipated in the free expansion (see Corollary 2) and
the corresponding microworks.

As interacting systems are also included in our analysis, we make a few
comments in passing about the MFTs, with special attention to the WFT, that
are derived for interacting systems and where trajectories are also exploited.
The formulation invariably uses exchange quantities $\Delta_{\text{e}}W$ and
$\Delta_{\text{e}}Q$ directly but $\Delta_{\text{i}}W$ and
$\Delta_{\text{i}}Q$ are not part of the formulation. Our comments basically
summarize the results already available in the literature.

The MFTs are claimed to describe NEQ processes, because of which they have
attracted a lot of attention. However, despite being part of an highly active
field, we find that they do not provide a useful methodology for our NEQ
consideration here. There is no direct proof of for their NEQ nature that we
are aware. The only indirect proof for the WFT is through the application of
the Jensen’s inequality to demonstrate its compliance with the second law
since the inequality leads to

$\Delta_{\text{e}}W_{\text{J}}\leq-\Delta\overline{F},$ (3)

where $\Delta_{\text{e}}W_{\text{J}}$ (”J” for Jarzynski’s formulation) is a
particular ”average exchange” work (properly defined in Eq. (58) later) that
is obtained by using the initial probability over the entire process. It turns
out to be a _non-thermodynamic average_ Gujrati-JensenInequality , and
$\Delta\overline{F}$ is the difference of the equilibrium (and, therefore,
thermodynamic) Helmholtz free energies in the process; see also comment (f)
below in the subsection. The above inequality looks very similar to the
following _thermodynamic_ inequality involving thermodynamic average (exchange
macrowork) $R=-\Delta_{\text{e}}W$, where $R$ is the exchange work
$\Delta_{\text{e}}\widetilde{W}$ done by $\widetilde{\Sigma}$ on $\Sigma$,

$R\geq\Delta\overline{F},\text{ }$ (4)

a well-known consequence of the second law, but only if $T_{0}$ remains a
constant in the process Landau . In the latter case, the dissipated work
defined as

$\Delta_{\text{diss}}W\doteq R-\Delta\overline{F}=T_{0}\Delta_{\text{i}}S\geq
0.$ (5)

To provide an ”indirect proof” that the JE is a nonequilibrium result,
Jarzynski sets without any proof that

$\Delta_{\text{e}}W_{\text{J}}\overset{\text{conjecture}}{=}-R$ (6)

to turn Eq. (3) into Eq. (4). However, as shown recently Gujrati-
JensenInequality , Jensen’s inequality applied to the MFTs does not prove
compliance with the second law inequality so $\Delta_{\text{e}}W_{\text{J}}$
in Eq. (3) _cannot_ be equated with $\Delta_{\text{e}}W$ even when $T_{0}=$
$const$.

There are other concerns about the MFTs, which raise doubts about their
usefulness for our investigation. (a) They do not include any internal
variables, necessary for irreversibility; see Sec. II. (b) The external
macroforce (such as the pressure $P_{0}$) is always assumed to be equal to the
macroforce (such as the pressure $P$) in the system; hence, they implicitly
assume that $d_{\text{e}}W=dW$, which results in $d_{\text{i}}W\equiv 0$; see
Eq. (16a). This was first pointed out in Ref. Gujrati-GeneralizedWork . Thus,
they do not include any thermodynamic macroforce
$\mathbf{F}_{\text{t}}^{\text{w}}$ necessary for $d_{\text{i}}W$ and for
irreversibility Prigogine . (c) From $d_{\text{i}}W\equiv 0$ follows
$d_{\text{i}}Q\equiv 0$, see Eq. (11). If the temperature of the system is
always equal to $T_{0}$, i.e., $F_{\text{t}}^{\text{h}}=T-T_{0}=0$, then it
follows from Eqs. (17a, 17b) that $d_{\text{i}}S\equiv 0$. Cohen and Mauzerall
Cohen0 ; Cohen00  were the first to raise concern that $T$ may not even exist
for a NEQ process; see also Jarzynski-Cohen  for counter-arguments, some of
which we will discuss later. The concern was justified as correct later by
Muschik Muschik  so $T=T_{0}$ will make the MFTs unsuitable for a NEQ process.
(d) MFTs are based on a _fixed_ set of classical microstates
$\left\\{\mathfrak{m}_{k}\right\\}$ or trajectories
$\left\\{\gamma_{k}\right\\}$ as the use of Hamiltonian dynamics is
consistently prevalent. Thus, their applicability is limited to the situation
$d\left|\Gamma\right|=\Delta\left|\Gamma\right|=0$; see Sec. II.4. This was
first pointed out by Sung Sung . Unfortunately, this limitation is not well
recognized in the field. (e) The WFT should also apply to an isolated system
JarzynskiNote . Because $d_{\text{e}}W=0$ in this case, they do not. (f) The
free expansion in Fig. 2 refers to an isolated system so the WFT should be
applicable in this case KestinNote , but does not as first pointed out by Sung
Sung ; see also Gross ; Jarzynski-Gross  for the ensuing debate. We will come
back to this issue later when we discuss free expansion. (g) In addition, the
averaging in the WFT is not a thermodynamic averaging over the process as
first hinted by Cohen and Mauzerall Cohen0 , and established rigorously
recently by us Gujrati-JensenInequality , whereas we require a thermodynamic
averaging in our investigation.

Because of all these limitations, the MFTs are not of central interest to us
in this study except to draw attention to the differences with our approach.
Therefore, we will discuss and substantiate the above points again later in
Sec. V.4 within our theoretical framework; we focus on the WFT for simplicity.

There have been several numerical attempts to study restricted expansion in an
interacting system Lua0 ; Lua1 ; Baule ; Crooks-Jarzynski ; Davie ; Jarzynski1
but with a goal only to verify the WFT. Because of this, these numerical
studies are also not helpful to us for the reasons stated above.

In conclusion, it is not a surprise that we are left to exclusively use the
MNEQT and $\mu$NEQT in this study of interacting and isolated systems. We have
already applied the MNEQT to briefly study free expansion Gujrati-Heat-Work0 ;
Gujrati-Heat-Work . Here, we wish to go beyond the earlier study to
demonstrate how the $\mu$NEQT can be used to study expansion/contraction with
special attention to free expansion by including internal variables also. The
$\mu$NEQT has also been recently applied successfully to provide a
_thermodynamic alternative_ to study Brownian motion without using the
mechanical approach involving the Langevin equation Gujrati-LangevinEq . The
macroscopic friction emerges as a consequence of the relative motion of the
Brownian particle, an internal variable, with respect to the medium. We do not
need to postulate the Langevin noise term; it emerges as a consequence of
thermodynamic averaging mentioned above.

Our methodology and theory will be formulated for any arbitrary process in
both interacting and isolated systems. The theory is derived from the MNEQT so
it is always consistent with classical thermodynamics. The process will also
include expansion and contraction as special cases but the main focus will be
mostly on the spontaneous process of unrestricted, i.e., free expansion for
the reason explained above. Whenever we study free expansion, we will consider
the gas as a closed system $\Sigma$, which is in a medium $\widetilde{\Sigma}$
that happens to be the vacuum; see Fig. 2. Their combination forms the
isolated system shown by $\Sigma_{0}$ in Fig. 1. For the set up for free
expansion, we follow Kestin KestinNote  as we want to make the system
($\Sigma$) and the vacuum ($\widetilde{\Sigma}$) independent. As
$\widetilde{\Sigma}$ is devoid of matter and radiation, $\Sigma_{0}$ is
nothing but $\Sigma$. We can replace the partition by a piston exerting an
external pressure $P_{0}$ from $\widetilde{\Sigma}$ for a general
expansion/contraction process. For $P_{0}<P$, the gas will expand; for
$P_{0}>P$, the gas will contract. As the piston is an insulator,
$\widetilde{\Sigma}$ only acts as a working medium
$\widetilde{\Sigma}_{\text{w}}$. We need to bring in a thermal medium
$\widetilde{\Sigma}_{\text{h}}$ to bring about thermal equilibrium. Such an
expansion/contraction process is covered by the WFT Jarzynski . As
$P_{0}\rightarrow 0$, and $P>0$, we obtain the limiting case of free
expansion. Thus, free expansion is merely a limiting case of
expansion/contraction in our approach, and does not require a separate
approach. In all these cases, we require the gas particles initially to be
always confined in the left chamber.

Figure 2: Free expansion of a gas. The gas is confined to the left chamber,
which is separated by a hard partition (shown by a solid black vertical line)
from the vacuum in the right chamber as shown in (a). At time $t=0$, the
partition is removed abruptly as shown by the broken line in its original
place in (b). The gas expands in the empty space, devoid of matter and
radiation, on the right but the expansion is gradual as shown by the solid
front, which separates it from the vacuum on its right. We can also think of
the hard partition in (a) as a piston, which maintains the volume of the gas
on its left. The piston can be moved slowly or rapidly to the right within the
right chamber with a pressure $P_{0}<P$ to change this volume. The free
expansion occurs when the piston moves extremely (infinitely) fast by letting
$P_{0}\rightarrow 0$.

###  I.3 Layout

We briefly review some useful new concepts in the next section. We begin the
discussion with the need for internal variables in a NEQ process. The nature
of the parameters in the Hamiltonian of a NEQ system is discussed after that,
which is then followed by the definition of Hamiltonian trajectories. We close
this section by introducing the central concept of internal equilibrium. We
follow this section by a brief introduction to the MNEQT in Sec. III. Here, we
show the importance of internal variables for an isolated system to determine
the irreversible macrowork; see Theorem 1. This partially answers one of the
motivating questions. This is then followed by an introduction to the
$\mu$NEQT in Sec. IV, where we introduce the concepts of various microworks
and microheats. We introduce the moment generating function in Sec. V, which
gives all the moments including fluctuations of microworks from this single
function. We then turn to the free expansion of a classical gas and study it
by restricting to use only two internal variables in the MNEQT in Sec. VI. We
study the free expansion in the $\mu$NEQT in the next section. We first study
it in a quantum case in Sec. VII.1 and then in the classical case in Sec.
VII.2, where we introduce the important trick that allows us to consider free
expansion for any arbitrary expansion. The moment generating function is used
to directly demonstrate that the trick does not affect thermodynamics. The
trick can be extended to a cyclic process during which the phase space volume
changes nonmonotonically or to restricted expansion and contraction. A brief
discussion of our results is presented in the last section.

##  II Basic Concepts

###  II.1 Need for an Internal Variable

Consider a simple example of a NEQ system of $N$ particles, each of which can
be in two levels, forming an isolated system $\Sigma$ of volume $V$. Let
$\rho_{l}$ and $e_{l}(V),l=1,2$ denote the probabilities and energies of the
two levels of a particle in a NEQ macrostate so that $\rho_{1},\rho_{2}$ keep
changing. We have $e=\rho_{1}e_{1}+\rho_{2}e_{2}$ for the average energy per
particle, which is a constant, and $d\rho_{1}+d\rho_{2}=0$ as a consequence of
$\rho_{1}+\rho_{2}=1$. Using $de=0$, we get

$d\rho_{1}+d\rho_{2}e_{2}/e_{1}=0,$

which, for $e_{1}\neq e_{2}$, is inconsistent with the second equation (unless
$d\rho_{1}=0=d\rho_{2}$, which corresponds to EQ). Thus, $e_{l}(V)$ cannot be
treated as constant in determining $de$. In other words, there must be an
extra dependence in $e_{l}$ so that

$e_{1}d\rho_{1}+d\rho_{2}e_{2}+\rho_{1}de_{1}+\rho_{2}de_{2}=0,$

and the inconsistency is removed. This extra dependence must be due to
_independent_ internal variables that are not controlled from the outside
(isolated system) so they continue to relax in $\Sigma$ as it approaches EQ.
Let us imagine that there is a single internal variable $\xi$ so that we can
express $e_{l}$ as $e_{l}(V,\xi)$ in which $\xi$ continues to change as the
system comes to equilibrium. The above equation then relates $d\rho_{1}$ and
$d\xi$; they both vanish simultaneously as EQ is reached. We also see that
without any $\xi$, the isolated system cannot equilibrate.

The above discussion is easily extended to a $\Sigma$ with many energy levels
of a particle with the same conclusion that at least a single internal
variable is required to express $e_{l}=e_{l}(V,\xi)$ for each level $l$. We
can also visualize the above system in terms of microstates. A microstate
$\mathfrak{m}_{k}$ refers to a particular distribution of the $N$ particles in
any of different levels with energy
$E_{k}={\textstyle\sum\nolimits_{l}}N_{l}e_{l}$, where $N_{l}$ is the number
of particles in the $l$th level, and is obviously a function of $N,V,\xi$ so
we will express it as $E_{k}(N,V,\xi)$. This makes the average energy of the
system also a function of $N,V,\xi$, which we express as $E(N,V,\xi)$.

An EQ system is uniform. Thus, the presence of $\xi$ suggests some sort of
nonuniformity in the system. To appreciate its physics, we consider a slightly
different situation below as a possible example of nonuniformity.

We consider as a simple NEQ example a composite isolated system $\Sigma$
consisting of two identical subsystems $\Sigma_{1}$ and $\Sigma_{2}$ of
identical volumes and numbers of particles but at different temperatures
$T_{1}$ and $T_{2}$ at any time $t\leq\tau_{\text{eq}}$ before EQ is reached
at $t=\tau_{\text{eq}}$ so the subsystems have different time-dependent
energies $E_{1}$ and $E_{2}$, respectively. We assume a diathermal wall
separating $\Sigma_{1}$ and $\Sigma_{2}$. Treating each subsystem in EQ at
each $t$, we write their entropies as $S_{1\text{eq}}(E_{1},V/2,N/2)$ and
$S_{2\text{eq}}(E_{2},V/2,N/2)$. The entropy $S$ of $\Sigma$ is a function of
$E_{1},E_{2},V$, and $N$. Obviously, $\Sigma$ is in a NEQ macrostate at each
$t<\tau_{\text{eq}}$. From $E_{1}$ and $E_{2}$, we form two independent
combinations $E=E_{1}+E_{2}=$constant and $\xi=E_{1}-E_{2}$ so that we can
express the entropy as $S(E,V,N,\xi)$. Here, $\xi$ plays the role of an
internal variable, which continues to relax towards zero as $\Sigma$
approaches EQ. For given $E$ and $\xi$, $S(E,V,N,\xi)$ has the maximum
possible values since both $S_{1\text{eq}}$ and $S_{2\text{eq}}$ have their
maximum value. As we will see below, this is the idea behind the concept of
internal equilibrium in which $S(E,V,N,\xi)$ is a state function of state
variables and continues to increase as $\xi$ decreases and vanishes in EQ.

We assume $\Sigma$ to be in IEQ at each $t$ in this simple example. From
$1/T=\partial S/\partial E$ and $A/T=\partial S/\partial\xi$, $A$ being the
_activity_ associated with $\xi$, we find that

$T=2T_{1}T_{2}/(T_{1}+T_{2}),A=(T_{2}-T_{1})/(T_{1}+T_{2}).$

As EQ is attained, $T\rightarrow T_{\text{eq}}$, the EQ temperature of both
subsystems and $A\rightarrow A_{0}=0$ as expected. We see that in this simple
example $Ad\xi/T$ is the contribution due to irreversiblity in $dS$, which
also shows that $(-Ad\xi)$ is the contribution due to irreversiblity in $dE$.

In general, the activity $\mathbf{A}$ controls the behavior of
$\boldsymbol{\xi}$ in a NEQ macrostate and vanishes when EQ is reached. Here,
we will take a more general view of $\mathbf{A}$, and extend its definition to
$\mathbf{X}$ also Gujrati-I . Comparing $A$ with $F_{\text{t}}^{\text{h}}$, we
clearly see that $F_{\text{t}}^{\text{h}}$ also plays the role of an activity.
The same reasoning also shows that $\mathbf{F}_{\text{t}}^{\text{w}}$ plays
the role of an activity.

The example can be easily extended to the case of expansion and contraction by
replacing $E,E_{1}$, and $E_{2}$ by $N,N_{\text{L}}$, and $N_{\text{R}}$, see
Fig. 2, to describe the diffusion of particles. The role of $\beta$ and $E$,
etc. are played by $\beta\mu$ and $N$, etc.

###  II.2 Hamiltonian with Internal Variables

It is clear that in order to capture a NEQ process, internal variables are
_necessary_. Another way to appreciate this fact is to realize that for an
isolated system, all the observables in $\mathbf{X=(}E,V,N,\cdots\mathbf{)}$
are fixed so if the entropy is a function of $\mathbf{X}$ only, it cannot
change Gujrati-I ; Gujrati-II ; Guj-entropy-2018 ; Gujrati-Entropy1 ; Gujrati-
Entropy2 . Thus, we need additional independent variables to ensure the law of
increase of entropy for a NEQ system. An EQ macrostate is represented by a
point in the state space $\mathfrak{S}_{0}$ spanned by $\mathbf{X}$, but a NEQ
macrostate by a point in an _enlarged_ state space $\mathfrak{S}$ spanned by
$\mathbf{Z\doteq}(\mathbf{X,}\boldsymbol{\xi})$, where $\boldsymbol{\xi}$ is
the set formed by internal variables. Internal variables cannot be controlled
from the outside of the system; they are only controlled by the processes
within the system. On the other hand, the observables in $\mathbf{X}$ are
controlled from the outside. We will call $\mathbf{X}$ a set of observables
and $\mathbf{Z}$ a set of state variables. In EQ, internal variables are no
longer independent of the observables. Consequently, their affinities (see
later) vanish in EQ. It is common to define the internal variables so their EQ
values vanish.

As we will be dealing with the Hamiltonian of the system, it is useful to
introduce the sets $\mathbf{X}^{E}\doteq\mathbf{X\backslash}E=(V,N,\cdots)$,
and
$\mathbf{Z}^{E}\doteq\mathbf{Z\backslash}E=(V,N,\cdots,\boldsymbol{\xi})=(\mathbf{X}^{E}\mathbf{,}\boldsymbol{\xi})$.
Then, $E$ and $E_{k}$ become a function of $\mathbf{Z}^{E}$ as we saw in Sec.
II.1. Here, $\mathbf{Z}^{E}$ appears as a _parameter_ in the Hamiltonian,
which we will write as $\mathcal{H}(\left.\mathbf{z}\right|\mathbf{Z}^{E})$,
where $\mathbf{z}$ is a point (collection of coordinates and momenta of the
particles) in the phase space $\Gamma(\mathbf{Z}^{E})$ specified by
$\mathbf{Z}^{E}$. As an example, $N,V$, and $\xi$ are the parameters in the
previous section. When the system moves about in the phase space
$\Gamma(\mathbf{Z}^{E})$, $\mathbf{z}$ changes but $\mathbf{Z}^{E}$ as a
parameter remains fixed in a state subspace
$\mathfrak{S}^{E}\doteq\mathfrak{S}\mathbf{\backslash}E$.

###  II.3 Hamiltonian Trajectories

Traditional formulation of statistical thermodynamics Landau ; Gibbs ;
Gujrati-Symmetry  takes a mechanical approach in which $\mathfrak{m}_{k}$
follows its classical or quantum mechanical evolution dictated by its SI
Hamiltonian $\mathcal{H}(\left.\mathbf{z}\right|\mathbf{Z}^{E})$. The quantum
microstates are specified by a set of good quantum numbers, which we have
denoted by $k$ above as a single quantum number for simplicity; we take
$k\in\mathbb{N},\mathbb{N}$ denoting the set of natural numbers. We will see
below that $k$ does not change as $\mathbf{Z}^{E}$ changes. In the classical
case, we can use a small cell $\delta\mathbf{z}_{k}$ of size
$(2\pi\hbar)^{3N}$ around $\mathbf{z}_{k}=\mathbf{z}$ as the microstate
$\mathfrak{m}_{k}$. In the rest of the work, we will keep $N$ fixed to fix the
size of the system. Therefore, from now on, $\mathbf{X}$ and $\mathbf{Z}$ will
not contain it. The Hamiltonian gives rise to a purely mechanical evolution of
individual $\mathfrak{m}_{k}$’s, which we will call the _Hamiltonian
evolution_ , and suffices to provide their mechanical description. The change
in $\mathcal{H}(\left.\mathbf{z}\right|\mathbf{Z}^{E})$ in a process is

$d\mathcal{H}=\frac{\partial\mathcal{H}}{\partial\mathbf{z}}\cdot
d\mathbf{z}+\frac{\partial\mathcal{H}}{\partial\mathbf{Z}^{E}}\cdot
d\mathbf{Z}^{E}.$ (7a) The first term on the right vanishes identically due to
Hamilton’s equations of motion for any $\mathfrak{m}_{k}$. Thus, for fixed
$\mathbf{Z}^{E}$, the energy
$E_{k}=\mathcal{H}_{k}\doteq\mathcal{H}(\left.\mathbf{z}_{k}\right|\mathbf{Z}^{E})$
remains constant as $\mathfrak{m}_{k}$ moves about in
$\Gamma(\mathbf{Z}^{E})$. Only the variation $d\mathbf{Z}^{E}$ in
$\mathfrak{S}$ generates any change in $E_{k}$. Consequently, we do not worry
about how $\mathbf{z}_{k}$ changes in
$\mathcal{H}(\left.\mathbf{z}\right|\mathbf{Z}^{E})$ in the phase space, and
focus, instead, on the state space $\mathfrak{S}$, in which can write
$dE_{k}=\frac{\partial E_{k}}{\partial\mathbf{Z}^{E}}\cdot
d\mathbf{Z}^{E}=-dW_{k},$ (7b) where $dW_{k}$ denotes the _generalized
microwork_ produced by the _generalized microforce_ $\mathbf{F}_{k}^{E}$:
$dW_{k}=\mathbf{F}_{k}^{E}\cdot d\mathbf{Z}^{E},\
\mathbf{F}_{k}^{E}\doteq-\partial E_{k}/\partial\mathbf{Z}^{E}.$ (7c) We can
now identify $\mathbf{Z}^{E}$ as the _work parameter_ , whose variation
$d\mathbf{Z}(t)\doteq(dE(t),d\mathbf{Z}^{E}(t))$ in $\mathfrak{S}$ defines not
only the microworks $\left\\{dW_{k}\right\\}$, but also a thermodynamic
process $\mathcal{P}$. The trajectory $\gamma_{k}$ in $\mathfrak{S}$ followed
by $\mathfrak{m}_{k}$ as a function of time will be called the _Hamiltonian
trajectory_ during which $\mathbf{Z}^{E}$ varies from its initial (in) value
$\mathbf{Z}_{\text{in}}^{E}$ to its final (fin) value
$\mathbf{Z}_{\text{fin}}^{E}$ during $\mathcal{P}$. The variation produces the
generalized microwork $dW_{k}$; $p_{k}$ plays no role so $dW_{k}$ is purely
mechanical, which simplifies its determination in our theory. The microwork
$dW_{k}$ also does not change the index $k$ of $\mathfrak{m}_{k}$ as said
above.

Being purely mechanical in nature, a trajectory is completely _deterministic_
and cannot describe the evolution of a macrostate $\mathfrak{M}$ during
$\mathcal{P}$ unless supplemented by thermodynamic stochasticity, which
requires $p_{k}(\mathfrak{M})$ as discussed above Landau , and is related to
$dQ_{k}$ as shown later; see Eq. (38). Thermodynamics emerges when quantities
pertaining to the trajectories are averaged over the trajectory ensemble
$\left\\{\gamma_{k}\right\\}$ with appropriate probabilities that will usually
change during the process. In this sense, our approach is different from
approaches using stochastic trajectories Seifert ; Broeck , where $dE_{k}$ is
identified with the exchange microwork $dR_{k}\doteq-d_{\text{e}}W_{k}$; see
Remark 6.

The development of the $\mu$NEQT requires pursuing individual trajectory
$\gamma_{k}$ of $\mathfrak{m}_{k}(\mathbf{Z}^{E})$. In the following,
$\mathfrak{m}_{k}$ will usually stand for classical microstates unless
specified otherwise, and follows its deterministic trajectory $\gamma_{k}$ as
$\mathfrak{m}_{k\text{,in}}\doteq\mathfrak{m}_{k}(\mathbf{Z}_{\text{in}}^{E})$
evolves into
$\mathfrak{m}_{k\text{,fin}}\doteq\mathfrak{m}_{k}(\mathbf{Z}_{\text{fin}}^{E})$.
By reversing the change in $\mathbf{Z}^{E}$ from $\mathbf{Z}_{\text{fin}}^{E}$
to $\mathbf{Z}_{\text{in}}^{E}$,
$\mathfrak{m}_{k}(\mathbf{Z}_{\text{fin}}^{E})$ comes back to
$\mathfrak{m}_{k}(\mathbf{Z}_{\text{in}}^{E})$. Thus, $\gamma_{k}$ defines an
identity map $\mathcal{T}:$

$\mathcal{T}:\mathfrak{m}_{k}(\mathbf{Z}_{\text{in}}^{E})\rightleftarrows\mathfrak{m}_{k}(\mathbf{Z}_{\text{fin}}^{E})$
(8)

without altering the index $k$ so it is a one-to-one ($1$-to-$1$) or identity
mapping of microstates. The two arrows mean that the mapping can be _inverted_
without altering the index $k$ so it is a one-to-one ($1$-to-$1$) or identity
mapping of microstates.

###  II.4 Phase Space Volume Change

However, Hamiltonian trajectories for classical microstates are not suitable
for processes that involve expansion and contraction in the volume $V$ and/or
other parameters in $\mathbf{Z}^{E}$ of the system with a corresponding change
in the phase space volume $\left|\Gamma\right|$. In the following, we will
think of $V$ as the varying work parameter for simplicity. Then, during
expansion, the initial volume
$\left|\Gamma\right|_{\text{in}}\doteq\left|\Gamma(\mathbf{Z}_{\text{in}}^{E})\right|$
is smaller than the final volume
$\left|\Gamma^{\prime}\right|_{\text{fin}}\doteq\left|\Gamma(\mathbf{Z}_{\text{fin}}^{E})\right|$
as shown in Fig. 3. This means that there are microstates such as
$\delta\mathbf{\zeta}^{\prime}$ ($\neq\delta\mathbf{z}^{\prime}$) in
$\Gamma_{\text{fin}}^{\prime}\doteq\Gamma(\mathbf{Z}_{\text{fin}}^{E})$ that
cannot be reached from any of the microstate $\delta\mathbf{z}$ in
$\Gamma_{\text{in}}\doteq\Gamma(\mathbf{Z}_{\text{in}}^{E})$ along Hamiltonian
trajectories; the latter take $\delta\mathbf{z}$ into
$\delta\mathbf{z}^{\prime}$ inside the broken horizontal ellipse. The converse
is true for contraction.

Note also that we are not interested in the cardinality of the initial and
final sets of microstates $\left\\{\mathfrak{m}_{\text{in}}\right\\}$ and
$\left\\{\mathfrak{m}_{\text{fin}}\right\\}$,respectively. We are interested
in how they map under Hamiltonian evolution; see Sec. VIII for more
clarification.

It is clear that in the classical case, we require a new approach to overcome
the loss of the $1$-to-$1$ mapping if we confine ourselves to only Hamiltonian
trajectories. To the best of our knowledge, the problem of how to overcome
this hurdle of phase space volume change using Hamiltonian trajectories has
not been solved.

Figure 3: The evolution of a microstate
$\delta\mathbf{z}\in\mathbf{\Gamma}_{\text{in}}$ into
$\delta\mathbf{z}^{\prime}\in\mathbf{\Gamma}_{\text{fin}}^{\prime}$ following
the variation in $\mathbf{Z}^{E}$ (green arrows). The initial and final phase
spaces are $\mathbf{\Gamma}_{\text{in}}$ and
$\mathbf{\Gamma}_{\text{fin}}^{\prime}$ shown by the interiors of the red
solid ellipses. By changing $\mathbf{Z}^{E}$ in the reverse order maps
$\delta\mathbf{z}^{\prime}$ into $\delta\mathbf{z}$ as implied by the reverse
green arrows. The microstates $\delta\mathbf{\zeta}^{\prime}$ and
$\delta\mathbf{\zeta}$, and other quantities are explained in the text.

We overcome the hurdle by introducing a novel but simple trick. The clue for
the new approach comes from considering trajectories in quantum mechanics. As
quantum microstates form a denumerable set with index $k\in\mathbb{N}$ as
$\mathbf{Z}^{E}$ changes, there is a $1$-to-$1$ mapping as in Eq. (8) between
$\mathfrak{m}_{k\text{,in}}$ and $\mathfrak{m}_{k\text{,fin}}$ during
expansion and contraction, which helps us remedy the lack of one-to-one
correspondence due to volume change in the classical case. The trick is to
enlarge the smaller phase space to become equal to the larger phase space by
adding missing microstates that appear with nonzero but vanishingly small
probabilities. As the work along deterministic trajectories in Eq. (7b) is
oblivious to their probabilities (even though they continue to change in
thermodynamics), we can add trajectories initiating at the missing microstates
to obtain a _enlarged_ $1$-to-$1$_trajectory ensemble_
$\left\\{\gamma\right\\}$. At the end of the computation of ensemble averages,
a formal limit of vanishing probabilities of missing initial microstates is
taken.

###  II.5 Internal Equilibrium (IEQ)

The central concept of the NEQT exploited here is that of the _internal
equilibrium_(IEQ) according to which the entropy $S$ of a NEQ macrostate is a
_state function_ of the state variables in the enlarged state space
$\mathfrak{S}$ Gujrati-I ; Gujrati-II ; Gujrati-III  relative to the EQ state
space $\mathfrak{S}_{0}$ due to independent internal variables Coleman ;
deGroot ; Prigogine ; Maugin ; Gujrati-I ; Gujrati-II  that are required to
describe a NEQ macrostate as explained above. In EQ, the internal variables
are no longer independent of the observables forming the space
$\mathfrak{S}_{0}$. As a consequence, their affinities vanish in EQ. In
general, the temperature $T$ of the system in IEQ is identified in the
standard manner by the relation

$1/T=\partial S/\partial E$ (9)

using the fact that $S$ is state function in $\mathfrak{S}$.

An important property of IEQ macrostates is the following: It is possible in
an IEQ macrostate to have different degrees of freedom or different parts of a
system to have different temperatures than $T$. For example, in a glass, it is
well known that the vibrational degrees of freedom have a different
temperature than the configurational degrees of freedom Debenedetti ; Gujrati-
Hierarchy . In the viscous drag problem, the CM-motion of the Brownian
particle can have a different temperature than $T$ of the rest of the
particles in the fluid Gujrati-LangevinEq . This observation is easily
verified in MNEQT based on the concept of IEQ as done elsewhere (Gujrati-
Hierarchy, , see Sec. 8.1 and Eq. (58)). By taking a larger and larger set of
internal variables, we can treat almost all NEQ macrostates as if in IEQ.
Thus, the MNEQT is an extremely useful thermodynamics for NEQ systems.

##  III The MNEQT

An EQ macrostate is described by $\mathbf{X=(}E,V,\cdots\mathbf{)}$, and its
entropy is a state function $S(\mathbf{X})$. Away from EQ, $S(\mathbf{Z})$
becomes a state function for NEQ macrostates in IEQ. In the following, we will
focus on $V$ and $\xi$ as members of
$\mathbf{Z}^{E}\doteq\mathbf{Z\backslash}E$ for simplicity but the discussion
is general and applies to any $\mathbf{Z}^{E}$. Indeed, we use two internal
variables when we study free expansion. The microstate $\mathfrak{m}$ follows
its evolution dictated by its (SI) Hamiltonian
$\mathcal{H}(\left.\mathbf{z}\right|V,\boldsymbol{\xi})$; the interaction with
$\widetilde{\Sigma}$ is usually treated as a very weak stochastic
perturbation, which immediately suggests adopting a SI description.

###  III.1 Macrowork and Macroheat

There are two kinds of macrowork,the SI and MI macroworks $dW$ and
$d_{\text{e}}W$, respectively, in NEQT; see Eq. (1a). The irreversible
macrowork $d_{\text{i}}W\geq 0$ vanishes only for a reversible process. We
similarly have two kinds of macroheat, the SI macroheat $dQ$ and the MI
exchange macroheat $d_{\text{e}}Q$; see Eq. (1b). The irreversible heat
$d_{\text{i}}Q\geq 0$ vanishes only for a reversible process. The first law
can be equivalently expressed in terms of SI and MI quantities, respectively:

$\displaystyle dE$ $\displaystyle=dQ-dW,$ (10a) $\displaystyle dE$
$\displaystyle=d_{\text{e}}Q-d_{\text{e}}W;$ (10b) the first equation follows
from Eq. (31a) and the discussion following it. From the above two equations
follows the important identity  $d_{\text{i}}Q\equiv d_{\text{i}}W,$ (11)

which establishes that internal processes ensure that irreversible macroheat
and macrowork within $\Sigma$ are identically equal in magnitude. This
equality is very general and will be used extensively in this study. It is a
consequence of a very general result of NEQT that no irreversible process can
generate any internal change in the (average) energy of the system, i.e.,

$d_{\text{i}}E\equiv 0,$ (12)

so that $d_{\text{e}}E=dE$. From now on, we will refer to $dW$ and $dQ$
($d_{\text{e}}W$ and $d_{\text{e}}Q$) simply as macrowork and macroheat
(exchange macrowork and exchange macroheat), respectively, so no confusion can
arise.

The discussion above is valid for any arbitrary process, but, from now on, we
restrict to the case when $S=S(\mathbf{Z})$ is a state function in
$\mathfrak{S}$ for simplicity, i.e., for the macrostate to be an IEQ
macrostate Gujrati-I ; Gujrati-II  to define $T$. The discussion to an
arbitrary process, which can be done, will be avoided here.

For a process requiring pressure-volume macrowork only, we have $dW=PdV$ done
_by_ $\Sigma$ (SI) or __ $d\widetilde{W}=-P_{0}dV$ done _by_
$\widetilde{\Sigma}$ (MI) in terms of the instantaneous pressure $P=-\partial
E/\partial V$ of $\Sigma$ or $P_{0}~{}$of $\widetilde{\Sigma}$, and their
volume change $dV$ or $-dV$, respectively. The exchange macrowork is
$d_{\text{e}}W=-d\widetilde{W}=P_{0}dV$. For irreversibility, $P\neq P_{0}$,
with $P-P_{0}$ playing the role of an activity Gujrati-I . The irreversible or
dissipated work is $d_{\text{i}}W=(P-P_{0})dV$ Kestin ; Woods ; Prigogine .

Comparing Eq. (10a) for an IEQ macrostate with

$dS=(\partial S/\partial\mathbf{X})\cdot d\mathbf{X+}(\partial
S/\partial\boldsymbol{\xi})\cdot d\boldsymbol{\xi},$ (13)

allows us to identify the macroheat and macrowork as

$\displaystyle dW$ $\displaystyle=T(\partial S/\partial\mathbf{Z}^{E})\cdot
d\mathbf{Z}^{E}=PdV+\cdots+\mathbf{A}\cdot d\boldsymbol{\xi},$ (14a)
$\displaystyle dQ$ $\displaystyle=TdS\boldsymbol{;}$ (14b) where $T\doteq
1/(\partial S/\partial E)=1/\beta,$ $\beta\mathbf{A}\doteq(\partial
S/\partial\boldsymbol{\xi})$ identifies the affinity $\mathbf{A}$, and
$\cdots$ refers to other elements in $\mathbf{X}$. Recalling that for
$\widetilde{\Sigma}$, $T=T_{0},P=P_{0},\cdots,\mathbf{A}_{0}=0$, we have in
general,  $\displaystyle d_{\text{e}}W$
$\displaystyle=-d\widetilde{W}=P_{0}dV+\cdots,$ (15a) $\displaystyle
d_{\text{e}}Q$ $\displaystyle=-d\widetilde{Q}=T_{0}d_{\text{e}}S.$ (15b) We
can identify the irreversible macrowork due to
$\mathbf{F}_{\text{t}}^{\text{w}}$:
$d_{\text{i}}W=(P-P_{0})dV+\cdots+\mathbf{A}\cdot
d\boldsymbol{\xi=}\mathbf{F}_{\text{t}}^{\text{w}}\cdot d\mathbf{Z}^{E}\geq
0,$ (16a) where
$\mathbf{F}_{\text{t}}^{\text{w}}=\left\\{P-P_{0},\cdots,\mathbf{A}=\mathbf{A-A}_{0}\right\\}$
(16b) is the _thermodynamic force_ , which also include the affinity
$\mathbf{A}$, driving the system towards EQ when $d_{\text{i}}W\rightarrow 0$
as $\mathbf{F}_{\text{t}}^{\text{w}}\rightarrow 0$. For irreversibility,
$d_{\text{i}}W>0$, which requires $\mathbf{F}_{\text{t}}^{\text{w}}$ to be
non-zero as asserted earlier in Sec. I.1. Each component
$(P-P_{0})dV,\cdots,A_{1}d\xi_{1},A_{2}d\xi_{2},\cdots$ of
$\mathbf{F}_{\text{t}}^{\text{w}}\cdot d\mathbf{Z}^{E}$ must be positive
separately for irreversibility.

Using $d_{\text{e}}Q=T_{0}d_{\text{e}}S$ in Eq. (14b), we find

$d_{\text{i}}Q=\left\\{\begin{array}[c]{c}(T-T_{0})d_{\text{e}}S+Td_{\text{i}}S\\\
(T-T_{0})dS+T_{0}d_{\text{i}}S\end{array}\right.\geq 0.$ (17a) Using
$d_{\text{i}}Q=d_{\text{i}}W$ and Eq. (16a), we also obtain
$d_{\text{i}}S=\left\\{\begin{array}[c]{c}\left\\{(T_{0}-T)d_{\text{e}}S+\mathbf{F}_{\text{t}}^{\text{w}}\cdot
d\mathbf{Z}^{E}\right\\}/T\\\
\left\\{(T_{0}-T)dS+\mathbf{F}_{\text{t}}^{\text{w}}\cdot
d\mathbf{Z}^{E}\right\\}/T_{0}\end{array}\right.\geq 0.$ (17b) We see that
$F_{\text{t}}^{\text{h}}=T_{0}-T$ can also be thought of as a ”thermodynamic
force” due to thermal imbalance driving the system towards EQ via heat
transfer; it ensures that $F_{\text{t}}^{\text{h}}d_{\text{e}}S\geq 0$, in
accordance with the second law. Thus, both contributions to $d_{\text{i}}S$
are always nonnegative as expected. In the absence of any heat exchange
($d_{\text{e}}S=0$) or for an isothermal system ($T=T_{0}$), we have
$d_{\text{i}}Q=Td_{\text{i}}S=d_{\text{i}}W,$ (18)

where $d_{\text{i}}W$ is given by Eq. (16a).

We can use

$\mathbf{F}_{\text{t}}\doteq(F_{\text{t}}^{\text{h}},\mathbf{F}_{\text{t}}^{\text{w}})$
(19)

as the _generalized thermodynamic force_ , which includes the thermal
imbalance $F_{\text{t}}^{\text{h}}$ and the work imbalance
$\mathbf{F}_{\text{t}}^{\text{w}}$. It should be obvious that
$F_{\text{t}}^{\text{h}}$ is meaningless for an isolated or an isothermal
system, while $\mathbf{F}_{\text{t}}^{\text{w}}$ is meaningful for all NEQ
systems, interacting or not.

###  III.2 Internal Variables and the Isolated System

The above formulation of MNEQT is perfectly suited for considering an isolated
system $\Sigma$ ($d_{\text{e}}W=d_{\text{e}}Q\equiv 0$) so that Eqs. (11-12)
becomes the most important thermodynamic equality. For an isolated system,
$d\mathbf{X}=0$ so that $d_{\text{i}}W=\mathbf{A}\cdot d\boldsymbol{\xi}$ as
seen from Eq. (16a).

###### Theorem 1

The irreversible entropy generated within an isolated system is still related
to the dissipated work performed by the internal variables.

Proof. As $E$ remains fixed for an isolated system ($dQ=Td_{\text{i}}S$), we
have from Eq. (10a)

$Td_{\text{i}}S=d_{\text{i}}W=\mathbf{A}\cdot d\boldsymbol{\xi}\geq 0$ (20)

in accordance with the second law.

Note that the above equation, though it is identical to Eq. (18) in form, is
very different in that $d_{\text{i}}W$ here is simply $\mathbf{A}\cdot
d\boldsymbol{\xi}$ and not the full expression in Eq. (16a). Same conclusion
is also obtained when we apply Eq. (17b) to an isolated system.

The above theorem thus clarifies the unsettling fact about the significance of
dissipated macrowork that motivated this study; see also Eq. (26). The
dissipated macrowork $d_{\text{i}}W$ in an isolated system is performed by the
internal variable $\boldsymbol{\xi}$, and can be identified with
$d_{\text{i}}S$ as noted in Sec. I.1.

###### Corollary 2

Neither the entropy can increase nor will there be any dissipated work unless
some internal variables are present in an isolated system. If no internal
variables are used to describe an isolated system, then thermodynamics
requires it to be in EQ.

Proof. The proof follows trivially from Eq. (20).

###  III.3 Cumulative Macroquantities in a Thermodynamic Process

Let us consider a thermodynamic process $\mathcal{P}$ between two macrostates
$\mathfrak{M}_{\text{in}}\doteq\mathfrak{M}(T_{\text{in}},\mathbf{Z}_{\text{in}}^{E})$
and
$\mathfrak{M}_{\text{fin}}\doteq\mathfrak{M}(T_{\text{fin}},\mathbf{Z}_{\text{fin}}^{E})$
at temperatures $T_{\text{in}}$ and $T_{\text{fin}}$, respectively. The system
may be isolated and in IEQ so its temperature is well defined. It may be very
different from the temperature $T_{0}$ of the medium, if $\Sigma$ is not
isolated. In each case, the cumulative macroquantities of $\Sigma$ are
obtained by simple integration along the process:

$\displaystyle\Delta_{\alpha}E$
$\displaystyle={\textstyle\int\nolimits_{\mathcal{P}}}d_{\alpha}E,\Delta_{\alpha}S={\textstyle\int\nolimits_{\mathcal{P}}}d_{\alpha}S,$
(21a) $\displaystyle\Delta_{\alpha}W$
$\displaystyle={\textstyle\int\nolimits_{\mathcal{P}}}d_{\alpha}W,\Delta_{\alpha}Q={\textstyle\int\nolimits_{\mathcal{P}}}d_{\alpha}Q.$
(21b) Similar definitions also apply to $\widetilde{\Sigma}$ and $\Sigma_{0}$.
Above we have used the compact notation $d_{\alpha}=d,d_{\text{e}}$, and
$d_{\text{i}}$ to indicate various infinitesimal forms, which we can treat as
linear operators. They will be useful in the rest of the work.

We now consider an interacting system and determine $\Delta W$ between two EQ
macrostates
$\mathfrak{M}_{\text{in,eq}}\doteq\mathfrak{M}(T_{0},\mathbf{Z}_{\text{in}}^{E})$
and
$\mathfrak{M}_{\text{fin,eq}}\doteq\mathfrak{M}(T_{0},\mathbf{Z}_{\text{fin}}^{E})$
at the same temperature. We denote the corresponding process by
$\mathcal{P}_{0}$, which may possibly be irreversible. We recall that the
Helmholtz free energy

$\overline{F}=E-T_{0}S$ (22a) (conventionally written as $F$ but we will use
that for a SI free energy here) is also the Helmholtz free energy of a NEQ
system Gujrati-I  in a canonical ensemble in a medium at temperature $T_{0}$;
the temperature $T$ of the system does not appear in $\overline{F}$
explicitly. It is this free energy that follows the second law Gujrati-I  and
not $F$, which is the SI free energy $F=E-TS+\mathbf{A}\cdot\boldsymbol{\xi.}$
(22b) It depends only on the system and is very different from $\overline{F}$.
It will be useful later. In terms of the difference $\Delta\overline{F}$ for
the two macrostates, we have $\Delta W=\Delta Q-\Delta E=\Delta
Q-\Delta\overline{F}-\Delta(T_{0}S)$. Thus,

$\displaystyle\Delta W$
$\displaystyle={\textstyle\int\nolimits_{\mathcal{P}}}[(T-T_{0})dS-
SdT_{0}]-\Delta\overline{F},$ (23a) $\displaystyle\Delta_{\text{e}}W$
$\displaystyle=-{\textstyle\int\nolimits_{\mathcal{P}}}[T_{0}d_{\text{i}}S+SdT_{0}]-\Delta\overline{F},$
(23b) $\displaystyle\Delta_{\text{i}}W$
$\displaystyle={\textstyle\int\nolimits_{\mathcal{P}}}[TdS-
T_{0}d_{\text{e}}S]$ (23c) The corresponding infinitesimal form is
$\displaystyle dW$ $\displaystyle=(T-T_{0})dS-SdT_{0}-d\overline{F},$ (24a)
$\displaystyle d_{\text{e}}W$
$\displaystyle=-[T_{0}d_{\text{i}}S+SdT_{0}]-d\overline{F},$ (24b)
$\displaystyle d_{\text{i}}W$ $\displaystyle=TdS-T_{0}d_{\text{e}}S.$ (24c) If
the medium is maintained at a fixed temperature during $\mathcal{P}$, we must
remove the $dT_{0}$ term above.

We see that if and only if $T=T_{0}=const$ over the entire process
$\mathcal{P}$ so that it is _isothermal_ , we have

$\Delta
W_{\text{isoth}}=-\Delta\overline{F},\Delta_{\text{e}}W_{\text{isoth}}=-T_{0}\Delta_{\text{i}}S-\Delta\overline{F},$
(25a) in terms of the Helmholtz free energy difference so that
$\Delta_{\text{i}}W_{\text{isoth}}=T_{0}\Delta_{\text{i}}S.$ (25b) This is a
very strong requirement when $\widetilde{\Sigma}$ remains in continuous
contact with $\Sigma$, since it requires complete thermal equilibrium at all
times. In this case, $d_{\text{i}}Q=T_{0}d_{\text{i}}S$ and
$\Delta_{\text{i}}Q=T_{0}\Delta_{\text{i}}S$; see also Eq. (18).

We now consider an isolated system for which $\Delta E=0$ so that

$\Delta_{\text{i}}W={\textstyle\int\nolimits_{\mathcal{P}}}Td_{\text{i}}S=T_{0}\Delta_{\text{i}}S+{\textstyle\int\nolimits_{\mathcal{P}}}[(T-T_{0})d_{\text{i}}S\geq
0,$ (26)

which is in accordance with Theorem 1 and finally explains the physical
meaning of the irreversible macrowork, which was one of the questions that had
prompted this investigation. It should be clear that the derivation is not
restricted to a process between two EQ macrostates.

###  III.4 A Simple Study Case

As remarked earlier, we use a single internal variable $\xi$ in addition to
$V$ for $\mathbf{Z}^{E}$ for simplicity so that we have

$dW=PdV+Ad\xi,d_{\text{e}}W=P_{0}dV.$ (27a) The dissipated work is
$d_{\text{i}}W=(P-P_{0})dV+Ad\xi\geq 0;$ (27b) in the absence of $\xi$. We
also have

$Td_{\text{i}}S=(T-T_{0})d_{\text{e}}S+(P-P_{0})dV+Ad\xi\geq 0,$ (28)

where we have used $d_{\text{i}}Q=d_{\text{i}}W$. If no exchange macrowork is
done, $dV=0$ and $Td_{\text{i}}S=(T-T_{0})d_{\text{e}}S+Ad\xi\geq 0$. In the
absence of any exchange macroheat, we have
$Td_{\text{i}}S=(P-P_{0})dV+Ad\xi\geq 0$. In an isolated system, we have

$Td_{\text{i}}S=Ad\xi\geq 0,$ (29)

which is a special case of the general result in Eq. (20).

##  IV The $\mu$NEQT

We will closely follow Refs. Gujrati-GeneralizedWork ; Gujrati-LangevinEq  to
provide a brief pedagogical review of the $\mu$NEQT for the sake of continuity
and demonstrate its successful application to the free expansion, which has
not been attacked by any other approach so far. The main idea is to cast any
macroquantity in the MNEQT as a thermodynamic average, see Eq. (30a), over
microstates. Then, we can identify corresponding microquantities. Some care
must be exercised to ensure their uniqueness as we will see. These microstate
microquantities can be used to identify the contribution along a trajectory by
simple integration.

###  IV.1 General Setup

The theory was first presented in a very condensed form in Ref. Gujrati-
GeneralizedWork . It was successfully applied Gujrati-LangevinEq  to provide
an alternative, but a much simpler, approach (using deterministic microforces
$\mathbf{F}_{k}^{E}$) to study Brownian motion without the use of Langevin’s
stochastic noise term so it does not require the use of the stochastic theory.
The microforce responsible for the Brownian motion is associated with the
relative motion of the center of mass of the Brownian particle with respect to
the medium. A good description of the salient features of the $\mu$NEQT is
available there.

It follows from $\mathcal{H}(\left.\mathbf{z}\right|\mathbf{Z}^{E})$ that
$E_{k}$ also depend on $\mathbf{Z}^{E}$. A macroquantity $O$ (except the
temperature) in the MNEQT appear as a thermodynamic average $\left\langle
O\right\rangle$ over $\left\\{\mathfrak{m}_{k}\right\\}$:

$O=\left\langle O\right\rangle\doteq{\textstyle\sum\nolimits_{k}}p_{k}O_{k},$
(30a) where $O_{k}$ is the value of $O$ associated with $\mathfrak{m}_{k}$.
Thus,
$E={\textstyle\sum\nolimits_{k}}p_{k}E_{k},P={\textstyle\sum\nolimits_{k}}p_{k}P_{k},S={\textstyle\sum\nolimits_{k}}p_{k}S_{k},$
(30b) etc. Here, $E_{k},P_{k}\doteq-\partial E_{k}/\partial V,S_{k}\doteq-\ln
p_{k}$, etc. are the values that are only determined by $\mathfrak{m}_{k}$.
However, $p_{k}$ or $S_{k}$, although associated with $\mathfrak{m}_{k}$ is
not determined by it alone because of the constraint
${\textstyle\sum\nolimits_{k}}p_{k}=1$, which makes it depend on the
macrostate also.

From $E$, we have the change in the macroenergy

$dE={\textstyle\sum\nolimits_{k}}E_{k}dp_{k}+{\textstyle\sum\nolimits_{k}}p_{k}dE_{k}$
(31a) between two neighboring macrostates. We will use the compact notation
$\displaystyle dE_{\text{h}}$
$\displaystyle\doteq{\textstyle\sum\nolimits_{k}}E_{k}dp_{k}=\left\langle
Ed\eta\right\rangle,$ $\displaystyle dE_{\text{w}}$
$\displaystyle\doteq{\textstyle\sum\nolimits_{k}}p_{k}dE_{k}=\left\langle
dE\right\rangle,$ (32)

where we have introduced $\eta_{k}\doteq\ln p_{k}$ so that
$d\eta_{k}=dp_{k}/p_{k}$. As $\left\\{E_{k}\right\\}$ does not change but
$\left\\{p_{k}\right\\}$ changes in $dE_{\text{h}}$, it must depend on $dS$;
see Eq. (14b). As $\left\\{p_{k}\right\\}$ is not changed in $dE_{\text{w}}$,
it is evaluated at _fixed entropy_ $S$__Gujrati-Heat-Work0 ; Gujrati-Entropy2
. Comparing $dE$ with the first law in Eq. (10a), we identify

$\displaystyle dQ$ $\displaystyle=dE_{\text{h}}\doteq\left\langle
Ed\eta\right\rangle,$ (33a) $\displaystyle dW$
$\displaystyle=-dE_{\text{w}}\doteq-\left\langle dE\right\rangle$ (33b) Thus,
the generalized macroheat $dQ$ is a contribution proportional to $dS$ and the
generalized macrowork $dW$ is an isentropic contribution as noted in Sec. I.1.
The identification of macroheat and macroworks above also explains the choice
of h for heat and w for work as suffix above and superfix in
$F_{\text{t}}^{\text{h}}$ and $\mathbf{F}_{\text{t}}^{\text{w}}$.

We can now identify the (generalized) microheat and microwork along
$\gamma_{k}$; they are given by

$dQ_{k}^{\prime}\doteq E_{k}d\eta_{k},dW_{k}=-dE_{k,}$ (34)

respectively. The reason for the prime in $d_{\alpha}Q_{k}^{\prime}$ will
become clear below. The second equation above is simply the previously derived
mechanical identity in Eq. (7b). We summarize this important result, which is
not properly appreciated in the field Gujrati-GeneralizedWork , in the form of
a

###### Theorem 3

The mechanical microwork $dW_{k}$ done by the system in the $k$th microstate
is the negative of the change $dE_{k}=(\partial
E_{k}/\partial\mathbf{Z}^{E})\cdot d\mathbf{Z}^{E}$:

$dW_{k}=-dE_{k}$ (35)

Proof. See the derivation of Eq. (7b).

As shown in Eq. (7b), the temporal evolution of $\mathfrak{m}_{k}$ is due to
$\mathbf{F}_{k}^{E}$, which changes its microenergy $E_{k}$ but does not
change $\mathfrak{m}_{k}$. The average over all
$\left\\{\mathfrak{m}_{k}\right\\}$ of $dW_{k}$ due to $\mathbf{F}_{k}^{E}$,
see Eq. (7c), gives the generalized macrowork $dW$, see Eq. (33b), due to the
macroforce

$\mathbf{F}^{E}\doteq\left\langle\mathbf{F}^{E}\right\rangle=T(\partial
S/\partial\mathbf{Z}^{E}),$

see Eq. (14a), so that

$dW=\mathbf{F}^{E}\cdot d\mathbf{Z}^{E}$

as expected. We can summarize the above result as the following theorem
because it plays a central role in the $\mu$NEQT.

In general, macroworks $d_{\alpha}W$ are thermodynamic averages of microworks
$d_{\alpha}W_{k}$:

$d_{\alpha}W=\left\langle
d_{\alpha}W\right\rangle\doteq{\textstyle\sum\nolimits_{k}}p_{k}d_{\alpha}W_{k}.$
(36)

As it is easy to determine the mechanical microworks $d_{\alpha}W_{k}$, we can
extend the _identity_ $dE_{k}=-dW_{k}$ to introduce microenergies

$d_{\text{e}}E_{k}=-d_{\text{e}}W_{k},d_{\text{i}}E_{k}=-d_{\text{i}}W_{k};$
(37)

they define what is meant by $d_{\text{e}}E_{k}$ and $d_{\text{e}}E_{k}$.

Shifting $E_{k\text{ }}$by a constant does not affect $dW$ and $dQ$, showing
their unique nature. While $dW_{k}$ is also not affected by the shift, it does
affect $dQ_{k}^{\prime}$. Therefore, instead of using $dQ=dE_{\text{h}}$ to
identify $dQ_{k}$, we instead use the identity $dQ=TdS$ in Eq. (14b) to
identify $dQ_{k}$ as there is no ambiguity in the definition of the
statistical entropy $S$ in Eq. (2). We thus find that

$dQ_{k}\doteq T\left.dS\right|_{k}\doteq-T(\eta_{k}+1)d\eta_{k},$ (38)

where $d\left.S\right|_{k}=-(\eta_{k}+1)d\eta_{k}$, not to be confused with
$dS_{k}=-d\eta_{k}$, is the microquantity corresponding to the macroquantity
$dS$

$dS=-{\textstyle\sum\nolimits_{k}}(\eta_{k}+1)dp_{k}\doteq{\textstyle\sum\nolimits_{k}}p_{k}\left.dS\right|_{k}.$

Similarly, we use Eqs. (15b) and (17a) to identify $d_{\text{e}}Q_{k}$ and
$d_{\text{i}}Q_{k}$, respectively. We are not going to be directly involved
with microheats in our investigation here so we will not spend time with them
further; we will treat them in a separate publication.

###  IV.2 The Simple Case

However, we need various microworks in this investigation as our focus is to
understand the dissipated work. Therefore, we give the results for them. For
the simple case $\mathbf{Z}^{E}=(V,\xi)$, compare with Sec. III.4, we have

$dW_{k}=P_{k}dV+A_{k}d\xi.$ (39a) From $d_{\text{e}}W=P_{0}dV$, we obtain
$d_{\text{e}}W_{k}=P_{0}dV$, which defines $d_{\text{e}}E_{k}=-P_{0}dV$. We
also find $d_{\text{i}}W_{k}=(P_{k}-P_{0})dV+A_{k}d\xi$ (39b) which identifies
$d_{\text{i}}E_{k}=-(P_{k}-P_{0})dV-A_{k}d\xi$ and explains how it becomes
nonzero due to internal processes. For an isolated NEQ system, we must
$d_{\text{i}}W_{k}=A_{k}d\xi=dW_{k}$. For free expansion, we must set
$P_{0}=0$ so
$d_{\text{i}}W_{k}^{\text{free}}=P_{k}dV+A_{k}d\xi=-d_{\text{i}}E_{k}.$ (39c)

###  IV.3 Hamiltonian Trajectory and Microworks

There have been several attempts to formulate microstate trajectory
thermodynamics Ritort ; Esposito  based on utilizing the work fluctuation
theorem so they are not directly applicable to an isolated system. Our own
attempt that includes an isolated system was briefly outline in Ref. Gujrati-
GeneralizedWork  and elaborated recently in Ref. Gujrati-JensenInequality .
Here, we briefly summarize it for continuity. We will assume the medium
$\widetilde{\Sigma}$ to consist of two noninteracting media
$\widetilde{\Sigma}_{\text{h}}$ that controls macroheat exchange and
$\widetilde{\Sigma}_{\text{w}}$ that controls macrowork exchange. We are
interested in a NEQ work process $\mathcal{P}_{0}$ as the system evolves from
one EQ macrostate to another by changing $\mathbf{X}^{E}$ from
$\mathbf{X}_{\text{in}}^{E}$ to $\mathbf{X}_{\text{fin}}^{E}$ by manipulating
the medium $\widetilde{\Sigma}_{\text{w}}$. It is usually the case that when
$\mathbf{X}^{E}=\mathbf{X}_{\text{fin}}^{E}$, the system is not yet in EQ so
the internal variables have not come to their EQ values. We denote this part
of $\mathcal{P}_{0}$ by $\mathcal{P}$. It takes a while during
$\mathcal{\bar{P}}$ ($\mathcal{P}_{0}=$ $\mathcal{P\cup\bar{P}}$) for the
system to reach the final EQ macrostate. We may allow the temperature $T_{0}$
of $\widetilde{\Sigma}_{\text{h}}$ to change during $\mathcal{P}_{0}$, or
disconnect $\widetilde{\Sigma}_{\text{h}}$ from $\Sigma$ during $\mathcal{P}$.
In both cases, the temperature $T$ of the system in the final macrostate
$\mathfrak{M}_{\text{fin}}\doteq\mathfrak{M}(T_{\text{fin}},\mathbf{Z}_{\text{fin}}^{E})$
may be different from that of the initial macrostate
$\mathfrak{M}_{\text{in}}\doteq\mathfrak{M}(T_{\text{in}},\mathbf{Z}_{\text{in}}^{E})$.
While the microstate maintains its identity ($k$ does not change) as shown in
Eq. (8), the microenergy $E_{k}$ changes during the entire evolution over
$\mathcal{P}_{0}$ in accordance with Eq. (7b). Let us focus on
$dW_{k}=-dE_{k}$ during $t$ and $t+dt$ along $\gamma_{k}$. Its integral along
$\gamma_{k}$ determines the accumulated microwork $\Delta_{\alpha}W_{k}$

$\Delta
W_{k}\mathbf{\doteq}{\textstyle\int\nolimits_{\gamma_{k}}}dW_{k}=-{\textstyle\int\nolimits_{\gamma_{k}}}dE_{k}.$
(40)

The integral is not affected by how $p_{k}$ changes during $\mathcal{P}_{0}$
so it is the same for all processes between $\mathbf{Z}_{\text{in}}^{E}$ and
$\mathbf{Z}_{\text{fin}}^{E}$. Thus, we can evaluate $\left\\{\Delta
W_{k}\right\\}$ for a single process such as an EQ process
$\mathcal{P}_{0\text{,eq}}$ but can us it for every other possible process
$\mathcal{P}_{0}(\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E})$.
On the other hand, the accumulated macrowork $\Delta W$ over
$\mathcal{P}_{0}$, see Eq. (21b), is affected by $p_{k}$, see Eq. (36), so it
is different for different processes
$\mathcal{P}_{0}(\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E})$.

###### Theorem 4

Let $\Delta\mathbf{Z}^{E}$ denote the change in $\mathbf{Z}^{E}$ in the
process
$\mathcal{P}_{0}(\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E})$.
The cumulative microwork $\Delta W_{k}=-\Delta
E_{k}=E_{k}(\mathbf{Z}_{\text{in}}^{E})-E_{k}(\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E})$
is the same for all processes, including the reversible one, that undergo the
same net change $\Delta\mathbf{Z}^{E}$:
$\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E}=\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E}$.
However, the cumulative macrowork $\Delta W$ depends on the process.

Proof. As $E_{k}$ is specific to the microstate $\mathfrak{m}_{k}$, the
integral in Eq. (40) is the required difference:

$\Delta W_{k}=-\Delta
E_{k}=E_{k}(\mathbf{Z}_{\text{in}}^{E})-E_{k}(\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E}).$
(41)

Let us consider various processes that occur when changing
$\mathbf{Z}_{\text{in}}^{E}$ by $\Delta\mathbf{Z}^{E}$ to
$\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E}$, regardless of the process.
As $E_{k}$ is a microproperty, the net difference $\Delta E_{k}$ is the same
for all these processes. As $\left\\{p_{k}\right\\}$ is different for
different processes, the macrowork $\Delta W$, see Eq. (21b), is usually
different for different processes as expected.

The theorem has far-reaching consequences. According to this, we can evaluate
$\left\\{\Delta W_{k}\right\\}$ for a single process such as an EQ process
$\mathcal{P}_{0\text{,eq}}$ but can use it for every other possible process
$\mathcal{P}_{0}(\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E})$.
On the other hand, the accumulated macrowork $\Delta W$ over $\mathcal{P}_{0}$
is affected by $p_{k}$, see Eq. (36), so it is different for different
processes
$\mathcal{P}_{0}(\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{fin}}^{E})$.

The identification in Eq. (35) or in (41) is the most important feature of the
$\mu$NEQT that distinguishes it from current stochastic thermodynamic
approaches, which invariably identifies $d_{\text{e}}W_{k}$ with $-dE_{k}$ or
$\Delta_{\text{e}}W_{k}$ with $-\Delta E_{k}\ $Gujrati-GeneralizedWork ; see
also Remark 6.

As the set $\left\\{\Delta W_{k}\right\\}$ is the same in all possible
processes with
$\mathbf{Z}_{\text{in}}^{E}\rightarrow\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E}$,
we can introduce a _random variable_ $\mathsf{W}$ such that it takes the value
(outcome) $W_{k}=-E_{k}$, or a random variable $d\mathsf{W}$ that takes the
value $dW_{k}=-dE_{k}$ in $\mathfrak{m}_{k}$ with probability $p_{k}$.

###### Corollary 5

For a spontaneous process in an isolated system,

$\Delta_{\text{i}}W_{k}\geq 0,\Delta_{\text{i}}W\geq 0.$ (42)

Proof. We see from the definition of generalized forces in Eq. (7c) that
$E_{k}$ acts like the potential energy of a mechanical system. According to
the principle of (potential) energy minimization, these forces spontaneously
cause the isolated mechanical system to decrease $E_{k}$. Therefore, for a
spontaneous process, $E_{k}(\mathbf{Z}_{\text{in}}^{E})\geq
E_{k}(\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E})$ in Eq. (41). As
$\Delta W_{k}=\Delta_{\text{i}}W_{k}$ for an isolated system, we have
$\Delta_{\text{i}}W_{k}\geq 0$; hence $\Delta_{\text{i}}W\geq 0$. This proves
the corollary.

###### Remark 6

As $\Delta_{\text{e}}W_{k}$ is associated with the MI macrowork
$\Delta_{\text{e}}W$, it cannot be related to the difference $dE_{k}$ of
microstate energies that are SI quantities. This clearly shows that the
$\mu$NEQT is very different from current stochastic thermodynamic approaches
as noted above.

The most important aspect of the Hamiltonian trajectory is the identity nature
of $\mathcal{T}$, which ensures that every initial microstate
$\mathfrak{m}_{k},k\in\mathbb{N}$, is mapped onto itself at the end of
$\mathcal{P}$, although its energy will have changed. Thus, each $\gamma_{k}$
is unique and a sum over its ensemble $\left\\{\gamma_{k}\right\\}$ is the
same as the sum over $k\in\mathbb{N}$. This is a major simplification of our
approach, and plays a major role in the rest of the study.

##  V Moment Generating Function

###  V.1 Trajectory Probability

Let us consider an arbitrary process $\mathcal{P}$ between
$\mathfrak{M}_{\text{in}}\ $and $\mathfrak{M}_{\text{fin}}$. We consider two
terminal microstates $\mathfrak{m}_{k\text{,in}}\ $and
$\mathfrak{m}_{k\text{,fin}}$ along $\mathcal{P}$ and introduce the following
trajectory probability Gujrati-JensenInequality  between them for any system,
interacting or isolated,

$p_{\gamma_{k}}\equiv\frac{{\textstyle\int\nolimits_{\gamma_{k}}}p_{k}dW_{k}}{{\textstyle\int\nolimits_{\gamma_{k}}}dW_{k}}={\textstyle\int\nolimits_{\gamma_{k}}}p_{k}dw_{k};$
(43)

here, $dw_{k}\doteq dW_{k}/\Delta W_{k}$ and is independent of $p_{k}$. We can
generalize the above definition to introduce $p_{\gamma_{k},\alpha}$ by
replacing $dW_{k}$ by $d_{\alpha}W_{k}$; see Ref. Gujrati-JensenInequality .
We see that

${\textstyle\sum\nolimits_{k}}p_{\gamma_{k}}\Delta
W_{k}={\textstyle\int\nolimits_{\gamma_{k}}}dW=\Delta W,$ (44)

the macrowork as introduced in Eq. (21b). It is clear that
${\textstyle\sum\nolimits_{k}}p_{\gamma_{k}}=1$ as expected, and that
$p_{\gamma_{k}}$ is nothing but the (thermodynamic) probability of having a
particular value $\Delta W_{k}=E_{k,\text{in}}-E_{k,\text{fin}}$, determined
only by $\Delta\mathbf{Z}^{E}$ as seen from Theorem 4; here
$E_{k,\text{in}}=E_{k}(\mathbf{Z}_{\text{in}}^{E})$ and
$E_{k,\text{fin}}=E_{k}(\mathbf{Z}_{\text{in}}^{E}+\Delta\mathbf{Z}^{E})$. It
is clear that $p_{\gamma_{k}}$ is the joint probability of
$\mathfrak{m}_{k\text{,in}}\ $and $\mathfrak{m}_{k\text{,fin}}$, which can be
expressed in terms of the conditional probability
$p\left(\left.\mathfrak{m}_{k\text{,fin}}\right|\mathfrak{m}_{k\text{,in}}\right)$:

$p_{\gamma_{k}}=p(\mathfrak{m}_{k\text{,in}})p\left(\left.\mathfrak{m}_{k\text{,fin}}\right|\mathfrak{m}_{k\text{,in}}\right),$
(45a) where $p_{k\text{,in}}\doteq p(\mathfrak{m}_{k\text{,in}})$ (45b) is the
probability of $\mathfrak{m}_{k\text{,in}}$. If $\mathfrak{M}_{\text{in}}$ is
an EQ one at $T_{\text{in}}=T_{0}$, then the EQ probability
$p_{k\text{,in}}^{(0)}$ in the canonical ensemble is
$p_{k\text{,in}}^{(0)}\doteq
e^{\beta_{0}(\overline{F}_{\text{in}}-E_{k,\text{in}})},$ (45c) where
$\overline{F}_{\text{in}}$ is the initial EQ free energy and
$\beta_{0}=1/T_{0}$; for an isolated system, $T_{0}$ denotes its EQ
temperature.

For a system in IEQ, with $\mathbf{Z}^{E}=(V,\boldsymbol{\xi})$, the
microstate probability looks very similar to the above Boltzmann probability

$p_{k}=\exp\left\\{\beta[\Phi-(E_{k}+P_{k}V+\mathbf{A}_{k}\cdot\boldsymbol{\xi})]\right\\},$
(46)

as given in Ref. (Gujrati-LangevinEq, , Eq. (20) with
$\mathbf{F}_{k\text{,BP}}\cdot\mathbf{R}$ replaced by
$\mathbf{A}_{k}\cdot\boldsymbol{\xi}$); here $\Phi$ is the SI thermodynamic
potential obtained by ensuring ${\textstyle\sum\nolimits_{k}}p_{k}=1$.

Alternatively, we first determine $S$ using Eq. (2). This gives

$S=\beta\left(E+PV+\mathbf{A}\cdot\boldsymbol{\xi-}\Phi\right).$ (47)

Using $p_{k}$ in a process involving IEQ macrostates, we determine
$p_{\gamma_{k}}$ above, which will be used in the rest of the paper. To obtain
the NEQ version of the canonical ensemble, the contribution from $V$ is
conventionally not included. In EQ, we must also remove the contribution from
$\boldsymbol{\xi}$ as it is not an independent variable anymore. This then
gives Eq. (45c).

We now determine $\Phi$ from Eq. (47) to obtain

$\Phi=E-TS+PV+\mathbf{A}\cdot\boldsymbol{\xi.}$

This is the SI potential in terms of system’s macroquantities, and should not
be confused with the corresponding conventional thermodynamic potential
$\overline{\Phi}=E-T_{0}S+P_{0}V$ in terms of the fields of the medium. In a
NEQ canonical ensemble ($V$ fixed), $\Phi$ reduces to the SI free energy $F$
in Eq. (22b), whereas the Helmholtz free energy is $\overline{F}=E-T_{0}S$. In
EQ, $\overline{\Phi}=$ $\Phi$ and $\overline{F}=F$; otherwise, they are
different macroquantities.

###  V.2 Moment Generating Function

We introduce the moment generating function (MGF) for the random variable
$\Delta\mathsf{W}$, with outcomes $\Delta W_{k}$ over $\mathfrak{m}_{k}$,
along $\mathcal{P}$:

$\mathcal{W}(\left.\dot{\beta}\right|\Delta\mathsf{W})=\left\langle
e^{\dot{\beta}\Delta
W}\right\rangle\doteq{\textstyle\sum\nolimits_{k}}p_{\gamma_{k}}e^{\dot{\beta}\Delta
W_{k}},$ (48)

where $\dot{\beta}$ is some independent parameter that $p_{\gamma_{k}}$ and
$\Delta W_{k}$ do not depend upon, and the sum is over the ensemble of all
trajectories $\gamma_{k}$ originating at $\mathfrak{m}_{k\text{,in}}$. The
definition is valid for any system, interacting or isolated, and for any
terminal macrostates $\mathfrak{M}_{\text{in}}\ $and
$\mathfrak{M}_{\text{fin}}$, neither of which has to be an EQ macrostate.
Implicit in the definition is the Hamiltonian characteristic of the process in
Eq. (8) and the _thermodynamic nature_ of the trajectory probability
$p_{\gamma_{k}}$ in Eq. (44). The latter makes $\mathcal{W}$ a thermodynamic
function. The independent parameter $\dot{\beta}$ should not be confused with
any inverse temperature of $\Sigma$.

Various moments of $\Delta\mathsf{W}$ are obtained by differentiating
$\mathcal{W}$ with respect to the parameter $\dot{\beta}$ and then setting
$\dot{\beta}=0$. For the first two moments, we have

$\displaystyle\left.d\mathcal{W}/d\dot{\beta}\right|_{\dot{\beta}=0}$
$\displaystyle={\textstyle\sum\nolimits_{k}}p_{\gamma_{k}}\Delta W_{k}=\Delta
W,$ (49a)
$\displaystyle\left.d^{2}\mathcal{W}/d\dot{\beta}^{2}\right|_{\dot{\beta}=0}$
$\displaystyle={\textstyle\sum\nolimits_{k}}p_{\gamma_{k}}\left(\Delta
W_{k}\right)^{2}=\Delta W^{2}>0;$ (49b) $\Delta W^{2}$ introduced above should
not be confused with $(\Delta W)^{2}$. Recalling Eq. (44), we see that $\Delta
W$ is the cumulative macrowork along the arbitrary process $\mathcal{P}$ for
which there is no sign restriction. Indeed, all moments derived from
$\mathcal{W}$ are thermodynamic in nature so they remain unchanged under any
transformation such as in Eq. (67) later that leaves $\mathcal{W}$ invariant.

The MGF, apart from yielding all the moments, is also quite useful in
establishing the following theorem for $\mathcal{P}_{0}$ between two EQ
macrostates $\mathfrak{M}_{\text{in,eq}}$ and $\mathfrak{M}_{\text{fin,eq}}$.

###### Theorem 7

For $\dot{\beta}=\beta_{0}=1/T_{\text{in}}$, the initial value of $\beta$ in
$\mathfrak{M}_{\text{in,eq}}$, the MGF becomes independence of the initial
values of the probabilities $\left\\{p_{k\text{,in}}^{(0)}\right\\}$ .

Proof. Recalling Eq. (45a) and setting
$p(\mathfrak{m}_{k\text{,in}})=p_{k\text{,in}}^{(0)}$, we see that the MGF
reduces to a function $\overline{\mathcal{W}}$ given by

$\overline{\mathcal{W}}(\left.\beta_{0}\right|\Delta\mathsf{W})=e^{\beta_{0}F_{\text{in}}}{\textstyle\sum\limits_{k}}p_{\text{eq}}\left(\left.\mathfrak{m}_{k\text{,fin}}\right|\mathfrak{m}_{k\text{,in}}\right)e^{-\beta_{0}E_{k,\text{fin}}},$
(50)

which clearly shows its independence from the initial probabilities
$\left\\{p_{k\text{,in}}^{(0)}\right\\}$.

The theorem proves extremely useful in the trick in Sec. VII.2 that is needed
to overcome phase space volume change. The bar on $\overline{\mathcal{W}}$ is
a reminder that we are considering a NEQ process $\mathcal{P}_{0}$ between two
EQ macrostates. We also note that since $\beta_{0}$ is no longer an
independent parameter, $\overline{\mathcal{W}}$ is no longer a MGF.

###  V.3 Some Special Cases

The sum in $\overline{\mathcal{W}}$ turns into an EQ partition function if we
set
$p\left(\left.\mathfrak{m}_{k\text{,fin}}\right|\mathfrak{m}_{k\text{,in}}\right)=1,$
$\forall k$, a poor approximation, in which case

$p_{k\text{,fin}}=p_{k\text{,in}}^{(0)},~{}p_{\gamma_{k}}(\mathcal{P}_{0})=p_{k\text{,in}}^{(0)},\forall
k.$ (51)

This leads to a new function $\widehat{\mathcal{W}}$ (the caret on top is a
reminder of the choice in Eq. (51)), defined by

$\widehat{\mathcal{W}}(\left.\beta_{0}\right|\Delta\mathsf{W})=\widehat{\left\langle
e^{\beta_{0}\Delta\mathsf{W}}\right\rangle}\doteq{\textstyle\sum\nolimits_{k}}p_{k\text{,in}}^{(0)}e^{\beta_{0}\Delta
W_{k}},$ (52)

where the ”average” denoted by $\widehat{\left\langle{}\right\rangle}$ is with
respect to $\left\\{p_{k\text{,in}}^{(0)}\right\\}$ as trajectory
probabilities. The new function simplifies to

$\widehat{\mathcal{W}}(\left.\beta_{0}\right|\left\\{\Delta\mathsf{W}\right\\})=e^{-\beta_{0}\Delta\overline{F}},$
(53)

where
$\Delta\overline{F}\doteq\overline{F}_{\text{fin}}-\overline{F}_{\text{in}}$.
Comparing the ”average” macrowork

$\widehat{\left\langle\Delta\mathsf{W}\right\rangle}\doteq{\textstyle\sum\nolimits_{k}}p_{k\text{,in}}^{(0)}\Delta
W_{k}.$ (54)

with Eq. (49a), we conclude that

$\Delta W\neq\widehat{\left\langle\Delta\mathsf{W}\right\rangle}.$

In general, the non-thermodynamic probability choice in
$\widehat{\mathcal{W}}$ will not result in a thermodynamic macrowork $\Delta
W$. It is clear that the assumption $p_{\gamma_{k}}=p_{k\text{,in}}^{(0)}$
does not result in thermodynamic average; see Gujrati-JensenInequality  for
more details. As $\widehat{\mathcal{W}}$ is a non-thermodynamic function, we
will refer to Eq. (53) as a _mathematical identity_ to differentiate it from a
thermodynamic identity. Despite this, it is an interesting function in that it
can be evaluated in a closed form as seen above. The assumption of a constant
$p_{\gamma_{k}},\forall k$ in Eq. (51) and the choice $\dot{\beta}=\beta_{0}$
has been popularized by Jarzynski through his WFT to be discussed below.

The issue of _non-thermodynamic_ averaging was first raised by Cohen and
Mauzerall Cohen0 ; Cohen00  in the context of MFTs. But its most significant
consequence is about justifying the second law as discussed earlier with
respect to Eqs. (3-6). The issue has been discussed and settled only recently
in Ref. Gujrati-JensenInequality . We will establish below that the non-
thermodynamic _identity_ in Eq. (53) is satisfied even for free expansion as
expected as the $\mu$NEQT is perfectly capable of describing isolated systems.
For free expansion, $\Delta W_{k}$ does not identically vanish for all $k$ as
we clearly see from Eq. (39c). In this regard, our approach is very different
from the one used in the WFT, to which we now turn.

###  V.4 The WFT

The non-thermodynamic function $\widehat{\mathcal{W}}$ is closely related to
the well-known Jarzynski’s WFT Jarzynski  given below in Eq. (57). We first
introduce the function used by Jarzynski

$\mathcal{W}_{\text{J}}(\left.\beta_{0}\right|\Delta_{\text{e}}\mathsf{W})\doteq{\textstyle\sum\nolimits_{k}}p_{k\text{,in}}^{(0)}e^{\beta_{0}\Delta_{\text{e}}W_{k}},$
(55a) which is obtained by replacing $\Delta\mathsf{W}$ by
$\Delta_{\text{e}}\mathsf{W}$ in
$\widehat{\mathcal{W}}(\left.\beta_{0}\right|\left\\{\Delta\mathsf{W}\right\\})$.
Here, $\Delta_{\text{e}}\mathsf{W}$ is assumed to be a random variable with
outcomes $\Delta_{\text{e}}W_{k}$ over $\mathfrak{m}_{k}$ with some
probability, and the suffix ”J” is a reminder for Jarzynski’s non-
thermodynamic average in which the probability is replaced by
$p_{k\text{,in}}^{(0)}$. Jarzynski further assume, without proof, that
$\Delta_{\text{e}}W_{k}\overset{\text{assump}}{=}\Delta W_{k}=-\Delta
E_{k},\forall k,$ (56)

as has been discussed recently for its validity Gujrati-GeneralizedWork . This
work-energy assumption is in addition to the non-thermodynamic average
conjecture in defining $\widehat{\mathcal{W}}$ in Eq. (52), and violates
Theorem 4. With the two assumptions, $\mathcal{W}_{\text{J}}$ becomes

$\mathcal{W}_{\text{J}}(\left.\beta_{0}\right|\Delta_{\text{e}}\mathsf{W})=e^{-\beta_{0}\Delta\overline{F}},$
(57)

which is the well-known WFT. Let us consider the following average

$\Delta_{\text{e}}W_{\text{J}}\doteq\widehat{\left\langle\Delta_{\text{e}}\mathsf{W}\right\rangle}={\textstyle\sum\nolimits_{k}}p_{k\text{,in}}^{(0)}\Delta_{\text{e}}W_{k}.$
(58)

From $\Delta_{\text{e}}W$ in Eq. (36), we see that in general

$\Delta_{\text{e}}W_{\text{J}}\neq\Delta_{\text{e}}W=-R.$

This should be contrasted with the fundamental assumption of the WFT in Eq.
(6), which has already been questioned earlier Cohen0 ; Gujrati-
JensenInequality . Thus, the conjecture in Eq. (6) cannot be justified.
Indeed, by considering an EQ process in an interacting system, for which
$\Delta_{\text{e}}W=\Delta W=-\Delta\overline{F}$, it has been shown (Gujrati-
JensenInequality, , see Eq.(20) there) that

$\Delta_{\text{e}}W_{\text{J}}<\Delta_{\text{e}}W,$

violating Eq. (6). For an isolated system undergoing free expansion for which
$\Delta_{\text{e}}W_{k}\equiv 0$, the ”mathematical identity” in Eq. (57)
obviously fails Sung ; Gross . Jarzynski Jarzynski-Gross  argues that such a
system does not start in EQ so the WFT should not apply there; however, see
KestinNote  for counter-argument.

The ”mathematical identity” in Eq. (57) is evidently different 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 identity satisfied for a class
of NEQ processes by most workers in the field who have not appreciated the
implications of Eq (53). Let us determine these implications within the
$\mu$NEQT. It follows from Eq. (56) that $\Delta_{\text{i}}W_{k}=0,\forall k$;
see Theorem 4. Consequently, we must have $\Delta_{\text{i}}W=0$ as noted in
Sec. I.2 so no irreversibility is captured by the WFT as was first concluded a
while back in Ref. Gujrati-GeneralizedWork . Using thermodynamic probabilities
$p_{\gamma_{k}}$ with the assumption in Eq. (56), we find $\Delta
W=\Delta_{\text{e}}W$ as a thermodynamic consequence. This says nothing about
$\widehat{\left\langle\Delta\mathsf{W}\right\rangle}$ or
$\Delta_{\text{e}}W_{\text{J}}$ that are not thermodynamic quantities.
However, it is commonly believed that
$\Delta_{\text{e}}W_{\text{J}}=\Delta_{\text{e}}W$, which cannot be justified
Gujrati-JensenInequality .

##  VI The Free Expansion in The MNEQT

Our derivation of the identity in Eq. (53) is exact so it should be valid for
all processes including free expansion $\mathcal{P}_{0}$ as we will now show.
In free expansion, there is no exchange of any kind so $d=d_{i}$. This
simplifies our notation as we do not need to use $d_{\text{i}}$ when referring
to $\Sigma$. The gas $\Sigma$ expands freely in a vacuum
($\widetilde{\Sigma}$) from $V_{\text{in}}$, the volume of the left chamber,
to $V_{0}=V_{\text{fin}}$, the volume of $\Sigma_{0}$; the volume of the right
chamber is $V_{\text{fin}}-V_{\text{in}}$. The vacuum exerts no pressure
($\widetilde{P}=P_{\text{vacuum}}=0$). The left (L) and right (R) chambers are
initially separated by an impenetrable partition, shown by the solid partition
in Fig. 2(a), to ensure that they are thermodynamically independent regions,
with all the $N$ particles of $\Sigma$ in the left chamber, which are
initially in an EQ macrostate $\mathfrak{M}_{\text{in,eq}}$ with entropy
$S_{\text{in}}$. For ideal gas, we have

$S_{\text{in}}=N\ln(eV_{\text{in}}/N);$

here, we are not including a temperature-dependent function Landau , which
does not play any role as we will be considering an isothermal free expansion.
The initial pressure and temperature of the gas prior to expansion at time
$t=0$ are $P_{\text{in}}$ and $T_{\text{in}}=T_{0}$, respectively, that are
related to $E_{0}=E_{\text{in}}$ and $V_{\text{in}}$ by its EQ equation of
state. As $\Sigma_{0}$ is isolated, the expansion occurs at _constant_ energy
$E_{0}$, which is also the energy of $\Sigma$.

It should be stated, which is also evident from Fig. 2(b), that while the
removal of the partition is instantaneous, the actual process of gas expanding
in the right chamber is continuous and gradually fills it. This is obviously a
very complex internal process in a highly inhomogeneous macrostate. As thus,
it will require many internal variables to describe different number of
particles, different energies, different pressures, different flow pattern
which may be even chaotic, etc. in each of the chambers. For example, we can
divide the volume $V_{\text{fin}}$ into many layers of volume parallel to the
partition, each layer in equilibrium with itself but need not be with others;
see the example in Sec. II.1. Here, we will simplify and take a single
internal variable $\xi$

$\xi\doteq N_{\text{L}}-N_{\text{R}}$ (59)

by considering only two layers to describe different numbers
$N_{\text{L}}=(N+\xi)/2$ and $N_{\text{R}}=(N-\xi)/2$ of particles as a
function of time. At each instant, we imagine a front of the expanding gas
shown by the solid vertical line in Fig. 2(b) containing all the particles to
its left. We denote this volume by a time-dependent $V=V(t)$ to the right of
which exists a vacuum. This means that at each instant when there is a vacuum
to the right of this front, the gas is expanding against zero pressure so that
$d_{\text{e}}W=0$. Since we have a NEQ expansion, $dW>0$. As $V(t)$ cannot be
controlled externally, it also represents an internal variable. The two
internal variables $\xi(t)$ and $V(t)$ allow us to distinguish between
$\mathcal{P}$ and $\mathcal{\bar{P}}$ as we will see below. We assume that the
expansion is isothermal so there is no additional internal variable associated
with temperature variation. As $dQ=dW\neq 0$, the expansion is irreversible so
the entropy continues to change (increase).

At $t=0$, the partition is suddenly removed, shown by the broken partition in
Fig. 2(b) and the gas expands freely to the final volume
$V(t^{\prime})=V_{\text{fin}}$ at time $t^{\prime}<\tau_{\text{eq}}$ during
$\mathcal{P}$. At $t^{\prime}$, the free expansion stops but there is no
reason a priori for $\xi=0$ so the gas is still inhomogeneous ($\xi\neq 0$).
This is in a NEQ macrostate until $\xi$ achieves its EQ value $\xi=0$ during
$\mathcal{\bar{P}}$, at the end of which at $t=\tau_{\text{eq}}$ the gas
eventually comes into $\mathfrak{M}_{\text{fin,eq}}$ isoenergetically. We
briefly review this expansion in the MNEQT Gujrati-Heat-Work .

We work in the state space $\mathfrak{S}$. Using Eq. (14b), we have

$dS(t)=dW(t)/T(t).$ (60a) Setting $P_{0}=0$ in Eq. (16a), we have
$dW(t)=\left\\{\begin{array}[c]{c}P(t)dV(t)+A(t)d\xi(t)\text{ for
}t<t^{\prime}<\tau_{\text{eq}},\\\ A(t)d\xi(t)\text{ for
}t^{\prime}<t\leq\tau_{\text{eq}};\end{array}\right.$ (60b) here, we have used
the fact that $V(t)$ does not change for
$\tau^{\prime}<t\leq\tau_{\text{eq}}$. Thus,  $\displaystyle\Delta S$
$\displaystyle=\int_{\mathcal{P}_{0}}\frac{dW(t)}{T(t)}>0,$
$\displaystyle\Delta Q$ $\displaystyle=\int_{\mathcal{P}_{0}}dW(t)=\Delta
W>0;$

the last equation is the fundamental identity in Eq. (11). The irreversible
entropy change $\Delta S$ from EQ macrostate from
$\mathfrak{M}_{\text{in,eq}}$ to $\mathfrak{M}_{\text{fin,eq}}$ during
$\mathcal{P}_{0}$ is the EQ entropy change $\Delta_{\text{i}}S$ is

$\Delta S\equiv S_{\text{fin}}-S_{\text{in}},$ (61)

and can be directly obtained if the EQ entropy $S(E,V)$ is known. The above
analysis is also valid for any arbitrary free expansion process $\mathcal{P}$;
we must carry out the integration over $\mathcal{P}$ above. We can evaluate
$\Delta_{\text{i}}S$ by using $S$ from Eq. (47).

The above exercise allows us to identify $\Delta W$ as the dissipated work
over the entire process even if the process. We have $\Delta
W=\Delta_{\text{i}}W=T_{0}\Delta_{\text{i}}S$ for an isothermal process
$T(t)=T_{0}$; see also Eq. (25b). However, with the inclusion of the internal
variables above, we are also able to determine $d_{\text{i}}S=dS=dW/T$ using
Eq. (60b) for any infinitesimal segment $\delta\mathcal{P}$ of the process in
the MNEQT that was one of our goals. Thus, $d_{\text{i}}W=Td_{\text{i}}S$ is
the infinitesimal ”dissipated work” over $\delta\mathcal{P}$ but the
relationship contains $T$ and not $T_{0}$ as it must; see Theorem 1.

Let us consider an ideal gas for which $V_{\text{fin}}=2V_{\text{in}}$ so that
$\Delta_{\text{i}}S=N\ln 2$, a well-known result Prigogine . Here, we provide
a more general result for the entropy for $t\leq t^{\prime}$, which can be
trivially determined:

$S(t,\xi)=N_{\text{L}}\ln(eV_{\text{in}}/N_{\text{L}})+N_{\text{R}}\ln(eV^{\prime}/N_{\text{R}}),$

with $\xi>0$; here $V^{\prime}=V-V_{\text{in}}$. Thus, for arbitrary $\xi$, we
have $\Delta_{\text{i}}S(t,\xi)=S(t,\xi)-S_{\text{in}}$. At EQ, not only
$V_{\text{fin}}^{\prime}=V_{\text{fin}}-V_{\text{in}}$, but also $\xi=0$ so
the EQ entropy is given by

$S_{\text{fin}}=(N/2)\ln(2eV_{\text{in}}/N)+(N/2)\ln(2eV_{\text{fin}}^{\prime}/N),$

which is consistent with $\Delta_{\text{i}}S=N\ln 2$, as expected. We can also
take the initial macrostate to be not an EQ one in $\mathcal{P}$ by using one
or more additional internal variables. Thus, the approach is very general.

##  VII The Free Expansion in The $\mu$NEQT

###  VII.1 Quantum Free Expansion

The expansion/contraction of a one-dimensional quantum _ideal_ gas with moving
walls has been treated in many different ways Rice ; Fojon ; Martino ; Cooney
but none deal with sudden expansion. The latter, however, has been studied
Bender ; Gujrati-QuantumHeat ; Gujrati-JensenInequality  quantum mechanically
(without any $\xi$) as a particle in an isolated box $\Sigma_{0}$ of length
$L_{\text{fin}}$, which we restrict to $2L_{\text{in}}$ here, with rigid,
insulating walls. We briefly revisit this study and expand on it by
introducing a $\xi$ to set the stage for the classical expansion using the
$\mu$NEQT in the following section. We will follow Ref. Gujrati-
JensenInequality  closely.

We make the very simplifying assumptions in the previous section to introduce
$\xi$. At time $t=0$, all the $N$ particles (or their wavefunctions) are
confined in EQ in the left chamber of length $L_{\text{in}}$ so that
$N_{\text{L}}=N$ initially. We can think of an intermediate length
$L_{\text{fin}}\geq L(t)>L_{\text{in}}$, in analogy with $V(t)$ in the
previous section, so that $N_{\text{R}}=N-N_{\text{L}}$ particles are
simultaneously confined in the intermediate chamber of size $L(t)$, while
$N_{\text{L}}$ particles are still confined in the left chamber for all $t>0$
as we did in the previous section. Eventually, at $t=\tau_{\text{eq}}$, all
the $N_{\text{R}}=N$ particles are confined in the larger chamber of size
$L_{\text{fin}}$ so that there are no particles in the initial chamber. We let
$\xi=N_{\text{L}}$, which gradually decreases from $\xi=N$ to $\xi=0$. Note
that this definition is different from the previous section but we make this
choice for the sake of simplicity. At some intermediate time
$\tau^{\prime}<\tau_{\text{eq}}$ that identifies $\mathcal{P}$,
$L(t)=L_{\text{fin}}$, but $N_{\text{R}}$ is still not equal to $N$ ($\xi\neq
0$). We then follow its equilibration during $\mathcal{\bar{P}}$ as the gas
come to EQ in the larger chamber at the end of $\mathcal{P}_{0}$ when $\xi=0$.
Again, there are two internal variables $L$ and $\xi$. The expansion is
isoenergetic at each instant. As we will see below, this means that it is also
isothermal. However, $dQ=dW\neq 0$ ensuring a irreversible process so the
microstate probabilities continue to change.

Since we are dealing with an ideal gas, we can focus on a single particle
whose energy levels are in appropriate units $E_{k}=k^{2}/l^{2}$, where $l$ is
the length of the chamber confining it. The single-particle partition function
for arbitrary $l$ and inverse temperature $\beta=1/T$ is given by

$Z(\beta,l)={\textstyle\sum\nolimits_{k}}e^{-\beta E_{k}(l)},$

from which we find that the single particle free energy is
$\overline{F}=-(T/2)\ln(\pi Tl^{2}/4)$ and the average single particle energy
is $E=1/2\beta$, which depends only on $\beta$ but not on $l$. Assuming that
the gas is in IEQ so that the particles in each of the two chambers are in EQ
(see the second example in Sec. II.1) at inverse temperatures
$\beta_{\text{L}}$ and $\beta$, we find that the $N$-particle partition
function is given by

$Z_{N}(\beta_{\text{L}},\beta)=\left[Z(\beta_{\text{L}},L_{\text{in}})\right]^{\xi}\left[Z(\beta,L)\right]^{N-\xi}$

so that the average energy is
$E_{N}(\beta_{\text{L}},\beta,L_{\text{in}},L,\xi)=\xi/2\beta_{\text{L}}+(N-\xi)/2\beta$.
As this must equal $N/2\beta_{0}$ for all values of $L$ and $\xi$, it is clear
that $\beta_{\text{L}}=\beta=\beta_{0}$, which proves the above assertion of
an isothermal free expansion at $T_{0}$.

To determine $\Delta W_{k}$, we merely have to determine the microenergy
change $\Delta E_{k}=E_{k\text{,fin}}-E_{k\text{,in}}$. It is trivially seen
that Eq. (53) is satisfied as was reported earlier Gujrati-GeneralizedWork ;
Gujrati-JensenInequality .

Below we will show that the quantum calculation here deals with an
irreversible $\mathcal{P}_{0}$. The single-particle energy change $\Delta
E_{k}$ is

$\Delta E_{k}=k^{2}(1/L^{2}-1/L_{\text{in}}^{2})<0,L>L_{\text{in}}.$

The micropressure

$P_{k}=-\partial E_{k}/\partial L=2E_{k}/L\neq 0$ (62)

determines the microwork

$\Delta
W_{k}={\displaystyle\int\nolimits_{L_{\text{in}}}^{L_{\text{fin}}}}P_{k}dL>0.$
(63)

It is easy to see that this microwork is precisely equal to $(-\Delta E_{k})$
as expected; see Eq. (40). It is also evident from Eq. (62) that for each $L$
between $L_{\text{in}}$ and $L_{\text{fin}}$,

$P={\textstyle\sum\nolimits_{k}}p_{k}P_{k}=2E/L\neq 0,$

We can use this average pressure to calculate the thermodynamic macrowork

$\Delta
W={\displaystyle\int\nolimits_{L_{\text{in}}}^{L_{\text{fin}}}}PdL=2{\textstyle\sum\nolimits_{k}}{\displaystyle\int\nolimits_{L_{\text{in}}}^{L_{\text{fin}}}}p_{k}E_{k}dL/L\neq
0.$

as expected. As $\Delta E=0$, this means that the irreversible macroheat and
macrowork are $\Delta Q=\Delta W>0$. This establishes that the expansion we
are studying is _irreversible_.

We now turn to the entire system in which the work is done by $N_{\text{R}}$
particles. We need to think of the microstate index $k$ as an $N$-component
vector $\mathbf{k}=\left\\{k_{i}\right\\}$ denoting the indices for the
single-particle microstates. For a given $\xi$, we have $\Delta
W_{\mathbf{k}}(L,\xi)=-{\textstyle\sum\nolimits_{i}}\Delta E_{k_{i}}$, where
$i$ runs over the $N_{\text{R}}$ particles. We can compute the macrowork,
which turns out to be $\Delta W_{N}(\xi)=(N-\xi)\Delta W>0$. The corresponding
change in the free energy is

$\displaystyle\Delta\overline{F}_{N}(L,\xi)$
$\displaystyle=(N-\xi)[\overline{F}(\beta_{0},L)-\overline{F}(\beta_{0},L_{\text{in}})]$
$\displaystyle=-\Delta W_{N}(\xi),$

which is consistent with Eq. (25a) for an isolated system for any $\xi$.

At the end of $\mathcal{P}_{0}$, $\Delta W_{N}(0)=N\Delta W>0$, and
$\Delta\overline{F}_{N}(0)=N[\overline{F}(\beta_{0},L_{\text{fin}})-\overline{F}(\beta_{0},L_{\text{in}})]$.
We can now set up the MGF
$\mathcal{W}(\left.\beta\right|\left\\{\Delta\mathsf{W}\right\\})$ for any $L$
and $\xi$ so that we can compute all the moments. However, we will only
consider the entire process $\mathcal{P}_{0}$ so that $\xi=0$ at the end. For
the first moment in Eq. (49a) we find that for the isothermal expansion

$\Delta W_{N}=-\Delta\overline{F}_{N}=T_{0}\Delta_{\text{i}}S_{N}>0,$ (64)

after using Eq. (25b). The same result is also obtained from 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 subscript for simplicity. The
benefit of using the $\mu$NEQT is that we get a much better perspective of the
dissipation in the free expansion. In terms of the SI microwork,
$\Delta_{\text{i}}W_{k}\neq 0,\forall k$ even though the SI microwork
$\Delta_{\text{e}}W_{k}\equiv 0,\forall k$. However, what is more revealing
about the free expansion is that $\Delta W_{k}>0$ for each Hamiltonian
trajectory $\gamma_{k}$ as seen from Eq. (63), which is in accordance with
Corollary 5. This is not true in a general process. For example,
$\Delta_{\text{i}}W=0$ in a reversible process even though
$\Delta_{\text{i}}W_{k}\neq 0,\forall k$. In fact, $\Delta_{\text{i}}W_{k}$
must be of either sign to ensure a vanishing average.

###  VII.2 Classical Free Expansion

The change in the phase space volume in classical statistical mechanics
destroys the required unique mapping in Eq. (8) as discussed in Sec. II.4, and
causes a problem with the use of the Hamiltonian trajectories that were used
in deriving the $\mu$NEQT and introducing the MGF $\mathcal{W}$. Their use for
classical expansion will requires some modification, which we describe below.
To introduce the required modification as simply as possible, we will first
consider $V_{\text{fin}}=2V_{\text{in}}$ that results in doubling the volume
after expansion; see Fig. 2(a). Later, we will generalize to any arbitrary
expansion/contraction. We will use the notation of Sec. VI, and restrict
ourselves to $\mathbf{Z}^{E}=\left(V_{\text{in}},V_{\text{fin}},V,\xi\right)$;
see Eq. (59) for $\xi$ and Fig. 3.

Let $\delta\mathbf{z}(t)\doteq\delta\mathbf{z}\left(V,\xi\right)$ denote a
microstate at some time $t$ with work variables $V\doteq V(t)$ and
$\xi\doteq\xi(t)$. Let $E_{k}(t)\doteq E(\delta\mathbf{z}(t))$ denote the
instantaneous microenergy of $\delta\mathbf{z}(t)$. Let $\delta\mathbf{z}$ be
some initial microstate at $t=0$ with
$N_{\text{L}}=N,N_{\text{R}}=0,V=V_{\text{in}}$, and $\xi=\xi_{\text{in}}=N$.
We denote the number of microstates in the initial phase space that is denoted
by the interior of the solid red ellipse
$\boldsymbol{\Gamma}_{\text{in}}=\boldsymbol{\Gamma}_{\text{in}}(\mathbf{Z}_{\text{in}}^{E})$
on the left by $\mathcal{N}$. The final phase space is shown by the solid red
ellipse
$\boldsymbol{\Gamma}_{\text{fin}}^{\prime}=\boldsymbol{\Gamma}_{\text{fin}}^{\prime}(\mathbf{Z}_{\text{fin}}^{E})$
on the right, which contains twice as many ($2\mathcal{N}$) microstates as are
in $\boldsymbol{\Gamma}_{\text{in}}$. We will assume that both the initial and
the final macrostates ($\mathcal{M}_{\text{in,eq}}$ and
$\mathcal{M}_{\text{fin,eq}}$) are in EQ for which $V$ and $\xi$ are no longer
independent state variables. Therefore, we will not use $V$ and $\xi$ in
$\mathbf{Z}^{E}$ for the two phase spaces at $t=0$ and $t=\tau_{\text{eq}}$.
We set $P_{0}=0$ in Eq. (39b) to obtain the microwork for $t>0$

$dW_{k}=P_{k}dV+A_{k}d\xi,$

which is nonzero, while $d_{\text{e}}W_{k}=0$. The EQ gas at $t=0$ has
microstate probabilities

$\displaystyle p_{\text{in}}(\delta\mathbf{z})$
$\displaystyle=e^{-\beta_{0}E(\delta\mathbf{z})}/Z_{\text{in}}(\beta_{0},V_{\text{in}})\geq
0,$ (65a) $\displaystyle Z_{\text{in}}(\beta_{0},V_{\text{in}})$
$\displaystyle\doteq{\textstyle\sum\limits_{\delta\mathbf{z}\in\boldsymbol{\Gamma}_{\text{in}}}}e^{-\beta_{0}E(\delta\mathbf{z})}.$
(65b)

Under $\mathcal{T}$,
$\delta\mathbf{z}(V_{\text{in}})\in\boldsymbol{\Gamma}_{\text{in}}$ maps onto
its image (see the green double arrow $\gamma_{\text{in}}$)
$\delta\mathbf{z}^{\prime}(V_{\text{fin}})\in\boldsymbol{\Gamma}_{\text{in}}^{\prime}=\boldsymbol{\Gamma}_{\text{in}}^{\prime}(V_{\text{fin}})$,
where $\boldsymbol{\Gamma}_{\text{in}}^{\prime}$ is shown schematically by the
broken red ellipse on the right. Thus,
$\boldsymbol{\Gamma}_{\text{in}}^{\prime}$ contains $\mathcal{N}$ microstates.
We then pick a microstate
$\delta\mathbf{\zeta}^{\prime}(V_{\text{fin}})\in\boldsymbol{\Gamma}_{\text{diff}}^{\prime}(V_{\text{fin}})\boldsymbol{\doteq\Gamma}_{\text{fin}}^{\prime}\backslash\boldsymbol{\Gamma}_{\text{in}}^{\prime}$;
here, $\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$ denotes
$\boldsymbol{\Gamma}_{\text{fin}}^{\prime}$ from which
$\boldsymbol{\Gamma}_{\text{in}}^{\prime}$ has been removed so it also
contains $\mathcal{N}$ microstates. Because of the uniqueness of
$\gamma_{\text{fin}}$,
$\delta\mathbf{\zeta}^{\prime}(V_{\text{fin}})\mathbf{\notin}\boldsymbol{\Gamma}_{\text{in}}^{\prime}$
is the image of a microstate
$\delta\mathbf{\zeta}(V_{\text{in}})\in\boldsymbol{\Gamma}_{\text{diff}}\boldsymbol{\doteq\Gamma}_{\text{fin}}\backslash\boldsymbol{\Gamma}_{\text{in}}$,
where $\boldsymbol{\Gamma}_{\text{fin}}$ is shown by the broken ellipse on the
left from which $\boldsymbol{\Gamma}_{\text{in}}$ has been taken out to obtain
$\boldsymbol{\Gamma}_{\text{diff}}$. Again,
$\boldsymbol{\Gamma}_{\text{diff}}$ contains $\mathcal{N}$ microstates. To
find $\delta\mathbf{\zeta}$, we follow the $inverse$ of $\gamma_{\text{fin}}$
along which $\mathbf{Z}_{\text{fin}}^{E}\rightarrow\mathbf{Z}_{\text{in}}^{E}$
($V_{\text{fin}}\rightarrow V_{\text{in}}$); this is shown by the left arrow
on $\gamma_{\text{fin}}$ in accordance with the reversibility of
$\mathcal{T}$. The same number of microstates in
$\boldsymbol{\Gamma}_{\text{diff}}$ and
$\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$ ensures the uniqueness of
$\gamma_{\text{fin}}$, a prerequisite for Hamiltonian trajectories.
Physically, the $\mathcal{N}$ microstates in$\
\boldsymbol{\Gamma}_{\text{diff}}$ corresponds to as if there are $N$
particles in the right chamber in Fig. 2(a), with the partition intact so that
each particle is confined in the volume $V$ of the right chamber only. But
note that there are no particles in the right chamber at $t=0$.

The situation was very simple for the uniqueness of $\gamma_{\text{fin}}$
because of the choice $V_{\text{fin}}=2V_{\text{in}}$:
$\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$ and
$\boldsymbol{\Gamma}_{\text{diff}}$ have the same number of $\mathcal{N}$
microstates. This will not be the case if
$V_{\text{in}}<V_{\text{fin}}<2V_{\text{in}}$, since the number of microstates
$\delta\mathbf{\zeta}^{\prime}(V_{\text{fin}})$ in
$\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$ is now
$\mathcal{N}^{\prime}<\mathcal{N}$, while $\boldsymbol{\Gamma}_{\text{diff}}$
has $\mathcal{N}$ microstates $\delta\mathbf{\zeta}(V_{\text{in}})$. To
identify the set
$\left\\{\delta\mathbf{\zeta}(V_{\text{in}})\right\\}\subset\boldsymbol{\Gamma}_{\text{diff}}$
of theses $\mathcal{N}^{\prime}$ microstates, we follow the inverse of
$\gamma_{\text{fin}}$ for each
$\delta\mathbf{\zeta}^{\prime}(V_{\text{fin}})\in\boldsymbol{\Gamma}_{\text{diff}}^{\prime}(V_{\text{fin}})$
to identify the image $\delta\mathbf{\zeta}(V_{\text{in}})$ as described
above. From now on, we will use
$\boldsymbol{\Gamma}_{\text{diff}}(\left.V_{\text{in}}\right|V_{\text{fin}})$
to denote the required set of $\mathcal{N}^{\prime}$ microstates. The
$\mathcal{N}-\mathcal{N}^{\prime}$ remaining microstates in
$\boldsymbol{\Gamma}_{\text{diff}}$ are superfluous. We can similarly extend
the above discussion to $V_{\text{fin}}=3V_{\text{in}}$, and then to
$2V_{\text{in}}<V_{\text{fin}}<2V_{\text{in}}$, and so on.

With the above understanding of
$\boldsymbol{\Gamma}_{\text{diff}}(\left.V_{\text{in}}\right|V_{\text{fin}})$,
there will be no confusion to simplify the notation and use
$\boldsymbol{\Gamma}_{\text{diff}}\ $for it. With this understanding,
$\boldsymbol{\Gamma}_{\text{fin}}$ as the union of
$\boldsymbol{\Gamma}_{\text{in}}$ and $\boldsymbol{\Gamma}_{\text{diff}}$ only
contains the required $\mathcal{N}+\mathcal{N}^{\prime}$ microstates. This
will be understood below.

Let us pick two microstates
$\delta\mathbf{z}(V_{\text{in}})\in\boldsymbol{\Gamma}_{\text{in}}$ and
$\delta\mathbf{\zeta}(V_{\text{in}})\in\boldsymbol{\Gamma}_{\text{diff}}$.
Their image microstates
$\delta\mathbf{z}^{\prime}(V_{\text{fin}})\in\boldsymbol{\Gamma}_{\text{in}}^{\prime}$
and
$\delta\mathbf{\zeta}^{\prime}(V_{\text{fin}})\in\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$
are obtained by the deterministic evolution along
$\gamma_{\text{in}}=\gamma(\delta\mathbf{z})$ and
$\gamma_{\text{fin}}=\gamma(\delta\mathbf{\zeta})$, respectively. The
corresponding microworks are given by

$\begin{array}[c]{c}\Delta
W(\delta\mathbf{z})=-(E(\delta\mathbf{z}^{\prime})-E(\delta\mathbf{z})),\\\
\Delta
W(\delta\mathbf{\zeta})=-(E(\delta\mathbf{\zeta}^{\prime})-E(\delta\mathbf{\zeta})).\end{array}$
(66)

However, the situation is still not fully resolved as
$\delta\mathbf{\zeta(Z}_{\text{in}}^{E})\in\boldsymbol{\Gamma}_{\text{diff}}$
does not represent a physical initial microstate in the left chamber; instead,
it refers to a microstate in the right chamber so its probability vanishes:

$p_{\text{in}}(\delta\mathbf{\zeta})=0,\delta\mathbf{\zeta}\in\boldsymbol{\Gamma}_{\text{diff}}.$

We recall that we do not need $p_{k}$ to determine $\Delta W_{k}$ so it does
not matter if $p_{\text{in}}(\delta\mathbf{\zeta})=0$. During expansion,
$p(\delta\mathbf{\zeta})$ at $t>0$ is not going to remain zero. Therefore, we
_formally assume_ that the initial probability distribution
$p_{\text{in}}(\delta\mathbf{\zeta})$ is _infinitesimally small_ for
$\delta\mathbf{\zeta}$ by shifting its initial energy by a very large positive
contribution

$E(\delta\mathbf{\zeta})\rightarrow
E(\delta\mathbf{\zeta})+e(\delta\mathbf{\zeta})/\varepsilon>0,\delta\mathbf{\zeta}\in\boldsymbol{\Gamma}_{\text{diff}}\text{
at }t=0$ (67)

using an infinitesimal quantity $\varepsilon>0$. At the end of the
calculation, the limit $\varepsilon\rightarrow 0^{+}$ will be taken to ensure
$p_{\text{in}}(\delta\mathbf{\zeta})\overset{\varepsilon\rightarrow
0^{+}}{\rightarrow}0$. This trick of energy shift transforms $\mathcal{W}$ as

$\displaystyle\mathcal{W}(\left.\dot{\beta},\varepsilon\right|\Delta\mathsf{W})$
$\displaystyle\rightarrow{\textstyle\sum\limits_{\delta\mathbf{z}\in\boldsymbol{\Gamma}_{\text{in}}}}p(\gamma_{\text{in}})e^{\dot{\beta}\Delta
W(\delta\mathbf{z})}$
$\displaystyle+{\textstyle\sum\limits_{\delta\mathbf{\zeta}\in\boldsymbol{\Gamma}_{\text{diff}}}}p(\gamma_{\text{fin}})e^{\dot{\beta}\Delta
W(\delta\mathbf{\zeta})};$ (68)

however, the energy shift leaves $\Delta W(\delta\mathbf{\zeta})$ invariant.
Thus, we see that letting $\varepsilon\rightarrow 0^{+}$ makes the second term
vanish so
$\mathcal{W}(\left.\dot{\beta},\varepsilon\right|\Delta\mathsf{W})\overset{\varepsilon\rightarrow
0^{+}}{\rightarrow}\mathcal{W}(\left.\dot{\beta}\right|\Delta\mathsf{W})$. As
$\mathcal{W}$ and all the moments remain invariant, the _trick_ (energy shift
followed by $\varepsilon\rightarrow 0^{+}$) does not affect thermodynamics in
anyway. In particular, it leaves the first law unaffected. We will verify this
below by direct manipulation.

It is clear that the two sums in Eq. (68) can be expressed as a single sum
over _all_ microstates
$\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}$, which refers to
microstates in $\boldsymbol{\Gamma}_{\text{fin}}$. For example, the shifted
initial partition function also remains invariant under the trick:

$Z_{\text{in}}(\beta_{0},V_{\text{in}},\varepsilon)\doteq{\textstyle\sum\limits_{\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}}}e^{-\beta_{0}E(\delta\mathbf{\bar{z}})}\overset{\varepsilon\rightarrow
0^{+}}{\rightarrow}Z_{\text{in}}(\beta_{0},V_{\text{in}}).$ (69)

This is consistent with Theorem 7. Thus, we can focus on
$\boldsymbol{\Gamma}_{\text{fin}}$ from the start instead of
$\boldsymbol{\Gamma}_{\text{in}}$. This allows us to basically use the
identity mapping $\mathcal{T}$ between initial microstates
$\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}$ and final
microstates
$\delta\mathbf{\bar{z}}^{\prime}\in\boldsymbol{\Gamma}_{\text{fin}}^{\prime}\
$using Hamiltonian trajectories $\gamma$ as
$(V_{\text{in}},\xi_{\text{in}})\rightarrow(V_{\text{fin}},\xi_{\text{fin}}=0)$.
In the following, we will use $\left\\{\delta\mathbf{\bar{z}}\right\\}$ and
$\left\\{\delta\mathbf{\bar{z}}^{\prime}\right\\}$ to show these sets. We will
use $\left\\{\overline{\gamma}(t)\right\\}$ to denote the ensemble of
Hamiltonian trajectories at any time $t$.

We can combine the two equations in Eq. (66) in a single equation $\Delta
W(\delta\mathbf{\bar{z}})=-(E(\delta\mathbf{\bar{z}}^{\prime})-E(\delta\mathbf{\bar{z}}))$,
with $E(\delta\overline{\mathbf{z}})$ playing the role of $E_{k}$ along
$\overline{\gamma}$. Let us consider $\overline{\mathcal{W}}$ for
$\mathcal{P}_{0}$

$\overline{\mathcal{W}}(\left.\beta_{0}\right|\left\\{\Delta\mathsf{W}\right\\})\doteq\lim_{\varepsilon\rightarrow
0^{+}}{\textstyle\sum\nolimits_{\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}}}p_{\text{eq}}(\overline{\gamma})e^{\beta_{0}\Delta
W(\delta\mathbf{\bar{z}})}.$

Recalling Eqs. (45a) and (65a), the summand becomes
$p_{\text{eq}}(\left.\delta\mathbf{\bar{z}}^{\prime}\right|\delta\mathbf{\bar{z}})e^{-\beta_{0}E(\delta\mathbf{\bar{z}}^{\prime})}/Z_{\text{in}}(\beta_{0},V_{\text{fin}},\varepsilon)$,
in which $e^{\beta_{0}E(\delta\mathbf{\bar{z}})}$ from
$p_{\text{in,eq}}(\delta\mathbf{\bar{z}})$ exactly cancels with
$e^{\beta_{0}E(\delta\mathbf{\bar{z}})}$ coming from $\Delta
W(\delta\mathbf{\bar{z}})$. As the conditional probability
$p(\left.\delta\mathbf{\bar{z}}^{\prime}\right|\delta\mathbf{\bar{z}})$ is not
affected by $p_{\text{in}}(\delta\mathbf{\bar{z}})$, taking the limit is
trivial as it affects only
$Z_{\text{in}}(\beta_{0},V_{\text{fin}},\varepsilon)$ in accordance with Eq.
(69) so the above summand converges to
$p(\left.\delta\mathbf{\bar{z}}^{\prime}\right|\delta\mathbf{\bar{z}})e^{-\beta_{0}E(\delta\mathbf{\bar{z}}^{\prime})}/Z_{\text{in}}(\beta_{0},V_{\text{in}})$.
Thus,

$\overline{\mathcal{W}}(\left.\beta_{0}\right|\Delta\mathsf{W})={\textstyle\sum\limits_{\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}}}\frac{p_{\text{eq}}(\left.\delta\mathbf{\bar{z}}^{\prime}\right|\delta\mathbf{\bar{z}})e^{-\beta_{0}E(\delta\mathbf{\bar{z}}^{\prime})}}{Z_{\text{in}}(\beta_{0},V_{\text{in}})}.$
(70)

Before proceeding further, we wish to confirm that the inclusion of
microstates in $\Gamma_{\text{diff}}$ does not violate the first law in MNEQT
by focusing on $\left\langle\Delta\mathsf{W}\right\rangle$. We consider a
microstate $\delta\mathbf{\bar{z}}(t)$ along an infinitesimal segment
$d\gamma$ on $\gamma$ between $t>0$ and $t+dt$. Over this segment,
$\mathbf{Z}^{E}(t)=(V,\xi)\rightarrow\mathbf{Z}^{E}(t+dt)=(V+dV,\xi+d\xi)$.
Thus,

$dW(t)=-{\textstyle\sum\limits_{\left\\{\delta\mathbf{\bar{z}}(t)\right\\}}}p(\delta\mathbf{\bar{z}}(t))[E(\mathbf{Z}^{E}(t+dt))-E(\mathbf{Z}^{E}(t))].$

Introducing $dp(t)\doteq$
$p(\delta\mathbf{\bar{z}}(t+dt))-p(\delta\mathbf{\bar{z}}(t))$, and
$dE(t)=E(t+dt)-E(t)$, we have

$\displaystyle dW(t)$
$\displaystyle=-dE(t)+{\textstyle\sum\limits_{\left\\{\delta\mathbf{\bar{z}}(t)\right\\}}}dp(t)E(\mathbf{Z}^{E}(t+dt))$
$\displaystyle\simeq-dE(t)+dQ(t),$

where we have neglected the second-order term $dp(t)dE(\mathbf{Z}^{E}(t))$ as
is a common practice and used Eq. (33a) to identify $dQ$. Thus, for any $t>0$,
we have satisfied Eq. (10a). To consider $t=0$, we need to take the limit
$\varepsilon\rightarrow 0^{+}$, which limits the sum in $dW$ above to
$\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{in}}$. This ensures that
Eq. (10a) remains satisfied at $t=0$ with our modification in Eq. (67). Thus,
we have established that our trick of using
$\forall\delta\mathbf{\bar{z}}\in\boldsymbol{\Gamma}_{\text{fin}}$ is
consistent with the MNEQT. It follows then that the discussion here in the
$\mu$NEQT will finally result in Eq. (61) once we realize that $dE=0$ so that
$dW=TdS$ as in Eq. (60a).

We now consider
$\widehat{\mathcal{W}}(\left.\beta_{0}\right|\Delta\mathsf{W})$, which can be
exactly evaluated. For this, we set
$p_{\text{eq}}(\left.\delta\mathbf{\bar{z}}^{\prime}\right|\delta\mathbf{\bar{z}})=1$
above and use the $1$-to-$1$ mapping
$\mathcal{T}:\delta\mathbf{\bar{z}\rightarrow}\delta\mathbf{\bar{z}}^{\prime}$
to replace the sum over $\delta\mathbf{\bar{z}}$ to a sum over
$\delta\mathbf{\bar{z}}^{\prime}$ to obtain

$\widehat{\mathcal{W}}(\left.\beta_{0}\right|\Delta\mathsf{W})={\textstyle\sum\limits_{\delta\mathbf{\bar{z}}^{\prime}\in\boldsymbol{\Gamma}_{\text{fin}}^{\prime}}}\frac{e^{-\beta_{0}E(\delta\mathbf{\bar{z}}^{\prime})}}{Z_{\text{in}}(\beta_{0},V_{\text{in}})}=\frac{Z_{\text{fin}}(\beta_{0},V_{\text{fin}})}{Z_{\text{in}}(\beta_{0},V_{\text{in}})},$
(71)

which is precisely Eq. (53). Notice that the initial EQ macrostate in proving
the above relation corresponds to the one with all the particles confined in
the left chamber; see, however, KestinNote .

Above, we have considered the case of free expansion. The same trick will also
work if the expansion is gradual and not abrupt. The only difference will be
that $V(t)$ will not be an internal variable as it is controlled externally.
We still will need the trick of inserting ”missing” microstates as above. By
interchanging the role of the initial and final phase spaces above, we can
also use the trick to investigate the case of contraction. The only difference
is that in the last two cases, the system is not isolated so we must make a
distinction between $dW$ (or $\Delta W$) and $d_{\text{i}}W$ (or
$\Delta_{\text{i}}W$).

If it happens that the phase space volume $\left|\Gamma\right|$ continues to
change during a process but
$\left|\Gamma_{\text{in}}\right|=\left|\Gamma_{\text{fin}}\right|$, such as a
cyclic process, we can treat it as a combination of expansion and contractions
processes. To see this, we look for the time $t_{\text{m}}$ when
$\left|\Gamma\right|$ is maximum (minimum). Then, we are dealing with
expansion (contraction) over $t>t_{\text{m}}$, and contraction (expansion)
over $t>t_{\text{m}}$. The same approach of a combination of
expansion/contraction can be taken when $\left|\Gamma\right|$ does not change
monotonically during a process, whether
$\left|\Gamma_{\text{in}}\right|=\left|\Gamma_{\text{fin}}\right|$ or not.

##  VIII Discussion and Conclusion

The current investigation was motivated by a desire to understand the
following two very important aspects of NEQ thermodynamics at the microstate
level

  1. 1.

how to use Hamiltonian trajectories to describe phase space volume changes in
a process to construct a microstate NEQ thermodynamics of a system,
_interacting or not_ , and

  2. 2.

the nature of microworks that give rise to the dissipated work in a
_noninteracting_ , i.e., an isolated system for which no exchange of macroheat
and macrowork is allowed ($\Delta_{\text{e}}Q=0,\Delta_{\text{e}}W=0$),

as these issues have not been addressed in the literature but lie at the heart
of the many common NEQ processes including free expansion taught to
undergraduates in macroscopic thermodynamics. Recently, we have developed a
macroscopic and microscopic SI NEQ thermodynamics (the MNEQT and the
$\mu$NEQT) that directly include the macroforce imbalance
$\mathbf{F}_{\text{t}}$ or the microforce imbalance
$\mathbf{F}_{\text{t,}k}^{\text{w}}$, an important concept introduced recently
by us, to ensure describing an interacting and a noninteracting system within
the same framework. As discussed in Sec. I.2, no other current NEQ
thermodynamics directly captures the microforce imbalance. In its absence
($\mathbf{F}_{\text{t,}k}^{\text{w}}=0,\forall k$), a situation common in
various MFTs noted in Sec. I.2 and discussed in Sec. V.4, there cannot be any
work irreversibility so we cannot overemphasize its importance for any NEQ
processes.

The phase space $\Gamma$ of a classical system with finite parameter
$\mathbf{Z}$ has a finite volume $\left|\Gamma\right|$. In particular, this
requires $\left\\{E_{k}\right\\}$ to have only _finite_ number of values. This
is the case for all numerical simulations so our approach here provides a
useful approach to carry out numerical simulation of a finite system.

Let us consider the following two cases that can arise.

1\. If the volume $\left|\Gamma\right|$ does not change at all during a
process ($\left|\Gamma_{\text{in}}\right|=\left|\Gamma_{\text{fin}}\right|$),
the finite number of microstates $\mathcal{N\doteq}\left|\mathfrak{m}\right|$
also does not change, and one can always use the identity property of the
Hamiltonian trajectory in Eq. (8) to follow the temporal evolution of each
microstate $\mathfrak{m}_{k}$ along $\gamma_{k}$.

2\. When the volume $\left|\Gamma\right|$ changes monotonically during a
process, which is a common situation such as during expansion or contraction
of a classical system, then
$\mathcal{N}_{\text{in}}\mathcal{\doteq}\left|\mathfrak{m}_{\text{in}}\right|\neq\mathcal{N}_{\text{fin}}\mathcal{\doteq}\left|\mathfrak{m}_{\text{fin}}\right|$.
Then these microstates cannot be connected $1$-to-$1$ by Hamiltonian
trajectories. The excess microstates
$\mathcal{N}_{\text{diff}}\doteq\left|\mathfrak{m}_{\text{diff}}\right|\
$between $\left\\{\mathfrak{m}_{\text{in}}\right\\}$ and
$\left\\{\mathfrak{m}_{\text{fin}}\right\\}$ will have no Hamiltonian
trajectories associated with them as seen in Fig. 3 for the case of expansion.
This is when our trick in Sec. VII.2 will be useful. We introduce excess
microstates in the phase space with the smaller number of microstates but with
extremely high positive microenergy shift to ensure that
$\left|\mathfrak{m}\right|$ is the same in the initial and final macrostates.
This trick is crucial and enables us to use only Hamiltonian trajectories to
connect the microstates in $\Gamma_{\text{in}}$ and $\Gamma_{\text{fin}}$ in a
$1$-to-$1$ manner. Eventually, the microenergy of the missing microstates is
taken to diverge to $+\infty$ to ensure their probabilities vanish in the
phase space of smaller volume. However, if $\left|\Gamma\right|$ changes
nonmonotonically during the process even if
$\left|\Gamma_{\text{in}}\right|=\left|\Gamma_{\text{fin}}\right|$, then, as
discussed in Sec. VII.2, we are dealing with a combination of expansion and
contraction so it belongs to this case.

The above argument suggests that when phase space volume $\left|\Gamma\right|$
is not finite, the Hamiltonian trajectories will not work, since there is no
way to argue that $\mathcal{N}_{\text{in}}\neq\mathcal{N}_{\text{fin}}$ when
both are infinitely large. This is not correct. Let us assume that
$\left|\mathfrak{m}\right|$ is infinitely large but the set
$\left\\{\mathfrak{m}_{k}\right\\}$ is denumerable as is the case with the
quantum expansion in Sec. VII.1. We still have $1$-to-$1$ Hamiltonian
trajectories connecting $\mathfrak{m}_{k\text{,in}}\ $with
$\mathfrak{m}_{k\text{,fin}}$,$\forall k\in\mathbb{N}$, where $\mathbb{N}$ is
the set of natural numbers, i.e. {1,2,3,…}. It is this property that allows us
to introduce the $\mu$NEQT in Sec. IV and the MGF in Sec. V. This is true
despite the sets $\left\\{\mathfrak{m}_{k\text{,in}}\right\\}$ and
$\left\\{\mathfrak{m}_{k\text{,fin}}\right\\}$ having ”equal” but infinite
large number of microstates so we can say that
$\left|\mathfrak{m}_{\text{in}}\right|\equiv\left|\mathfrak{m}_{\text{fin}}\right|$.

In the classical expansion/contraction, the equality of the numbers of
microstates is clearly not valid despite the fact that they both have the same
cardinality as that of $\mathbb{N}$. What is important for the cardinality
consideration is the idea of association and not the unique $1$-to-$1$
mapping. This is the same when we compare $\mathbb{N}$ with the set
$\mathbb{N}_{\text{o}}$ of odd natural numbers, i.e. {1,3,5,…} or the set
$\mathbb{N}_{\text{e}}$ of even natural numbers, i.e. {2,4,6,…}. All these
sets have the same cardinality, but $\mathbb{N}$ has twice as many members as
either of the sets $\mathbb{N}_{\text{o}}$ or $\mathbb{N}_{\text{e}}$. It is
the number of members that is important in the mapping in Eq. (8).

To understand it, we proceed as follows. Think of $\mathbb{N}$ as an example
of the set of microstates in the interior of the dotted ellipse
$\Gamma_{\text{fin}}$, $\mathbb{N}_{\text{o}}$ as an example of the set of
microstates in the interior of the solid ellipse $\Gamma_{\text{in}}$, and
$\mathbb{N}_{\text{e}}$ as an example of the set of microstates in the
difference $\boldsymbol{\Gamma}_{\text{diff}}$ of the two ellipses
$\Gamma_{\text{fin}}$ and $\Gamma_{\text{in}}$. We can similarly introduce
$\mathbb{N}^{\prime}$ for the interior of the solid ellipse
$\Gamma_{\text{fin}}^{\prime}$, $\mathbb{N}_{\text{o}}^{\prime}$ for the
interior of the broken ellipse $\Gamma_{\text{in}}^{\prime}$, and
$\mathbb{N}_{\text{e}}^{\prime}$ for the difference
$\boldsymbol{\Gamma}_{\text{diff}}^{\prime}$. Under the $1$-to-$1$ mapping,
$\mathbb{N}_{\text{o}}$ goes to $\mathbb{N}_{\text{o}}^{\prime}$, with no
mapping leading to $\mathbb{N}_{\text{e}}^{\prime}$. This clearly shows that
the concept of cardinality is irrelevant as noted in Sec. II.4. Now we can use
our trick introducing excess microstates to ensure
$\mathcal{N}_{\text{in}}=\mathcal{N}_{\text{fin}}$. Using the MGF
$\mathcal{W}$, we have also established that the trick does not destroy the
consistency of the $\mu$NEQT with the MNEQT so the trick is thermodynamically
consistent. It is clear that the trick is very general and will work in all
cases of _missing_ microstates.

The most important signature of a NEQ process is the existence of
(generalized) thermodynamic force $\mathbf{F}_{\text{t}}$, see Eq. (19), which
drives the system towards EQ. It is common to study a NEQ isothermal process
($T=T_{0}\Rightarrow F_{\text{t}}^{\text{h}}=0$) so we must look for a
nonvanishing $\mathbf{F}_{\text{t}}^{\text{w}}$ in this case, regardless of
whether the system is isolated or interacting. In the former case,
$\mathbf{F}_{\text{t}}^{\text{w}}$ reduces to $\mathbf{A}$ as seen in Eq.
(20), and contains $\mathbf{A}$ in the latter case. The presence of
$\mathbf{F}_{\text{t}}^{\text{w}}$ in a NEQ system, including the free
expansion that we are interested in, is necessary along with the presence of