summarize this as a conclusion Prigogine :

###### Conclusion 39

To have dissipation, it is necessary and sufficient to have a nonzero
thermodynamic force. In its absence, there can be no dissipation.

We now prove one of the central results in the MNEQT in the following theorem.

###### Theorem 40

The proportionality parameter $T$ in Eq. (5a) or (62) satisfies all the
criteria (C1-C4) of a sensible temperature. Therefore, we identify $T$ as the
temperature of the system in any arbitrary macrostate
$\mathcal{M}_{\text{arb}}$.

Proof. As $dQ$ and $dS$ scale the same way with the size of $\Sigma$, $T$ is
an intensive quantity. When the entropy is a state function in
$\mathfrak{S}_{\mathbf{Z}}$ or
$\mathfrak{S}_{\mathbf{Z}^{\prime}}\supset\mathfrak{S}_{\mathbf{Z}}$, we have
a Gibbs fundamental relation given in Eq. (71a). So the temperature is defined
by a derivative in $\mathfrak{S}_{\mathbf{Z}}$ or
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$, the latter giving $T_{\text{arb}}$. This
shows that C1 is satisfied for any $\mathcal{M}_{\text{arb}}$. As we have not
imposed any restrictions on the signs of $dQ(t)$ and $dS(t)$, the parameter
$T(t)$ can be of any sign, which shows that C2 is satisfied. To demonstrate
consistency with the second law, we rewrite the top equation in Eq. (76b) to
express $d_{\text{i}}S$ as a sum of two independent contributions

$\displaystyle d_{\text{i}}S^{\text{Q}}(t)$
$\displaystyle\doteq(1/T-1/T_{0})d_{\text{e}}Q(t),$ (86) $\displaystyle
d_{\text{i}}S^{\text{W}}(t)$ $\displaystyle\doteq d_{\text{i}}W(t)/T\equiv
d_{\text{i}}Q(t)/T,$

so that

$d_{\text{i}}S(t)=d_{\text{i}}S^{\text{Q}}(t)+d_{\text{i}}S^{\text{W}}(t);$
(87)

here, $d_{\text{i}}S^{\text{Q}}(t)$ is generated solely by exchange macroheat
$d_{\text{e}}Q(t)$ at different temperatures, and
$d_{\text{i}}S^{\text{W}}(t)$ by the irreversible macrowork or macroheat
$d_{\text{i}}W(t)\equiv d_{\text{i}}Q(t)$. The two contributions are
_independent_ of each other. Accordingly, both contributions individually must
be nonnegative in accordance with the second law. In particular, the
inequality

$d_{\text{i}}S^{\text{Q}}(t)\geq 0.$ (88)

For $T(t)>T_{0}$, $d_{\text{e}}Q(t)<0$ so that the macroheat flows from the
system to the medium. For $T(t)<T_{0}$, $d_{\text{e}}Q(t)>0$ so that the
macroheat flows from the medium to the system. This establishes that C3 is
satisfied. As $dQ(t)$ and $dS(t)$ are global quantities, the parameter $T(t)$
is also a global parameter, which means that C4 is also satisfied. This proves
the theorem.

Because of the importance of C4, we give many example in Sect. X to justify
that $T$ acts as a global temperature of the system even if it is composite
with different temperatures. These examples leave no doubt that C4 is
satisfied.

We conclude this subsection by considering a special case, also studied by
Landau and Lifshitz (Landau, , Sect. 13 and specifically Eq. (13.4)): it deals
with the irreversibility generated _only by macroheat exchange_ _at different
temperatures but no internal (macrowork) dissipation_. It follows from Eq.
(87) that $d_{\text{i}}W=0$ in this case, even though there is irreversibility
($d_{\text{i}}S(t)>0$) due to the macroheat exchange. If there are internal
variables also, then $\mathbf{A=A}_{0}=0$ to ensure $d_{\text{i}}W=0$. This
example is important in that it shows that just because there is
irreversibility in the system, we do not have $d_{\text{i}}W(t)\equiv
d_{\text{i}}Q(t)\neq 0$. We see from Eq. (87) that
$d_{\text{i}}S(t)=d_{\text{i}}S^{\text{Q}}(t)$, which can be rewritten, using
$d_{\text{e}}Q(t)=T_{0}d_{\text{e}}S(t)$, as

$d_{\text{e}}Q(t)=T(t)dS(t)=T_{0}d_{\text{e}}S(t),$ (89)

a result also derived by Landau and Lifshitz; note that they use $dQ$ for
$d_{\text{e}}Q$. For $d_{\text{i}}Q=0$, we have $dQ(t)=d_{\text{e}}Q(t)$ so
that Eq. (89) is consistent with Eq. (5a), as it must. In the presence of
nonzero $d_{\text{i}}Q$, Eq. (89) gets modified: one must subtract
$d_{\text{i}}Q$ from the right side.

###  VI.8 Cyclic Process

For a general body that is not isolated, the concept of its internal
equilibrium state plays a very important role in that the body can come back
to this macrostate $\mathcal{M}_{\text{ieq}}$ several times in a
nonequilibrium process. In a cyclic nonequilibrium process, such a macrostate
can repeat itself in time after some cycle time $\tau_{\text{c}}$ so that all
state variables and functions including the entropy repeat themselves:

$\mathbf{Z}(t+\tau_{\text{c}})=\mathbf{Z}(t),~{}\mathcal{M}(t+\tau_{\text{c}})=\mathcal{M}(t),~{}S(t+\tau_{\text{c}})=S(t).$

This ensures

$\Delta_{\text{c}}S\equiv S(t+\tau_{\text{c}})-S(t)=0$

in a cyclic process. All that is required for the cyclic process to occur is
that the body must start and end in the same internal equilibrium state;
however, during the remainder of the cycle, the body need not be in internal
equilibrium.

The same argument also applies to a cyclic process that starts and returns to
$\mathcal{M}_{\text{eq}}$ after some cycle time $\tau_{\text{c}}$. However,
the body need not be in EQ macrostates during the rest of the cycle. We will
consider such a case when we consider a NEQ Carnot cycle in Sec. XII.

###  VI.9 Steady State

Consider a system between two different media as shown in Fig. 2. For example,
we can consider $\Sigma$ between two heat baths
$\widetilde{\Sigma}_{\text{h}}$ and $\widetilde{\Sigma}_{\text{h}}$ replacing
the two media in the figure. We will study this example in the MNEQT in Sec.
X.3 using a composite system $\Sigma$ between the two heat sources. In the
presence of two media, it is possible for $\Sigma$ to reach a steady state, in
which it satisfies conditions similar to that for a cyclic process above in
terms of MI-macroquantities:

$d\mathbf{Z}=0,~{}dS=0,$ (90a) where the changes are over the system $\Sigma$.
The above conditions in the MNEQT lead to important relations between exchange
and irreversible macroquantities:
$d_{\text{i}}\mathbf{Z}=-d_{\text{e}}\mathbf{Z},~{}d_{\text{i}}S=-d_{\text{e}}S;$
(90b) as usual, the irreversible contributions satisfy the second law
inequalities. For $E$, we have from Eq. (23a)  $dQ=dW=0,$

which follows from $dQ=TdS=0$ for $\mathcal{M}_{\text{ieq}}$ in
$\mathfrak{S}_{\mathbf{Z}}$ or for $\mathcal{M}_{\text{nieq}}$ in
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$. Therefore, in the MNEQT,

$d_{\text{i}}Q=-d_{\text{e}}Q\geq 0,d_{\text{i}}W=-d_{\text{e}}W\geq 0.$

As a consequence,

$d_{\text{e}}Q=d_{\text{e}}W\leq 0,$

a result that cannot be derived in the M̊NEQT by using the first law in Eq.
(23b).

It should be noted, as said earlier in Sect. I, that the steady state occurs
only over a short period $\tau\sim\tau_{\text{st}}$ compared to the time
$\tau_{\text{EQ}}$ required for the two media to equilibrate with each other.
The latter time period is extremely large compared to $\tau_{\text{st}}$
because of their extreme sizes. For a time period longer than
$\tau_{\text{st}}$, the steady state cannot be treated as steady as $\Sigma$
will begin the equilibrium process between them so that eventually at
$\tau_{\text{EQ}}$, $d_{\text{i}}S$ will vanish as $d_{\text{e}}Q\rightarrow
0$. We will not consider this possibility here, but can be studied in the
MNEQT.

###  VI.10 Intrinsic Adiabaticity Theorem

We now have a clear statement of the generalization of the adiabatic theorem
Landau  for nonequilibrium processes going on in a body in an arbitrary
macrostate in terms of the intrinsic quantity $dS$. We will call it the
_intrinsic adiabatic theorem_.

###### Definition 41

Intrinsic Adiabatic Process: _An intrinsic adiabatic process is an isentropic
process (_ $dS(t)=0$_and not necessarily_ $d_{\text{e}}S(t)=0$_)._

Such a process also includes the stationary limit, i.e. the steady macrostate
of a non-equilibrium process discussed in the previous section. However, the
extension goes beyond the conventional notion of an adiabatic process commonly
dealt with in the M̊NEQT, according to which an adiabatic process Landau  is
one for which $d_{\text{e}}Q(t)=0$, which is equivalent to
$d_{\text{e}}S(t)=0$. If $d_{\text{i}}S(t)=0$, it also represents a reversible
process in a thermally isolated system so that $d_{\text{e}}Q(t)=0$. One can
also have $dS(t)=0$ in an irreversible process during which

$d_{\text{i}}S(t)=-d_{\text{e}}S(t)>0;$ (91)

as usual, Eq. (22) always remains satisfied. If the system is in a
$\mathcal{M}_{\text{arb}}$, then we must also have

$d_{\text{i}}Q(t)=-d_{\text{e}}Q(t)=T_{0}d_{\text{i}}S(t)>0;$ (92)

recall Eq. (62) for a $\mathcal{M}_{\text{arb}}$.

###### Theorem 42

In an intrinsic adiabatic process, the sets of microstates and of their
probabilities $p_{k}$ do not change, but
$d_{\text{e}}p_{k}=-d_{\text{i}}p_{k}\neq 0$ for all $k$.

Proof. In terms $d_{\text{e}}p_{k}$ and $d_{\text{i}}p_{k}$, Eqs. (91) and
(92) become

$\displaystyle\sum_{k}\eta_{k}d_{\text{i}}p_{k}$
$\displaystyle=-\sum_{k}\eta_{k}d_{\text{e}}p_{k},$ (93a)
$\displaystyle\sum_{k}E_{k}d_{\text{i}}p_{k}$
$\displaystyle=-\sum_{k}E_{k}d_{\text{e}}p_{k}.$ (93b) Recognizing that there
is only macrowork in $dE$, which requires $p_{k}$ not to change, we conclude
that  $dp_{k}=0\text{ \ for }\forall k$

in an adiabatic process. As $d_{\text{i}}S(t)$ does not vanish in an
irreversible process, $d_{\text{i}}p_{k}(t)$ cannot vanish. Accordingly,
$d_{\text{e}}p_{k}=-d_{\text{i}}p_{k}\neq 0,\forall k$ for an irreversible
adiabatic process. The conditions in Eq. (108) remain valid as expected. As
$p_{k}$’s do not change, no microstate can appear or disappear. This proves
the theorem.

##  VII Clausius Equality

We recall that Eq. (5a), which we call the _Clausius equality_ , follows from
the Gibbs fundamental equation for $\mathcal{M}_{\text{ieq}}$ in
$\mathfrak{S}_{\mathbf{Z}}$ or $\mathcal{M}_{\text{nieq}}$ in
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$, see Eq. (71b). It is merely is a
consequence of the state function $S$ for a $\mathcal{M}_{\text{ieq}}$ or
$\mathcal{M}_{\text{nieq}}$ in respective state spaces so the equality is also
valid for any $\mathcal{M}_{\text{arb}}$, see Eq. (62). Here, we are only
concerned with some $\mathcal{M}_{\text{ieq}}$. The equality is very
interesting, and should be contrasted with the _Clausius inequality_

$d_{\text{e}}Q\leq T_{0}dS.$ (94)

First, it follows from Eq. (5a) that $dQ/T$ is nothing but the _exact
differential_ $dS$ for $\mathcal{M}_{\text{ieq}}$ so that

$\oint dQ(t)/T(t)\equiv 0$ (95)

for any cyclic process; here we have added the time argument for clarity. It
is only because of the use of $dQ(t)$ in place of $d_{\text{e}}Q(t)$ that the
Clausius inequality has become an equality. The equality should not be
interpreted as the absence of irreversibility ($\Delta_{\text{i}}S>0$) as is
clear from Eq. (96) obtained by using $d_{\text{i}}S(t)\equiv
dS(t)-d_{\text{e}}S(t)$for a cyclic process taking time $\tau$:

$N(t,\tau)\equiv{\textstyle\oint}d_{\text{i}}S(t)=-{\textstyle\oint}d_{\text{e}}Q(t)/T_{0}\geq
0,$ (96)

which is the second law for a cyclic process, and represents the irreversible
entropy generated in a cycle. This is the original Clausius inequality. The
quantity $N(t)$ is the _uncompensated transformation_ of Clausius Prigogine
that is directly related to $d_{\text{i}}S(t)$ Eu ; in contrast,
$N_{0}(t,\tau)$

$N_{0}(t,\tau)\equiv{\textstyle\oint}d_{\text{i}}Q(t)/T(t)\equiv{\textstyle\oint}d_{\text{i}}W(t)/T(t)\geq
0,$ (97)

where we have used the fundamental identity in Eq. (22), is determined by the
irreversible __ macroheat $d_{\text{i}}Q(t)$ or the irreversible __ macrowork
$d_{\text{i}}W(t)$, and represents a different quantity as is evident. In
terms of the two macroheats, we have

${\textstyle\oint}d_{\text{i}}Q(t)/T(t)=-{\textstyle\oint}d_{\text{e}}Q(t)/T(t)\geq
0,$ (98)

which results in a new Clausius inequality __

${\textstyle\oint}d_{\text{e}}Q(t)/T(t)\leq 0;$ (99)

compare with the original Clausius inequality in Eq. (96).

##  VIII Extended State Space and $\mathcal{M}_{\text{nieq}}$

###  VIII.1 Choice of $\mathbf{Z}$

We first discuss how to choose a particular state space for a unique
description of a macrostate $\mathcal{M}$ depending on the experimental setup.
To understand the procedure for this, we begin by considering a set
$\boldsymbol{\xi}_{n}$ of internal variables __
$\left(\xi_{1},\xi_{2},\cdots,\xi_{n}\right)$ and
$\mathbf{Z}_{n}\doteq\mathbf{X}\cup\boldsymbol{\xi}_{n}$ to form a sequence of
state spaces $\mathfrak{S}_{\mathbf{Z}}^{(n)}$. In general, one may need many
internal variables, with the value of $n$ increasing as $\mathcal{M}$ is more
and more out of EQ Gujrati-Hierarchy  relative to $\mathcal{M}_{\text{eq}}$.
We will take $n^{\ast}$ to be the maximum $n$ in this study, even though $n$
$<<n^{\ast}$ needed for $\mathfrak{S}_{\mathbf{Z}}^{(n)}$ will usually be a
small number in most cases. We refer to Sect. VI.4, where the choice of
$n^{\ast}$ is determined by the mathematical identity in Eq. (74b) in
$\mathfrak{S}_{\mathbf{Z}}^{(n^{\ast})}$. The two most important but distinct
time scales are $\tau_{\text{obs}}$, the time to make observations, and
$\tau_{\text{eq}}$, the equilibration time for a macrostate $\mathcal{M}$ to
turn into $\mathcal{M}_{\text{eq}}$. For $\tau_{\text{obs}}<\tau_{\text{eq}}$,
the system will be in a NEQ macrostate. Let $\tau_{i}\ $denote the relaxation
time of $\xi_{i}$ needed to come to its equilibrium value so that its affinity
$A_{i}\rightarrow 0$ deGroot ; Prigogine ; Gujrati-Hierarchy ; Prigogine0 ;
Gutzow ; Nemilov . For convenience, we order $\xi_{i}$ so that

$\tau_{1}>\tau_{2}>\cdots;$

we assume distinct $\tau_{i}$’s for simplicity without affecting our
conclusions. For $\tau_{1}<\tau_{\text{obs}}$, all internal variables have
equilibrated so they play no role in equilibration except thermodynamic forces
$T-T_{0},P-P_{0}$, etc. associated with $\mathbf{X}$ that still drive the
system towards EQ. We choose $n$ satisfying
$\tau_{n}>\tau_{\text{obs}}>\tau_{n+1}$ so that all of
$\xi_{1},\xi_{2},\cdots,\xi_{n}$ have not equilibrated (their affinities are
nonzero). They play an important role in the NEQT, while
$\xi_{n+1},\xi_{n+2},\cdots$ need not be considered as they have all
equilibrated. This specify $\mathcal{M}$ _uniquely_ in
$\mathfrak{S}_{\mathbf{Z}}^{(n)}$, which was earlier identified as in IEQ.

Note that NEQ macrostates with $\tau_{n+1}>\tau_{\text{obs}}>\tau_{n+2}$ are
not uniquely identifiable in $\mathfrak{S}_{\mathbf{Z}}^{(n)}$, even though
they are uniquely identifiable in $\mathfrak{S}_{\mathbf{Z}}^{(n+1)}$. Thus,
there are many NEQ macrostates that are not unique in
$\mathfrak{S}_{\mathbf{Z}}^{(n)}$. The unique macrostates
$\mathcal{M}_{\text{ieq}}$ are special in that its Gibbs entropy
$S(\mathbf{Z}_{n})$ is a state function of $\mathbf{Z}_{n}$ in
$\mathfrak{S}_{\mathbf{Z}}^{(n)}$. Thus, given $\tau_{\text{obs}}$, we look
for the window $\tau_{n}>\tau_{\text{obs}}>\tau_{n+1}$ to choose the
particular value of $n$. This then determines
$\mathfrak{S}_{\mathbf{Z}}^{(n)}$ in which the macrostates are in IEQ. From
now onward, we assume that $n$ has been found and
$\mathfrak{S}_{\mathbf{Z}}^{(n)}$ has been identified. We now suppress $n$ and
simply use $\mathfrak{S}_{\mathbf{Z}}$ below.

###### Remark 43

The linear sizes of various subsystems introduced in Sect. VI.4 must be larger
than the correlation length $\lambda_{\text{corr}}$ as discussed elsewhere
Gujrati-II  for the first time, and briefly revisited in Sect. II to ensure
entropy additivity; see also Sect. XV. Therefore, it is usually sufficient to
take the linear size of $\Sigma$ to be a small multiple (for example, $10$ to
$20$) of the correlation length to obtain a proper thermodynamics, which is
extensive. This means that we will usually need a theoretically manageable but
small number of internal variables $n$.

###  VIII.2 Microstate probabilities for $\mathcal{M}_{\text{ieq}}$

As $\mathcal{M}_{\text{ieq}}$ is unique in $\mathfrak{S}_{\mathbf{Z}}$, we
need to identify the unique set $\left\\{p_{k}\right\\}$. If we keep
$\mathbf{W}$ fixed in $\mathcal{M}_{\text{ieq}}$ as the parameter, then
$\mathbf{F}_{\text{w}k}$ are fluctuating microforces in
$\mathfrak{S}_{\mathbf{Z}}$ as we have seen in Sect. (V.2). In additions, we
have microstate energies $E_{k}$ also fluctuating. We need to maximize the
entropy $S(\mathbf{Z})$ at fixed

$E=\sum_{k}E_{k}p_{k},\mathbf{F}_{\text{w}}=\sum_{k}\mathbf{F}_{\text{w}k}p_{k}$

by varying $p_{k}$ without changing $\left\\{\mathfrak{m}_{k}\right\\}$ , i.e.
$E_{k}$ and $\mathbf{F}_{\text{w}k}$. This variation has nothing to do with
$dp_{k}$ in a physical process. Using the Lagrange multiplier technique, it is
easy to show that the condition for this in terms of three Lagrange
multipliers with obvious definitions is

$\eta_{k}=\lambda_{1}+\lambda_{2}E_{k}+\boldsymbol{\lambda}_{3}\cdot\mathbf{F}_{\text{w}k},$
(100)

from which follows the statistical entropy
$\mathcal{S}=-(\lambda_{1}+\lambda_{2}E+\boldsymbol{\lambda}_{3}\cdot\mathbf{F}_{\text{w}})$;
we have reverted back to the original symbol for the statistical entropy here.
It is now easy to identify
$\lambda_{2}=-\beta,\boldsymbol{\lambda}_{3}=-\beta\mathbf{W}$ by comparing
$d\mathcal{S}$ with $dS$ in Eq. (71a) by varying $E$ and $\mathbf{W}$ so we
finally have

$p_{k}=\exp[\beta(\widehat{G}-E_{k}-\mathbf{W}\cdot\mathbf{F}_{\text{w}k})],$
(101)

where $\lambda_{1}=\beta\widehat{G}$ with $\widehat{G}(t)$ is a normalization
constant and defines a NEQ partition function

$\exp(-\beta\widehat{G})\equiv\sum_{k}\exp[-\beta(E_{k}+\mathbf{W}\cdot\mathbf{F}_{\text{w}k})].$
(102a) It is easy to verify that
$\widehat{G}(T,\mathbf{W})=E+\mathbf{W}\cdot\mathbf{F}_{\text{w}}-TS,$ (102b)
so that if we neglect the fluctuations $E_{k}-E$ and
$\mathbf{F}_{\text{w}k}-\mathbf{F}_{\text{w}}$ and replace $E_{k}$ by $E$ and
$\mathbf{F}_{\text{w}k}$ by $\mathbf{F}_{\text{w}}$, then $p_{k}$ reduces to
the flat distribution
$p_{k}=1/W(E,\mathbf{W})=\exp[\beta(\widehat{G}-E-\mathbf{W}\cdot\mathbf{F}_{\text{w}})]=\exp(-S)$
in Remark 21, which can be identified as the microstate probability in the NEQ
microcanonical ensemble. It should be stressed that this is consistent with
the well-known fact that thermodynamics does not describe fluctuations; the
latter require using statistical mechanics Landau .

It should be remarked that the Lagrange multipliers in $p_{k}$ are determined
by comparing the resulting entropy to match exactly the Gibbs fundamental
relation, a thermodynamic relation. This then proves that $\mathcal{S}$ is the
same as the thermodynamic entropy $S$ up to a constant Gujrati-Entropy2 ,
which can be fixed by appeals to the third law, according to which $S$
vanishes at absolute zero. We avoid considering here the issue of a residual
entropy, which is discussed elsewhere Gujrati-ResidualEntropy ; Gujrati-
Hierarchy . The $p_{k}$ above clearly shows the effect of irreversibility and
is very different from its equilibrium analog $p_{k}^{\text{eq}}$

$p_{k}^{\text{eq}}=\exp[\beta_{0}(\widehat{G}(T_{0},\mathbf{w})-E_{k}-\mathbf{w}\cdot\mathbf{f}_{\text{w}k})],$

see Eq. (28), obtained by replacing $\mathbf{W}$ by $\mathbf{w}$,
$\mathbf{F}_{\text{w}k}$ by $\mathbf{f}_{\text{w}0k}$, and $\beta$ by
$\beta_{0}$. The fluctuating $E_{k},\mathbf{f}_{\text{w}k}$ satisfy

$E=\sum_{k}E_{k}p_{k}^{\text{eq}},\mathbf{f}_{\text{w}0}=\sum_{k}\mathbf{f}_{\text{w}0k}p_{k}^{\text{eq}}.$

The observation time $\tau_{\text{obs}}$ is determined by the way $T$ and
$\mathbf{W}$ are changed during a process. Thus, during each change,
$\tau_{\text{obs}}$ must be compared with the time needed for $\Sigma$ to come
to the next IEQ macrostate, and for the microstate probabilities to be given
by Eq. (101) with the new values of $T$ and $\mathbf{W}$.

###  VIII.3 $\mathcal{M}_{\text{nieq}}$ in $\mathfrak{S}_{\mathbf{Z}}$

We now focus on a non-unique macrostate $\mathcal{M}_{\text{nieq}}$ in
$\mathfrak{S}_{\mathbf{Z}}$. This will be needed if $\tau_{\text{obs}}$ is
reduced to make the process faster so that instead of falling in the window
($\tau_{n},\tau_{n+1}$), it now falls in a higher window such as
($\tau_{n+1},\tau_{n+2}$). As said above, $\mathcal{M}$ can now be treated as
a unique macrostate in a larger state space
$\mathfrak{S}_{\mathbf{Z}^{\prime}}\supset\mathfrak{S}_{\mathbf{Z}}$. Let
$\boldsymbol{\xi}^{\prime}(t)$ denote the set of additional internal variables
needed over $\mathfrak{S}_{\mathbf{Z}}$ so that

$\mathbf{Z}^{\prime}(t)=(\mathbf{Z}(t),\boldsymbol{\xi}^{\prime}(t)).$

The entropy $S(\mathbf{Z}^{\prime}(t))=S(\mathbf{Z}(t),t)$ for
$\mathcal{M}_{\text{ieq}}(t)$ in $\mathfrak{S}_{\mathbf{Z}^{\prime}}$
satisfies the Gibbs fundamental relation

$dS(\mathbf{Z}^{\prime}(t))\mathbf{=}\frac{\partial S}{\partial
E}dE\mathbf{+}\frac{\partial S}{\partial\mathbf{W}}\cdot
d\mathbf{W+}\frac{\partial S}{\partial\boldsymbol{\xi}^{\prime}}\cdot
d\boldsymbol{\xi}^{\prime},$

where $\mathbf{W}$ is the work variable in $\mathfrak{S}_{\mathbf{Z}}$.
Expressing the last term as

$\frac{\partial
S}{\partial\boldsymbol{\xi}^{\prime}}\cdot\frac{d\boldsymbol{\xi}^{\prime}}{dt}dt,$

we obtain the following generalization of the Gibbs fundamental relation for
$\mathcal{M}_{\text{nieq}}(t)$ in $\mathfrak{S}_{\mathbf{Z}}$:

$dS(\mathbf{Z}(t),t)=\frac{\partial S}{\partial E}dE\mathbf{+}\frac{\partial
S}{\partial\mathbf{W}}\cdot d\mathbf{W+}\frac{\partial S}{\partial t}dt,$
(103a) where $\frac{\partial S}{\partial t}\doteq\frac{\partial
S}{\partial\boldsymbol{\xi}^{\prime}}\cdot\frac{d\boldsymbol{\xi}^{\prime}}{dt}\geq
0.$ (103b) In $\mathfrak{S}_{\mathbf{Z}^{\prime}}$, we can identify the
temperature $T$ as the thermodynamic temperature in
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$ by the standard definition. But, it is
clear from the above discussion that $\partial
S(\mathbf{Z}^{\prime}(t))/\partial E$ in $\mathfrak{S}_{\mathbf{Z}^{\prime}}$
has the same value as $\partial S(\mathbf{Z}(t),t)/\partial E$ in
$\mathfrak{S}_{\mathbf{Z}}$. Therefore, we are now set to identify
$T_{\text{arb}}$ in Eq. (62) as a thermodynamic temperature.

###### Remark 44

$T_{\text{arb}}$ in Eq. (62) in $\mathfrak{S}_{\mathbf{Z}}$ is identified by
the same derivative in the Gibbs fundamental relation in
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$ as follows

$\beta_{\text{arb}}=1/T_{\text{arb}}=\partial
S(\mathbf{Z}^{\prime}(t))/\partial E=\partial S(\mathbf{Z}(t),t)/\partial E.$
(104)

###### Definition 45

As the presence of $\partial S/\partial t$ above in
$\mathfrak{S}_{\mathbf{Z}}$ is due to ”hidden” internal variables in
$\boldsymbol{\xi}^{\prime}$, we will call it the _hidden entropy generation
rate_ , and

$d_{\text{i}}S^{\text{hid}}(t)=\frac{\partial S}{\partial t}dt=\frac{\partial
S}{\partial\boldsymbol{\xi}^{\prime}}\cdot d\boldsymbol{\xi}^{\prime}\geq 0,$
(105a) the _hidden entropy generation_. It results in a _hidden irreversible
macrowork_ $d_{\text{i}}W^{\text{hid}}\doteq
Td_{\text{i}}S^{\text{hid}}=\mathbf{A}^{\prime}\cdot
d\boldsymbol{\xi}^{\prime},$ (105b) in $\mathfrak{S}_{\mathbf{Z}}$ due to the
hidden internal variable with affinity $\mathbf{A}^{\prime}$.

###### Remark 46

A macrostate $\mathcal{M}_{\text{nieq}}(t)$ with
$\mathcal{S}(\mathbf{Z}(t),t)$ can be converted to
$\mathcal{M}_{\text{ieq}}(t)$ with a state function
$\mathcal{S}(\mathbf{Z}^{\prime}(t))$ in an appropriately chosen state space
$\mathfrak{S}_{\mathbf{Z}^{\prime}}\supset\mathfrak{S}_{\mathbf{Z}}$ by
finding the appropriate window in which $\tau_{\text{obs}}$ lies. The needed
additional internal variable $\boldsymbol{\xi}^{\prime}$ determines the hidden
entropy generation rate $\partial S/\partial t$ in Eq. (103b) due to the non-
IEQ nature of $\mathcal{M}_{\text{nieq}}(t)$ in $\mathfrak{S}_{\mathbf{Z}}$,
and ensures validity of the Gibbs relation in Eq. (103a) for it, thereby
providing not only a new interpretation of the temporal variation of the
entropy due to hidden variables but also extends the MNEQT to
$\mathcal{M}_{\text{nieq}}(t)$ in $\mathfrak{S}_{\mathbf{Z}}$.

The above discussion strongly points towards the possible

###### Proposition 47

The MNEQT provides a very general framework to study any
$\mathcal{M}_{\text{nieq}}(t)$ in $\mathfrak{S}_{\mathbf{Z}}$, since it can be
converted into a $\mathcal{M}_{\text{ieq}}(t)$ in an appropriately chosen
state space $\mathfrak{S}_{\mathbf{Z}^{\prime}}$, with
$d_{\text{i}}S^{\text{hid}}(t)$ originating from hidden internal variable
$\boldsymbol{\xi}^{\prime}$.

###### Remark 48

In a process $\mathcal{P}$ resulting in $\mathcal{M}_{\text{nieq}}(t)$ in
$\mathfrak{S}_{\mathbf{Z}}$, it is natural to assume that the terminal
macrostates in $\mathcal{P}$ are $\mathcal{M}_{\text{ieq}}$ so the affinity
corresponding to $\boldsymbol{\xi}^{\prime}$ must vanish in them.

###### Remark 49

By replacing $\mathbf{Z}$ by $\mathbf{X}$, and $\mathbf{Z}^{\prime}$ by
$\mathbf{Z}$, we can also express the Gibbs fundamental relation for any NEQ
macrostate in $\mathfrak{S}_{\mathbf{X}}$ as

$dS(\mathbf{X}(t),t)=\frac{\partial S}{\partial E}dE\mathbf{+}\frac{\partial
S}{\partial\mathbf{w}}\cdot d\mathbf{w+}\frac{\partial S}{\partial t}dt,$
(106)

by treating $\mathcal{M}$ as $\mathcal{M}_{\text{ieq}}$ in
$\mathfrak{S}_{\mathbf{Z}}$. In a NEQ process $\overline{\mathcal{P}}$ between
two EQ macrostates but resulting in $\mathcal{M}_{\text{ieq}}(t)$ between them
in $\mathfrak{S}_{\mathbf{Z}}$, the affinity corresponding to
$\boldsymbol{\xi}$ must vanish in the terminal EQ macrostates of
$\overline{\mathcal{P}}$.

Eq. (106) proves extremely useful to describe $\mathcal{M}$ in
$\mathfrak{S}_{\mathbf{X}}$ as it may not be easy to identify
$\boldsymbol{\xi}$ in all cases.

###### Remark 50

The explicit time dependence in the entropy for $\mathcal{M}_{\text{neq}}$ in
$\mathfrak{S}_{\mathbf{X}}$ or $\mathcal{M}_{\text{nieq}}(t)$ in
$\mathfrak{S}_{\mathbf{Z}}$ is solely due to the internal variables, which do
not affect $dQ=dE_{\text{s}}$, Eq. (5a) remains valid, with $T$ defined as the
inverse of $\partial S/\partial E$ at fixed $\mathbf{w},t$ or $\mathbf{W},t$
in the two state spaces, respectively; see Eq. (72).

###  VIII.4 External and Internal Variations of $dp_{k}(t)$

We focus on $\mathcal{N}_{k}$ in Sect. IV.2, and partition its change
$d\mathcal{N}_{k}$ in accordance with the micropartition rule; see Definition
12. We take $\mathcal{N}$ fixed. Then, the macropartition results in the
partition for $dp_{k}$ given in Eq. (17a), where $d_{\text{e}}p_{k}$ is the
change due to exchanges with the medium and $d_{\text{i}}p_{k}$ the change due
to internal processes. It follows from the partition that

$d_{\text{e}}Q(t)\equiv{\textstyle\sum\nolimits_{k}}E_{k}d_{\text{e}}p_{k}(t),\
d_{\text{i}}Q(t)\equiv{\textstyle\sum\nolimits_{k}}E_{k}d_{\text{i}}p_{k}(t),$
(107)

where we have replaced $d$ by $d_{\alpha}$ in Eq. (61). As $d_{\alpha}Q(t)$
are thermodynamic quantities, they must not change their values if we change
$E_{k}$ by adding a constant to $\mathcal{H}$. This requires

${\textstyle\sum\nolimits_{k}}d_{\alpha}p_{k}(t)=0,\forall\alpha,$ (108)

and put a limitation on the possible variations $d_{\alpha}p_{k}$. We will
assume this to be true. Using this fact, we similarly have

$d_{\text{e}}S(t)\equiv-{\textstyle\sum\nolimits_{k}}\eta_{k}d_{\text{e}}p_{k}(t),\
d_{\text{i}}S(t)\equiv-{\textstyle\sum\nolimits_{k}}\eta_{k}d_{\text{i}}p_{k}(t).\
$ (109)

The relation $d_{\text{e}}Q(t)=T_{0}d_{\text{e}}S(t)$ can be expressed in
terms of $d_{\text{e}}p_{k}(t)$

${\textstyle\sum\nolimits_{k}}(\eta_{k}-\beta_{0}E_{k})d_{\text{e}}p_{k}=0.$

Similarly, the relation $d_{\text{i}}Q(t)=T(t)dS(t)-T_{0}d_{\text{e}}S(t)$ can
be written as

${\textstyle\sum\nolimits_{k}}(E_{k}-T_{0}\eta_{k})d_{\text{i}}p_{k}=(T(t)-T_{0}){\textstyle\sum\nolimits_{k}}\eta_{k}dp_{k},$

which acts as a constraint on possible variations $d_{\text{i}}p_{k}$, given
that $dp_{k}$ can be directly obtained from Eq. (101).

##  IX A Model Entropy Calculation

We consider a gas of non-interacting identical structureless particles with no
spin, each of mass $m$, in a fixed region confined by impenetrable walls
(infinite potential well). Initially, the gas is in a NEQ macrostate, and is
isolated in that region. In time, the gas will equilibrate and the microstate
probabilities change in a way that the entropy increases. We wish to
understand how the increase happens.

###  IX.1 1-dimensional ideal Gas:

In order to be able to carry out an _exact calculation_ , we consider the gas
in a $1$-dimensional box of initial size $L_{\text{in}}$. As there are no
interactions between the particles, the wavefunction $\Psi$ for the gas is a
product of individual particle wavefunctions $\psi$. Thus, we can focus on a
single particle to study the nonequilibrium behavior of the gas Gujrati-
JensenInequality ; Gujrati-QuantumHeat ; GujTyler ; GujBoyko ; Wu . The simple
model of a particle in a box has been extensively studied in the literature
but with a very different emphasis. Bender ; Doescher ; Stutz  The particle
only has non-degenerate eigenstates whose energies are determined by $L$, and
a quantum number $k$. We use the energy scale
$\varepsilon_{1}=\pi^{2}\hbar^{2}/2mL_{\text{in}}^{2}$ to measure the
eigenstate energies, and $\alpha=L/L_{\text{in}}$ so that

$\varepsilon_{k}(L)=k^{2}/\alpha^{2};$ (110)

the corresponding eigenfunctions are given by

$\psi_{k}(x)=\sqrt{2/L}\sin(k\pi x/L),\ \ k=1,2,3,\cdots.$ (111)

The pressure generated by the eigenstate on the walls is given by Landau-QM

$P_{k}(L)\equiv-\partial\varepsilon_{k}/\partial L=2\varepsilon_{k}(L)/L.$
(112)

In terms of the eigenstate probability $p_{k}(t)$, the average energy and
pressure are given by

$\displaystyle\varepsilon(t,L)$
$\displaystyle\equiv{\textstyle\sum\nolimits_{k}}p_{k}(t)\varepsilon_{k}(L),$
(113a) $\displaystyle P(t,L)$
$\displaystyle\equiv{\textstyle\sum\nolimits_{k}}p_{k}(t)P_{k}(L)=2\varepsilon(t,L)/L.$
(113b) The entropy follows from Eq. (47) and is given for the single particle
case by  $s(t,L)\equiv-{\textstyle\sum\nolimits_{k}}p_{k}(t)\ln p_{k}(t).$

The time dependence in $\varepsilon(t)$ or $P(t)$ is due to the time
dependence in $p_{k}$ and $\varepsilon_{k}(L)$. Even for an isolated system,
for which $\varepsilon$ remains constant, $p_{k}$ cannot remain constant as
follows directly from the second law Gujrati-Symmetry  and creates a
conceptual problem because the eigenstates are mutually orthogonal and there
can be no transitions among them to allow for a change in $p_{k}$.

As the gas is isolated, its energy, volume and the number of particles remain
constant. As it is originally not in equilibrium, it will eventually reach
equilibrium in which its entropy must increase. This requires the introduction
of some internal variables even in this system whose variation will give rise
to entropy generation by causing internal variations $d_{\text{i}}p_{k}(t)$ in
$p_{k}(t)$. Here, we will assume a single internal variable $\xi(t)$. What is
relevant is that the variation in $\xi(t)$ is accompanied by changes
$dp_{k}(t)$ occurring within the isolated system. According to our
identification of heat with changes in $p_{k}(t)$, these variations must be
associated with heat, which in this case will be associated with irreversible
heat $d_{\text{i}}Q(t)$.

###  IX.2 Chemical Reaction Approach

A way to change $p_{k}$ in an isolated system is to require the presence of
some stochastic interactions, whose presence allows for transitions among
eigenstates Gujrati-Entropy1 . As these transitions are happening within the
system, we can treat them as ”chemical reactions” between different
eigenstates DeDonder ; deGroot ; Prigogine  by treating each eigenstate $k$ as
a chemical species. During the transition, these species undergo chemical
reactions to allow for the changes in their probabilities.

We follow this analogy further and extend the traditional approach DeDonder ;
deGroot ; Prigogine  to the present case. For the sake of simplicity, our
discussion will be limited to the ideal gas in a box; the extension to any
general system is trivial. Therefore, we will use microstates
$\left\\{\mathfrak{m}_{k}\right\\}$ instead of eigenstates in the following to
keep the discussion general. Let there be $N_{k}(t)$ particles in
$\mathfrak{m}_{k}$ at some instant $t$ so that

$N={\textstyle\sum\nolimits_{k}}N_{k}(t)$

at all times, and $p_{k}(t)=N_{k}(t)/N$. We will consider the general case
that also includes the case in which final microstates refer to a box size
$L^{\prime}$ different from its initial value $L$. Let us use $A_{k}$ to
denote the reactants (initial microstates) and $A_{k}^{\prime}$ to denote the
products (final microstates). For the sake of simplicity of argument, we will
assume that transitions between microstates is described by a single chemical
reaction, which is expressed in stoichiometry form as

${\textstyle\sum\nolimits_{k}}a_{k}A_{k}\longrightarrow{\textstyle\sum\nolimits_{k}}a_{k}^{\prime}A_{k}^{\prime}.$
(114)

Let $N_{k}$ and $N_{k}^{\prime}$ denote the population of $A_{k}$ and
$A_{k}^{\prime}$, respectively, so that
$N={\textstyle\sum\nolimits_{k}}N_{k}={\textstyle\sum\nolimits_{k}}N_{k}^{\prime}$.
Accordingly, $p_{k}(t)=N_{k}(t)/N$ for the reactant and
$p_{k}(t+dt)=N_{k}^{\prime}(t)/N$ for the product. The single reaction is
described by a single extent of reaction $\xi$ and we have

$d\xi(t)\equiv-dN_{k}(t)/a_{k}(t)\equiv
dN_{k^{\prime}}^{\prime}(t)/a_{k^{\prime}}^{\prime}(t)\text{ \ \ for all
}k,k^{\prime}.$

It is easy to see that the coefficients satisfy an important relation

${\textstyle\sum\nolimits_{k}}a_{k}(t)={\textstyle\sum\nolimits_{k}}a_{k}^{\prime}(t),$

which reflects the fact that the change $\left|dN\right|$ in the reactant
microstates is the same as in the product microstates. The _affinity_ in terms
of the chemical potentials $\mu$ is given by

$A(t)={\textstyle\sum}a_{k}(t)\mu_{A_{k}}(t)-{\textstyle\sum}a_{k}^{\prime}(t)\mu_{A_{k}^{\prime}}(t),$

and will vanish only in ”equilibrium,” i.e. only when $p_{k}$’ s attain their
equilibrium values. Otherwise, $A(t)$ will remain non-zero. It acts as the
thermodynamic force in driving the chemical reaction DeDonder ; deGroot ;
Prigogine . But we must wait long enough for the reaction to come to
completion, which happens when $A(t)$ and $d\xi/dt$ both vanish. The extent of
reaction $\xi$ is an example of an internal variable. There may be other
internal variables depending on the initial NEQ macrostate. This will be
discussed in the following section.

##  X Simple Applications

###  X.1 Isothermal Expansion

Let us first consider an isothermal expansion of an ideal gas in which the
temperature $T$ of the gas remains constant and equal to that of the medium
$T_{0}$. During an irreversible isothermal expansion, energy is pumped into
the gas isothermally from outside so $E(t)$ remains constant. The pumping of
energy will result in the change $d_{\text{e}}p_{k}(t)$. This will determine
$d_{\text{e}}S(t)=d_{\text{e}}Q(t)/T_{0}$. In addition, the gas may undergo
transitions among various energy levels, as discussed in Sect. IX.2, without
any external energy input, which will determine the change
$d_{\text{i}}p_{k}(t)$. From Eq. (76a), we determine
$d_{\text{i}}Q(t)=T_{0}d_{\text{i}}S(t)$, and consequently $d_{\text{i}}W(t)$.
Thus,

$[P(t)-P_{0}]dV(t)+A(t)d\xi(t)=T_{0}dS(t)-d_{\text{e}}Q(t).$

Such a calculation will not be possible using the first law in Eq. (23b) in
the M̊NEQT.

###  X.2 Intrinsic Adiabatic Expansion

In a nonequilibrium intrinsic adiabatic process, we have $d_{\text{i}}W(t)=-$
$d_{\text{e}}Q(t)$ so the heat exchange $\left|d_{\text{e}}Q(t)\right|=$
$T_{0}\left|d_{\text{e}}S(t)\right|$ is converted into the irreversible work.
We can use this to determine the work $d_{\text{i}}W_{\xi}(t)$ due to the
single internal variable

$A(t)d\xi(t)=-d_{\text{e}}Q(t)-(P(t)-P_{0})dV>0.$

The identification $d_{\text{i}}W(t)=-$ $d_{\text{e}}Q(t)$ and the calculation
of $A(t)d\xi(t)$ and of $d_{\text{i}}S(t)$ cannot be done in the traditional
formulation of the first law in the M̊NEQT, in which $d_{\text{e}}Q(t)=0$ for
the traditional adiabatic process so that $dE=-d_{\text{e}}W(t)$.

###  X.3 Composite $\Sigma$ with Temperature Inhomogeneity.

Here, we will show by examples that the thermodynamic temperature $T$ of
$\Sigma$ allows us to treat it as a ”black box” $\Sigma_{\text{B}}$ without
knowing its detailed internal structure such as its composition in terms of
two subsystems $\Sigma_{1}$ and $\Sigma_{2}$. Alternatively, we can treat
$\Sigma$ as a combination $\Sigma_{\text{C}}$ of $\Sigma_{1}$ at $T_{1}$ and
$\Sigma_{2}$ at at $T_{2}<T_{1}$, and obtain same thermodynamics. Thus, both
approaches are equivalent, which justifies the usefulness and uniqueness (see
Theorem 33) of $T$ as a thermodynamically appropriate global temperature.

In the following, we will consider various cases that can be obtained as
special cases of the following general situation (2): $\Sigma_{1}$ in thermal
contact with the medium $\widetilde{\Sigma}_{\text{h1}}$ at temperature
$T_{01}$, and $\Sigma_{2}$ in thermal contact with the medium
$\widetilde{\Sigma}_{\text{h2}}$ at temperature $T_{02}$, with the two media
having no mutual interaction.

We will consider the two realizations for $\Sigma$: $\Sigma_{\text{B}}$ and
$\Sigma_{\text{C}}$ to compare their predictions. As discussed for the case
(b) in Sec. III, $\Sigma_{1}$ and $\Sigma_{2}$ are always taken to be in EQ,
but $\Sigma$ in IEQ. The entropies in the two realizations are

$S_{\text{B}}(t)=S(E(t),\xi(t));S_{\text{C}}=S_{1}(E_{1}(t))+S_{2}(E_{2}(t)),$
(115)

and have the same value; recall that $E(t)=E_{1}(t)+E_{2}(t)$, and
$\xi(t)=E_{1}(t)-E_{2}(t)$ for $\Sigma(t)$; see Eq. (36). For clarity, we will
often use the argument $t$ to emphasize the variations in time $t$ in this
section. In general, the irreversible entropy generation is given by

$d_{\text{i}}S(t)=d\widetilde{S}_{1}(t)+d\widetilde{S}_{2}(t)+dS(t),$ (116)

where $dS$ should be replaced by $dS_{\text{B}}$ or $dS_{\text{C}}$ as the
case may be:

$dS_{\text{B}}(t)=\beta(t)dE(t)+\beta(t)A(t)d\xi(t),$
---
$dS_{\text{C}}(t)=\beta_{1}(t)dE_{1}(t)+\beta_{2}(t)dE_{2}(t),$
(117)

where we are using the inverse temperatures for various bodies. Let
$d_{\text{e}}Q_{l}(t),l=1,2$ be the energy or macroheat transferred to
$\Sigma_{l}(t)$ from $\widetilde{\Sigma}_{\text{h}}^{(l)}$, and
$dE_{\text{in}}(t)=d_{\text{e}}Q_{\text{in}}(t)$ the energy or macroheat
transferred from $\Sigma_{1}(t)$ to $\Sigma_{2}(t)$. We have, using
$\delta_{1}=+1$ and $\delta_{2}=-1,$

$\displaystyle dE_{l}(t)$
$\displaystyle=d_{\text{e}}Q_{l}(t)+\delta_{l}dE_{\text{in}}(t),$
$\displaystyle dE(t)$
$\displaystyle=d_{\text{e}}Q_{1}(t)+d_{\text{e}}Q_{2}(t),$ (118a)
$\displaystyle d\widetilde{S}_{l}(t)$
$\displaystyle=-d_{\text{e}}S_{l}(t)=-\beta_{0l}d_{\text{e}}Q_{l}(t).$ We see
that $dE(t)$ is unaffected by the internal energy transfer
$dE_{\text{in}}(t)$, while
$d\xi(t)=d_{\text{e}}Q_{1}(t)-d_{\text{e}}Q_{2}(t)+2dE_{\text{in}}(t),$ (118b)
is affected by the macroheat exchange disparity
$d_{\text{e}}Q_{1}(t)-d_{\text{e}}Q_{2}(t)$ along with $dE_{\text{in}}(t)$.

We finally have

$d_{\text{i}}S(t)=-{\textstyle\sum\nolimits_{l}}\beta_{0l}d_{\text{e}}Q_{l}(t)+dS.$
(119)

We now consider various cases to make our point.

####  X.3.1 Isolated $\Sigma$

We first consider the realization $\Sigma_{\text{B}}$. Using
$dE(t)=dE_{1}(t)+dE_{2}(t),d\xi(t)=dE_{1}(t)-dE_{2}(t)$, see Eqs. (36), and
(117) for $dS_{\text{B}}(t)$ above, we obtain

$\beta(t)=\frac{\beta_{1}(t)+\beta_{2}(t)}{2},\beta(t)A(t)=\frac{\beta_{1}(t)-\beta_{2}(t)}{2}.$
(120a) This identifies $T\left(t\right)$ in terms of $T_{1}(t)$ and
$T_{2}(t)$. As EQ is attained, $T(t)\rightarrow T_{0}$, the EQ temperature
between $\Sigma_{1}$ and $\Sigma_{2}$, and $A(t)\rightarrow A_{0}=0$ as
expected. In the following, we will use $A^{\prime}(t)$ for $\beta(t)A(t)$ for
simplicity. In terms of $\beta$ and $A^{\prime}$, we also have
$\beta_{1}=\beta+A^{\prime},\beta_{2}=\beta-A^{\prime}.$ (120b)

We now justify that in this simple example, $A^{\prime}(t)d\xi(t)$ determines
$d_{\text{i}}S(t)$ due to irreversibilty in $\Sigma(t)$; see Eq. (80). Setting
$dE(t)=0$ in $dS_{\text{B}}(t)$, we have by direct evaluation,

$d_{\text{i}}S(t)=A^{\prime}(t)d\xi(t)=\beta(t)d_{\text{i}}W(t).$ (121)

The last equation follows from the general result in Eq. (81). It should be
emphasized that the existence of $d_{\text{i}}S(t)\geq 0$ due to $\xi$ in
$\mathcal{M}_{\text{ieq}}$ is consistent with $\mathcal{M}_{\text{ieq}}$ as a
NEQ macrostate, even though its entropy is a state function in the extended
state space.

We now consider $\Sigma_{\text{C}}$, which is also very instructive to
understand the origin of $d_{\text{i}}S(t)$ in a different way. Considering
internal energy or macroheat transfer
$dE_{\text{in}}(t)=d_{\text{e}}Q_{\text{in}}(t)$ between $\Sigma_{1}(t)$ and
$\Sigma_{2}(t)$ at some instant $t$, we have

$dS_{1}(t)=\frac{dE_{\text{in}}(t)}{T_{1}(t)},dS_{2}(t)=-\frac{dE_{\text{in}}(t)}{T_{2}(t)},$
(122a) due to this transfer. This results in
$d_{\text{i}}S(t)=\left[\beta_{1}(t)-\beta_{2}(t)\right]dE_{\text{in}}(t)=A^{\prime}d\xi(t),$
(122b) since $d\xi(t)=dE_{1}(t)-dE_{2}(t)=2dE_{\text{in}}(t)$. Thus, the
physical origin of $d_{\text{i}}S(t)$ is the internal entropy change of the
subsystems.

####  X.3.2 $\Sigma$ Interacting with $\widetilde{\Sigma}_{\text{h}}$

To further appreciate the physical significance of the NEQ $T(t)$ of the above
composite system $\Sigma(t)$, we allow it to interact with
$\widetilde{\Sigma}_{\text{h}}$, a heat bath, at the EQ temperature $T_{0}$.
For this, we take $\widetilde{\Sigma}_{\text{h1}}$ and
$\widetilde{\Sigma}_{\text{h2}}$ at the same common temperature
$T_{0}=T_{01}=T_{02}$ so that we can treat them as a single medium
$\widetilde{\Sigma}_{\text{h}}$ with macroheat exchange $d_{\text{e}}Q(t)$. We
thus obtain from Eq. (119)

$d_{\text{i}}S(t)=-\beta_{0}d_{\text{e}}Q(t)+dS.$

We will consider two different kinds of interaction below:

(i) We first consider $\Sigma_{\text{B}}(t)$ in $\mathcal{M}_{\text{ieq}}$ at
$T(t)$ so we use $dS_{\text{B}}(t)$ above. We thus have (using the identity
$d_{\text{e}}S(t)=\beta_{0}d_{\text{e}}Q(t)$)

$d_{\text{i}}S(t)=[\beta(t)-\beta_{0}]d_{\text{e}}Q(t)+A^{\prime}(t)d\xi(t),$
(123)

which is consistent with the general identity given by the top equation in Eq.
(76a), a result which was derived for a single system at temperature
$T\left(t\right)$. This confirms that the composite $\Sigma_{\text{C}}$ here
can be treated as a noncomposite $\Sigma_{\text{B}}$ at $T\left(t\right)$. To
be convinced that the above $d_{\text{i}}S(t)$ includes the internally
generated irreversibility in Eq. (121) due to macroheat transfer between
$\Sigma_{1}(t)$ and $\Sigma_{2}(t)$, we only have to set $d_{\text{e}}S(t)=0$
to ensure the isolation of $\Sigma$. We reproduce Eq. (121) as
$d_{\text{i}}Q(t)=d_{\text{i}}W(t)$. The remaining source of irreversibility
$T(t)d_{\text{i}}S^{\text{Q}}(t)$ given by the first term above is due to
external macroheat exchange between $\Sigma$ and
$\widetilde{\Sigma}_{\text{h}}$

$d_{\text{i}}S^{\text{Q}}(t)=[T_{0}\beta(t)-1]d_{\text{e}}S(t),$ (124a) as
expected; see the first term on the right in Eq. (85b).

(ii) We take treat $\Sigma(t)$ as $\Sigma_{\text{C}}(t)$ in contact with
$\widetilde{\Sigma}_{\text{h}}$. We deal directly with the two macroheat
exchanges $d_{\text{e}}Q_{l}(t),l=1,2$ to $\Sigma_{l}(t)$ from
$\widetilde{\Sigma}_{\text{h}}$, and the internal energy transfer
$dE_{\text{in}}(t)$. Using $dE_{l}(t)$ from Eq. (118a) in $dS_{\text{C}}$
given in Eq. (119), we find that

$d_{\text{i}}S(t)={\textstyle\sum\nolimits_{l}}[\beta_{l}(t)-\beta_{0}]d_{\text{e}}Q_{l}(t)+\left[\beta_{1}(t)-\beta_{2}(t)\right]dE_{\text{in}}(t).$

Using Eq. (120b) to express $\beta_{l}$, we can rewrite the above equation as

$d_{\text{i}}S(t)=[\beta(t)-\beta_{0}]d_{\text{e}}Q(t)+A^{\prime}d\xi,$

where we have used the identity

$d_{\text{e}}Q(t)=d_{\text{e}}Q_{1}+d_{\text{e}}Q_{2},$ (125)

and have found

$d\xi=d_{\text{e}}Q_{1}-d_{\text{e}}Q_{2}+2dE_{\text{in}}$ (126)

using its general definition $d\xi(t)=dE_{1}(t)-dE_{2}(t)$. We thus see that
$d_{\text{i}}S(t)$ obtained by both realizations are the same as they must.
However, the realization $\Sigma_{\text{C}}(t)$ allows us to also identify
$d\xi$.

Each exchange generates irreversible entropy following Eq. (124a). Using
$d_{\text{e}}Q(t)=d_{\text{e}}Q_{1}(t)+d_{\text{e}}Q_{2}(t)$ in
$dQ(t)=T(t)dS(t)$ to determine $d_{\text{i}}Q(t)$, we find the generalization
of Eq. (123):

$d_{\text{i}}S(t)=\left[\beta_{1}(t)-\beta_{2}(t)\right]d_{\text{e}}Q_{\text{in}}(t)+{\textstyle\sum\nolimits_{l}}[T_{0}\beta_{l}(t)-1]d_{\text{e}}S_{l}(t).$
(127)

It is easy to see that the last term above gives nothing but the sum of the
irreversible entropies due to external exchanges of macroheat by
$\Sigma_{1}(t)$ and $\Sigma_{2}(t)$ with $\widetilde{\Sigma}_{\text{h}}$:

$d_{\text{i}}S^{\text{Q}}(t)=d_{\text{i}}S_{1}^{\text{Q}}(t)+d_{\text{i}}S_{2}^{\text{Q}}(t),$
(128)

where

$d_{\text{i}}S_{l}^{\text{Q}}(t)=[T_{0}\beta_{l}(t)-1]d_{\text{e}}S_{l}(t),l=1,2$
(129)

is the external entropy exchange of $\Sigma_{l}(t)$ with
$\widetilde{\Sigma}_{\text{h}}$.

Thus, whether we treat $\Sigma$ as a system $\Sigma_{\text{B}}$ at temperature
$T(t)$ or a collection $\Sigma_{\text{C}}$ of $\Sigma_{1}(t)$ and
$\Sigma_{2}(t)$ at temperatures $T_{1}(t)$ and $T_{2}(t)$, respectively, we
obtain the same irreversibility. In other words, $T(t)$ is a sensible
thermodynamic temperature even in the presence of inhomogeneity.

###  X.4 $\Sigma$ Interacting with $\widetilde{\Sigma}_{\text{h1}}$ and
$\widetilde{\Sigma}_{\text{h2}}$

We now consider our composite $\Sigma$ in thermal contact with two distinct
and mutually noninteracting stochastic media $\widetilde{\Sigma}_{\text{h1}}$
and $\widetilde{\Sigma}_{\text{h2}}$ at temperatures $T_{01}$ and $T_{02}$. We
will again discuss the two different realizations as above.

(i) We first consider $\Sigma_{\text{B}}(t)$ at temperature $T(t)$, which
interacts with the two $\widetilde{\Sigma}_{\text{h}}$’s, and use the general
result in Eq. (119). A simple calculation using $dS_{\text{B}}$ generalizes
Eq. (123) and yields

$d_{\text{i}}S(t)={\textstyle\sum\nolimits_{l}}[\beta(t)-\beta_{0l}]d_{\text{e}}Q_{l}(t)+A^{\prime}(t)d\xi(t),$
(130a) since this reduces to that result when we set
$\beta_{01}=\beta_{02}=\beta_{0}$. As above,
$d_{\text{i}}Q(t)=d_{\text{i}}W(t)=A(t)d\xi(t)$; see Eq. (121), which gives
rise to the last term above. Thus, setting $d_{\text{e}}Q_{l}(t)=0,l=1,2$ to
make $\Sigma$ isolated, we retrieve $d_{\text{i}}S(t)$ in Eq. (121) as
expected. The first sum above gives the external entropy exchanges with the
two stochastic media as above.

(ii) We now consider $\Sigma_{\text{C}}$, and allow
$\widetilde{\Sigma}_{\text{h1}}$ to directly interact with $\Sigma_{1}(t)$ at
temperature $T_{1}(t)$ and $\widetilde{\Sigma}_{\text{h2}}$ to directly
interact with $\Sigma_{2}(t)$ at temperature $T_{2}(t)$. Using $dS_{\text{C}}$
generalizes Eq. (123) and yields

$d_{\text{i}}S(t)={\textstyle\sum\nolimits_{l}}[\beta_{l}(t)-\beta_{0l}]d_{\text{e}}Q_{l}(t)+\left[\beta_{1}(t)-\beta_{2}(t)\right]dE_{\text{in}}(t).$
(130b)

Again using Eq. (120b) to express $\beta_{l}$, we can rewrite the above
$d_{\text{i}}S(t)$ as the $d_{\text{i}}S(t)$ in Eq. (130a) for
$\Sigma_{\text{B}}$, and also find that $d\xi$ is given by Eq. (126).

It should be emphasized that the determination of $d_{\text{i}}S(t)$ in Eqs.
(130a-130b) is valid for all cases of $\Sigma$ interacting with
$\widetilde{\Sigma}_{\text{h1}}$ and $\widetilde{\Sigma}_{\text{h2}}$ as we
have not imposed any conditions on $T_{1}(t)$ and $T_{2}(t)$ with respect to
$T_{01}$ and $T_{02}$, respectively. Thus it is very general. The derivation
also applies to the NEQ stationary state, which happens when
$T_{1}(t)\rightarrow T_{01}$ and $T_{2}(t)\rightarrow T_{02}$. For the
stationary case, using Eq. (130b), we have

$d_{\text{i}}S^{\text{st}}=\left[\beta_{01}-\beta_{02}\right]dE_{\text{in}},$
(131)

where all quantities on the right have their steady values. Thus,
$d_{\text{i}}S^{\text{st}}$ is only determined by the stationary value of the
internal energy exchange $dE_{\text{in}}$. The reader can easily verify that
$d_{\text{i}}S(t)$ in Eqs. (130a) also reduces to the above result in the
stationary limit.

From the above examples, we see that we can consider $\Sigma$ in any of the
two realization $\Sigma_{\text{B}}$ and $\Sigma_{\text{C}}$ as we obtain the
same thermodynamics in that $d_{\text{i}}S(t)$ is identical. We emphasize this
important observation by summarizing it in the following conclusion.

###### Conclusion 51

If we consider $\Sigma(t)$ as a single system $\Sigma_{\text{B}}$ with an
uniform temperature $T(t)$ and with an internal variable $\xi(t)$, we do not
need to consider the energy transfer $dE_{\text{in}}(t)$ explicitly to obtain
$d_{\text{i}}S(t)$. If we consider $\Sigma(t)$ as a composite system
$\Sigma_{\text{C}}$ formed of $\Sigma_{1}(t)$ and $\Sigma_{2}(t)$ at their
specific temperatures, then we specifically need to consider the energy
transfer $dE_{\text{in}}(t)$ to obtain $d_{\text{i}}S(t)$ but no internal
variable.

This conclusion emphasizes the most important fact of the MNEQT that the
homogeneous thermodynamic temperature $T(t)$ of $\Sigma_{\text{B}}$ can also
describe an inhomogeneous system $\Sigma_{\text{C}}$. This observation
justifies using the thermodynamic temperature $T(t)$ for treating $\Sigma(t)$
as a single system $\Sigma_{\text{B}}$, a black box, without any need to
consider the internal energy transfers.

The above discussion can be easily extended to also include inhomogeneities
such as two different work media $\widetilde{\Sigma}_{\text{w}}^{(1)}$ and
$\widetilde{\Sigma}_{\text{w}}^{(2)}$ corresponding to different pressures
$P_{01\text{ }}$and $P_{02}$. We will not do that here.

###  X.5 $\Sigma$ Interacting with $\widetilde{\Sigma}_{\text{w}}$ and
$\widetilde{\Sigma}_{\text{h}}$

In this case, $\Sigma$ is specified by two observables $E$ and $V$ so to
describe any inhomogeneity will require considering at least two subsystems
$\Sigma_{1}$ and $\Sigma_{2}$ specified by $E_{1},V_{1}$ and $E_{2,}V_{2}$,
respectively. From these four observables, we construct the following four
combinations

$\displaystyle E_{1}+E_{2}$ $\displaystyle=E,\xi_{\text{E}}=E_{1}-E_{2},$
$\displaystyle V_{1}+V_{2}$ $\displaystyle=V,\xi_{\text{V}}=V_{1}-V_{2},$

to express the entropy of the system

$S(E,V,\xi_{\text{E}},\xi_{\text{V}})=S_{1}(E_{1},V_{1})+S_{2}(E_{2},V_{2})$

in terms of

$E_{1,2}=\frac{E\pm\xi_{\text{E}}}{2},V_{1,2}=\frac{V\pm\xi_{\text{V}}}{2}.$

Note that we have assumed that $\Sigma_{1}$ and $\Sigma_{2}$ are in EQ (no
internal variables for them). We now follow the procedure carried out in Sec.
X.3 to identify thermodynamic temperature $T$, pressure $P$, and affinities:

$\begin{array}[c]{c}\beta=\frac{(\beta_{1}+\beta_{2})}{2},\beta
P=\frac{(\beta_{1}P_{1}+\beta_{2}P_{2})}{2},\\\ \beta
A_{\text{E}}=\frac{(\beta_{1}-\beta_{2})}{2},\beta
A_{\text{V}}=\frac{(\beta_{1}P_{1}-\beta_{2}P_{2})}{2}.\end{array}$ (132)

All these quantities are SI-quantities and have the same values regardless of
whether $\Sigma$ is isolated or interacting. A more complicated
inhomogeneities will require more internal variables.

###### Remark 52

We now make an important remark about Eq. (85b) that contains only a single
internal variable. From what is said above, it must include at least two
internal variables if $\Sigma$ contains inhomogeneity. in both $E$ and $V$. If
it contains inhomogeneity. in only one variable, then and only then we will
have at least one internal variable. Thus, either we will $\xi_{\text{E}}$ or
$\xi_{\text{V}}$ as the case may be.

##  XI Tool-Narayanaswamy Equation

We consider a simple NEQ laboratory problem to model the situation in a glass
Gujrati-I . It is a system consisting of two ”interpenetrating” parts at
different temperatures $T_{1}$ and $T_{2}>T_{1}$, but insulated from each
other so that they cannot come to equilibrium. The two parts are like slow and
fast motions in a glass, and the insulation allows us to treat them as
independent, having different temperatures. This is a very simple model for a
glass. A more detailed discussion is given elsewhere Gujrati-I , where each
part was assumed to be in EQ macrostates. Here, we go beyond the earlier
discussion, and assume that the two parts are in some IEQ macrostates
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ with temperatures $T_{1}$ and $T_{2}$,
respectively; we have suppressed ”ieq” in the subscripts for simplicity. Thus,
there are irreversible processes going on within each part so that there are
nonzero irreversible macroheat $d_{\text{i}}Q_{1}$ and $d_{\text{i}}Q_{2}$
generated within each part. We wish to identify the temperature of the system
that we treat as a black box $\Sigma_{\text{B}}$. This will require
introducing its global temperature $T$. However, we also need to relate it to
$T_{1}$ and $T_{2}$ so that we need to treat $\Sigma$ as $\Sigma_{\text{C}}$.
We now imagine that each part is added a certain _infinitesimal_ amount of
exchange macroheat from outside, which we denote by $d_{\text{e}}Q_{1}$ and
$d_{\text{e}}Q_{2}$ so that $dQ_{1}=d_{\text{e}}Q_{1}+d_{\text{i}}Q_{1}$ and
$dQ_{2}=d_{\text{e}}Q_{2}+d_{\text{i}}Q_{2}$. This does not affect their
temperatures. We assume the entropy changes to be $dS_{1}$ and $dS_{2}$. Then,
we have for the net macroheat and entropy change

$dQ=dQ_{1}+dQ_{2},dS=dS_{1}+dS_{2}.$

We introduce the temperature $T$ by $dQ=TdS$. This makes it a thermodynamic
temperature of the black box. Using $dQ_{1}=T_{1}dS_{1},dQ_{2}=T_{2}dS_{2}$,
we immediately find

$dQ(1/T-1/T_{2})=dQ_{1}(1/T_{1}-1/T_{2}).$

By introducing $x=dQ_{1}/dQ$, which is determined by the setup, we find that
$T$ is given by

$\frac{1}{T}=\frac{x}{T_{1}}+\frac{1-x}{T_{2}}.$ (133)

As $x$ is between $0$ and $1$, it is clear that $T$ lies between $T_{1}$ and
$T_{2}$ depending on the value of $x$. For $x=1/2$, this heuristic model
calculation reduces to that in Eq. (120a) as expected. The derivation also
shows that the thermodynamic temperature $T$ is not affected by having two
nonoverlapping parts or overlapping parts. A similar relation also exists for
the pressure $P$ of a composite system; see Eq. (156).

If the insulation between the parts is not perfect, there is going to be some
energy transfer between the two parts, which would result in maximizing the
entropy of the system. As a consequence, their temperatures will eventually
become the same. During this period, $T$ will also change until all the three
temperatures become equal. This will require additional internal variable or
variables as in Sect. X.3.

Figure 5: An irreversible Carnot cycle running between two heat reservoirs
$\widetilde{\Sigma}_{\text{h1}}$ and $\widetilde{\Sigma}_{\text{h2}}$.

##  XII Irreversible Carnot Cycle

Let us consider an “irreversible” Carnot engine running between two heat
sources $\widetilde{\Sigma}_{\text{h1}}$ and $\widetilde{\Sigma}_{\text{h2}}$
as shown in Fig. 5 that are always maintained at fixed temperatures $T_{01}$
and $T_{02}$, respectively, during each cyclic process
$\mathcal{P}_{\text{cyc}}$. As $\Sigma$ needs to perform work, we also need to
consider it to be in constant with a work source
$\widetilde{\Sigma}_{\text{w}}$. We first observe the following features of a
reversible Carnot cycle. The system, which we take to be formed by an ideal
gas, starts in thermal contact with $\widetilde{\Sigma}_{\text{h1}}$ in
$\mathcal{A}_{\text{eq}}=\mathcal{M}_{\text{eq}}(T_{01},V_{1})$ as it expands
to $V_{2}>V_{1}$, and ends in $\mathcal{M}_{\text{eq}}(T_{01},V_{2})$ through
an isothermal process $\mathcal{P}_{1\text{eq}}$ resulting in
$\Delta_{\text{e}}Q_{1\text{eq}}=\Delta_{\text{e}}W_{1\text{eq}}>0$. It is
then detached from $\widetilde{\Sigma}_{\text{h1}}$ so no heat is exchanged
($\Delta_{\text{e}}Q_{2\text{eq}}=0$) but exchanges work
$\Delta_{\text{e}}W_{2\text{eq}}>0$ during the process
$\mathcal{P}_{2\text{eq}}$ as it expands to $V_{3}$ and ends in
$\mathcal{M}_{\text{eq}}(T_{02},V_{3})$ at temperature $T_{02}$. The system is
brought in thermal contact with $\widetilde{\Sigma}_{\text{h2}}$ now, and the
volume is compressed to $V_{4}$ isothermally during the process
$\mathcal{P}_{3\text{eq}}$ and ends in
$\mathcal{M}_{\text{eq}}(T_{02},V_{4})$. During $\mathcal{P}_{3\text{eq}}$,
$\Delta_{\text{e}}Q_{3\text{eq}}=\Delta_{\text{e}}W_{3\text{eq}}<0$. The
choice of $V_{4}$ is chosen so that $\Sigma$ comes back to
$\mathcal{A}_{\text{eq}}=\mathcal{M}_{\text{eq}}(T_{01},V_{1})$ along a
process $\mathcal{P}_{4\text{eq}}$ after detaching it from
$\widetilde{\Sigma}_{\text{h2}}$ during which
$\Delta_{\text{e}}Q_{4\text{eq}}=0$, but $\Delta_{\text{e}}W_{4\text{eq}}<0$.
The four segments bring back $\Sigma$ to its starting state
$\mathcal{A}_{\text{eq}}$, and form a cycle $\mathcal{P}_{\text{eq,cyc}}$. It
is well known that the EQ efficiency $\epsilon_{\text{eq}}$ of the Carnot
cycle is

$\epsilon_{\text{eq}}=1-T_{02}/T_{01},$ (134a) so that
$\Delta_{\text{e}}W_{\text{eq}}=\epsilon_{\text{eq}}\Delta_{\text{e}}Q_{1},$
(134b) the equilibrium macrowork obtained from the cycle for a given
$\Delta_{\text{e}}Q_{1}$.

We now consider an irreversible cyclic process $\mathcal{P}$, which consists
of the same four segments except that some or all may be irreversible. We have
discussed such a process in Sect. VI.8. However, to have a cyclic process, the
system must start and end in $\mathcal{M}_{\text{eq}}(T_{01},V_{1})$, which
does not require any internal variable. Being an irreversible process, there
is no guarantee that $\Sigma$ would be in EQ macrostates at the end of
$\mathcal{P}_{1},\mathcal{P}_{2}$, and $\mathcal{P}_{3}$; $\mathcal{P}_{4}$
must bring $\Sigma$ to the EQ initial macrostate
$\mathcal{M}_{\text{eq}}(T_{01},V_{1})$. However, we will simplify the
calculation here by assuming that the end states in
$\mathcal{P}_{1},\mathcal{P}_{2}$, and $\mathcal{P}_{3}$ are
$\mathcal{M}_{\text{eq}}(T_{01},V_{2}),\mathcal{M}_{\text{eq}}(T_{02},V_{3})$,
and $\mathcal{M}_{\text{eq}}(T_{02},V_{4})$, respectively. However, relaxing
this condition does not change the results below.

Being a cyclic process, we have

$\Delta_{\text{c}}E=\Delta_{\text{c}}S=0$ (135a) over $\mathcal{P}$. Thus,
over $\mathcal{P}$, $\Delta_{\text{e}}Q=\Delta_{\text{e}}W.$ (135b) In the
MNEQT, we also have over $\mathcal{P}$, $\Delta Q=\Delta W.$ (135c)

Let $\Delta_{\text{e}}Q_{1}=T_{01}\Delta_{\text{e}}S_{1}$ and
$\Delta_{\text{e}}Q_{3}=T_{02}\Delta_{\text{e}}S_{3}$ be the macroheat
exchanges during $\mathcal{P}_{1}$ and $\mathcal{P}_{3}$, and
$\Delta_{\text{e}}Q=\Delta_{\text{e}}Q_{1}+\Delta_{\text{e}}Q_{3}$. Similarly,
let $\Delta_{\text{e}}W_{l}$ be the macrowork exchanges during
$\mathcal{P}_{l},l=1,2,3,4$, and
$\Delta_{\text{e}}W={\textstyle\sum\nolimits_{l}}\Delta_{\text{e}}W_{l}$, the
net exchange macrowork. From $\Delta_{\text{c}}S=0$ follows

$\Delta_{\text{e}}Q_{1}/T_{01}+\Delta_{\text{e}}Q_{3}/T_{02}+\Delta_{\text{i}}S=0,$

which can be rewritten by simple manipulation as

$\Delta_{\text{e}}W=\Delta_{\text{e}}Q_{1}(1-T_{02}/T_{01})-T_{02}\Delta_{\text{i}}S,$
(136a) where we have used the identity in Eq. (135b). We can also express it
as
$\Delta_{\text{e}}W=\epsilon_{\text{eq}}\Delta_{\text{e}}Q_{1}-T_{02}\Delta_{\text{i}}S\leq\Delta_{\text{e}}W_{\text{eq}}.$
(136b) The efficiency of the irreversible Carnot cycle is given by
$\epsilon_{\text{irr}}=\epsilon_{\text{eq}}-T_{02}\Delta_{\text{i}}S/\Delta_{\text{e}}Q_{1}\leq\epsilon_{\text{eq}}.$
(137)

We remark that $\epsilon_{\text{irr}}$ above is similar to the result obtained
by Eu (Eu, , see Eq. (7.139)) in which the numerator of the last term is
identified as ”the total dissipation” by Eu; however, the analysis is tedious
compared to the one given here.

The determination of $\Delta_{\text{i}}S$ requires the extended state space
$\mathfrak{S}_{\mathbf{Z}}$ needed for the four process segments. We will
focus on Eq. (85b). Let $T_{l}(t)$ and $P_{l}(t)$ be the temperature and
pressure of $\Sigma$ along $\mathcal{P}_{l}$, respectively, with
$l=1,\cdots,4$ for $\mathcal{P}_{l}$. As seen from Eq. (132), we need at least
two internal variables $\xi_{\text{E}l}$ and $\xi_{\text{V}l}$ along
$\mathcal{P}_{l}$ that are usually different along the four segments. The
corresponding affinities must vanish at the end points of each segment because
they are EQ macrostates. We will assume this to be the case, and introduce

$\mathbf{A}_{l}(t)\cdot
d\boldsymbol{\xi}_{l}(t)=A_{\text{E}l}(t)d\xi_{\text{E}l}(t)+A_{\text{V}l}(t)d\xi_{\text{V}l}(t)$

for each segment. Then,

$\displaystyle\Delta_{\text{i}}S_{l}$
$\displaystyle=\int_{\mathcal{P}_{l}}[(\beta_{l}(t)-\beta_{0l})d_{\text{e}}Q_{l}(t)+(P_{l}(t)-P_{0l})dV(t)$
(138) $\displaystyle+\beta_{l}(t)\mathbf{A}_{l}(t)\cdot
d\boldsymbol{\xi}_{l}(t)],$

where $d_{\text{e}}Q_{l}(t)=0$ for $l=2,4$, and where $P_{0l}$ is the external
pressures of $\widetilde{\Sigma}_{\text{w}}$ along $\mathcal{P}_{l}$ and must
be the same as for the reversible Carnot engine. This then determines
$\Delta_{\text{i}}S$ so that $\epsilon_{\text{irr}}$ is determined.

Using $\Delta_{\text{i}}Q=\Delta Q-\Delta_{\text{e}}Q$, we also have

$\Delta_{\text{i}}W={\textstyle\sum\limits_{l=1}^{4}}{\textstyle\int\limits_{\mathcal{P}_{l}}}\left[T_{l}(t)d_{\text{i}}S_{l}(t)+(T_{l}(t)-T_{0l})d_{\text{e}}S_{l}(t)\right]$
(139)

Recognizing that $d_{\text{e}}S_{l}(t)$ is nonzero only for $l=1,3$, we can
also rewritten $\Delta_{\text{i}}W$ as

$\Delta_{\text{i}}W={\textstyle\sum\limits_{l=1,3}}{\textstyle\int\limits_{\mathcal{P}_{l}}}(\beta_{0l}/\beta_{l}(t)-1)d_{\text{e}}Q_{l}(t)+{\textstyle\int\limits_{\mathcal{P}}}T(t)d_{\text{i}}S(t).$
(140)

It should be noted that nowhere did we use the vanishing of the affinities in
the EQ states at the end of $\mathcal{P}_{1},\mathcal{P}_{2}$, and
$\mathcal{P}_{3}$ so the calculation above is not limited by this requirement.
Thus, the results in this section are general.

##  XIII Origin of Friction and Brownian Motion

It is well known, see for example Kestin (Kestin, , Secs. 4.7 and 5.12), that
there are several ways to incorporate friction in a system in thermodynamics.
This has to do with the difficulties in making an unambiguous distinction
between various possibilities of exchange macroheat in a process
$\mathcal{P}$. We overcome this problem by using the MNEQT in which this is
not an issue as both $dQ$ and $d_{\text{e}}Q$ are uniquely defined. We
identify the origin of friction in our approach Gujrati-Entropy1 ; Gujrati-
Entropy2 ; Gujrati-II ; Gujrati-Heat-Work0 ; Gujrati-Heat-Work  by considering
relative motion between parts of a system or between the system and the
medium; see also Gujrati-LangevinEq ; Landau . Such a situation arises during
sudden mixing of fluids or in a Couette flow or when friction is involved
between two bodies. The origin of friction is also applicable to NEQ terminal
states of the process in the MNEQT.

###  XIII.1 Piston-Gas System

We consider the piston gas system in Fig. 4(a). As discussed in Sec. III.5,
the system entropy in $\mathcal{M}_{\text{ieq}}$ is
$S(E,V,\mathbf{P}_{\text{gc}},\mathbf{P}_{\text{p}})$. Hence, the
corresponding Gibbs fundamental relation becomes

$dS=\beta[dE+PdV-\mathbf{V}_{\text{gc}}\mathbf{\cdot}d\mathbf{P}_{\text{gc}}-\mathbf{V}_{\text{p}}\mathbf{\cdot}d\mathbf{P}_{\text{p}}],$

where we have used the conventional conjugate fields

$\beta\doteq\partial S/\partial E,\beta P\doteq\partial S/\partial V,,$
---
$\beta\mathbf{V}_{\text{gc}}\doteq-\partial
S/\partial\mathbf{P}_{\text{gc}}\mathbf{,}\beta\mathbf{V}_{\text{p}}\doteq-\partial
S/\partial\mathbf{P}_{\text{p}}$
(141)

as shown by Landau and Lifshitz Landau-Fluid  and by us elsewhere (Gujrati-II,
, and references theirin). Using Eq. (40), we can rewrite this equation as

$dS=\beta[dE+PdV-\mathbf{V\cdot}d\mathbf{P}_{\text{p}}]$ (142)

in terms of the relative _velocity_ , also known as the _drift velocity_
$\mathbf{V\doteq V}_{\text{p}}-\mathbf{V}_{\text{gc}}$ of the piston with
respect to $\Sigma_{\text{gc}}$. We can cast the drift velocity term as
$\mathbf{V\cdot}d\mathbf{P}_{\text{p}}\equiv\mathbf{F}_{\text{p}}\mathbf{\cdot}d\mathbf{R}$,
where $\mathbf{F}_{\text{p}}\doteq d\mathbf{P}_{\text{p}}\mathbf{/}dt$ is the
_force_ (a macroforce in nature) and $d\mathbf{R=V}dt$ is the _relative
displacement_ of the piston.

The internal motions of $\Sigma_{\text{gc}}$ and $\Sigma_{\text{p}}$ is not
controlled by any external agent so the relative motion described by the
relative displacement $\mathbf{R}$ represents an _internal variable_Kestin  so
that the corresponding affinity $\mathbf{F}_{\text{p}0}=0$ for
$\widetilde{\Sigma}$. Because of this, the first law
$dE=T_{0}d_{\text{e}}S-P_{0}dV$ as given in Eq. (23b) does not involve the
relative displacement $\mathbf{R}$. We now support this claim using our
approach in the following. This also shows how
$\mathcal{H}(\left.\mathbf{x}\right|V,\mathbf{P}_{\text{gc}},\mathbf{P}_{\text{p}})$
develops a dependence on the internal variable $\mathbf{R}$. We manipulate
$dS$ in Eq. (142) by using the above first law for $dE$ so that

$TdS=T_{0}d_{\text{e}}S+(P-P_{0})dV-\mathbf{F}_{\text{p}}\mathbf{\cdot}d\mathbf{R,}$

which reduces to

$T_{0}d_{\text{i}}S=(T_{0}-T)dS+(P-P_{0})dV-\mathbf{F}_{\text{p}}\mathbf{\cdot}d\mathbf{R.}$

This equation expresses the irreversible entropy generation as sum of three
distinct and independent irreversible entropy generations. To comply with the
second law, we conclude that for $T_{0}>0$,

$(T_{0}-T)dS\geq 0,(P-P_{0})dV\geq
0,\mathbf{F}_{\text{p}}\mathbf{\cdot}d\mathbf{R\leq}0,$ (143)

which shows that each of the components of $d_{\text{i}}S$ is nonnegative. In
equilibrium, each irreversible component vanishes, which happens when

$T\rightarrow T_{0},P\rightarrow P_{0}\text{, and }\mathbf{V}\rightarrow
0\text{ or }\mathbf{F}_{\text{p}}\rightarrow 0.$ (144)

The inequality $\mathbf{F}_{\text{p}}\mathbf{\cdot}d\mathbf{R\leq}0$ shows
that $\mathbf{F}_{\text{p}}$ and $d\mathbf{R}$ are antiparallel, which is what
is expected of a _frictional_ force $\mathbf{F}_{\text{fr}}$. Thus, we can
identify $\mathbf{F}_{\text{p}}$ with $\mathbf{F}_{\text{fr}}$, and the
irreversible frictional macrowork $d_{\text{i}}W_{\text{fr}}$ done during this
motion by

$d_{\text{i}}W_{\text{fr}}\doteq-\mathbf{V}(t)\cdot
d\mathbf{P}(t)=-d\mathbf{R}(t)\cdot\mathbf{F}_{\text{fr}}(t)>0.$ (145)

This macrowork, which appears as part of $d_{\text{i}}W$ in our approach,
vanishes as the motion ceases so that the equilibrium value $\mathbf{V}_{0}$
of $\mathbf{V}(t)$ is $\mathbf{V}_{0}=0$ just as the equilibrium affinity
$\mathbf{A}_{0}=0$ for $\mathbf{\xi}$. This causes the piston to finally come
to rest. As $\mathbf{F}_{\text{fr}}$ and $\mathbf{V}$ vanish together, we can
express this force as

$\mathbf{F}_{\text{fr}}=-\mu\mathbf{V}f(\mathbf{V}^{2}),$ (146)

where $\mu>0$ and $f$ is an _even_ function of $\mathbf{V}$. The medium
$\widetilde{\Sigma}$ is specified by $T=T_{0},P=P_{0}$ and $\mathbf{V}_{0}=0$
or $\mathbf{F}_{\text{p}}=0$. We will take $\mathbf{F}_{\text{fr}}$ and
$d\mathbf{R}$ to be colinear and replace
$\mathbf{F}_{\text{fr}}\mathbf{\cdot}d\mathbf{R}$ by $-F_{\text{fr}}dx$
($F_{\text{fr}}dx\geq 0$), where $dx$ is the magnitude of the relative
displacement $d\mathbf{R}$. The sign convention is that $F_{\text{fr}}$ and
increasing $x$ point in the same direction. We can invert Eq. (142) to obtain

$dE=TdS-PdV-F_{\text{fr}}dx$ (147)

in which $dQ=TdS$ from our general result in Eq. (5a). Comparing the above
equation with the first law in Eq. (23a), we conclude that

$dW=PdV+F_{\text{fr}}dx.$ (148)

The important point to note is that the friction term $F_{\text{fr}}dx$
properly belongs to $dW$. As $d_{\text{e}}W=P_{0}dV$, we have

$d_{\text{i}}W=(P-P_{0})dV+F_{\text{fr}}dx.$ (149)

Both contributions in $d_{\text{i}}W$ are separately nonnegative.

We can determine the exchange macroheat $d_{\text{e}}Q=dQ-d_{\text{i}}W$

$d_{\text{e}}Q=TdS-(P-P_{0})dV-F_{\text{fr}}dx$ (150)

It should be emphasized that in the above discussion, we have not considered
any other internal motion such as between different parts of the gas besides
the relative motion between $\Sigma_{\text{gc}}$ and $\Sigma_{\text{p}}$.
These internal motions within $\Sigma_{\text{g}}$ can be considered by
following the approach outlined elsewhere Gujrati-II . We will not consider
such a complication here.

###  XIII.2 Particle-Spring-Fluid System

It should be evident that by treating the piston as a mesoscopic particle such
as a pollen or a colloid, we can treat its thermodynamics using the above
procedure. This allows us to finally make a connection with the system
depicted in Fig. 4(b) in which the particle (a pollen or a colloid) is
manipulated by an external force $F_{0}$. We need to also consider two
additional forces $F_{\text{s}}$ and $F_{\text{fr}}$, both pointing in the
same direction as increasing $x$; the latter is the frictional force induced
by the presence of the fluid in which the particle is moving around. The
analog of Eq. (149) for this case becomes

$d_{\text{i}}W=(F_{\text{s}}+F_{0})dx+F_{\text{fr}}dx=F_{\text{t}}dx,$ (151)

where $F_{\text{t}}=F_{\text{s}}+F_{0}+F_{\text{fr}}$ is the net force. The
other two macroworks are $dW=(F_{\text{s}}+F_{\text{fr}})dx$ and
$d\widetilde{W}=F_{0}dx=-d_{\text{e}}W$. In EQ, $F_{\text{fr}}=0$ and
$F_{\text{s}}+F_{0}=0$ ($F_{0}\neq 0$) to ensure $d_{\text{i}}W=0$.

###  XIII.3 Particle-Fluid System

In the absence of a spring in the previous subsection, we must set
$F_{\text{s}}=0$ so

$dW=F_{\text{fr}}dx,d\widetilde{W}=F_{0}dx=-d_{\text{e}}W,d_{\text{i}}W=(F_{0}+F_{\text{fr}})dx.$
(152)

In EQ, $F_{0}+F_{\text{fr}}=0$ so that $F_{\text{fr}}=-F_{0}$. This means that
in EQ, the particle’s nonzero terminal velocity is determined by $F_{0}$ as
expected.

Figure 6: 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
$P_{0}\rightarrow 0$.

##  XIV Free Expansion

What makes NEQ thermodynamics complicated than EQ thermodynamics is the
evaluation of nonzero irreversible entropy generation $d_{\text{i}}S(t)\geq
0$. As $d_{\text{i}}S^{\text{Q}}(t)$ and $d_{\text{i}}S^{\text{W}}(t)$ are
independent contributions, it is simpler to consider an isolated system for
which $d_{\text{i}}S^{\text{Q}}(t)\equiv 0$ so that we only deal with
$d_{\text{i}}S^{\text{W}}(t)$ in Eq. (87). Then the use of Eq. (81) allows us
to determine the temperature of the system in any arbitrary macrostate. 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
$\Sigma$, which we will do in this section.

###  XIV.1 Classical Free Expansion in $\mathfrak{S}_{\mathbf{Z}}$

The gas, which forms $\Sigma$, expands freely in a vacuum
($\widetilde{\Sigma}$) from $V_{\text{in}}$, the volume of the left chamber,
to $V_{\text{fin}}=2V_{\text{in}}$, the volume of $\Sigma_{0}$; the volume of
the right chamber is $V_{\text{fin}}-V_{\text{in}}=V_{\text{in}}$. The initial
and final macrostate are denoted by $\mathcal{A}_{\text{eq}}$ and
$\mathcal{B}_{\text{eq}}$. 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. 6(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 $\mathcal{A}_{\text{eq}}$ with entropy
$S_{\text{in}}$. For ideal gas, we have Landau

$S(E,V)=N\ln(eV/N)+f(E),$

where $N$ is kept as a suffix for a reason that will become evident below. The
initial pressure and temperature of the gas prior to expansion in
$\mathcal{A}_{\text{eq}}$ 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. A similar set of quantities
also pertain to $\mathcal{B}_{\text{eq}}$. 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. 6(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. III. As our aim is to show the feasibility of the
MNEQT in this investigation, we will simplify the situation by limiting to two
internal variables. The first internal variable $\xi_{\text{N}}$

$\xi_{\text{N}}\doteq N_{\text{R}}/N$ (153)

is obtained by considering only two layers to describe different numbers
$N_{\text{L}}=(1-\xi_{\text{N}})N$ particles to left and
$N_{\text{R}}=N\xi_{\text{N}}$ particles to the right of the chamber partition
in Fig. 6(a) as a function of time. Initially, $\xi_{\text{N}}=0$ and finally
at EQ, $\xi_{\text{N}}=1/2$. At each instant, we imagine a front of the
expanding gas shown by the solid vertical line in Fig. 6(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 can be used to determine another
internal variable by using $V^{\prime}=V-V_{\text{in}}$:

$\xi_{\text{V}}\doteq V^{\prime}/V=1-V_{\text{in}}/V,$

so that $V^{\prime}=\xi_{\text{V}}V$ and $V_{\text{in}}=(1-\xi_{\text{V}})V$.
Initially, $\xi_{\text{V}}=0$ and finally at EQ, $\xi_{\text{V}}=1/2$. The
choice of the two internal variables $\xi_{\text{N}}(t)$ and
$\xi_{\text{V}}(t)$ follows the procedure in Sect. XI for two subsystems of
different sizes, and allow us to distinguish between $\mathcal{P}^{\prime}$
and $\mathcal{P}^{\prime\prime}$ as we will see below. We assume that the
expansion is isothermal (which it need not be) so there is no additional
internal variable associated with temperature variation. As $dQ=dW>0$, the
expansion is irreversible so the entropy continues to increase.

At $t=0$, the partition is suddenly removed, shown by the broken partition in
Fig. 6(b) and the gas expands freely to the final volume
$V(t^{\prime})=V_{\text{fin}}$ at time $t^{\prime}<\tau_{\text{eq}}$ during
$\mathcal{P}^{\prime}$. At $t^{\prime}$, the free expansion stops but there is
no reason a priori for $\xi_{\text{N}}=0$ so the gas is still inhomogeneous
($\xi_{\text{N}}\neq 0$). This is in a NEQ macrostate until $\xi_{\text{N}}$
achieves its EQ value $\xi_{\text{N}}=0$ during $\mathcal{P}^{\prime\prime}$,
at the end of which at $t=\tau_{\text{eq}}$ the gas eventually comes into
$\mathcal{B}_{\text{eq}}$ isoenergetically. The complete process is
$\overline{\mathcal{P}}=\mathcal{P}^{\prime}\cup\mathcal{P}^{\prime\prime}$
between $\mathcal{A}_{\text{eq}}$ and $\mathcal{B}_{\text{eq}}$. We briefly
review this expansion in the MNEQT Gujrati-Heat-Work .

We work in the extended state space with the two internal variables, which we
denote simply by $\mathfrak{S}$ here. Using Eq. (23a), we have

$dS(t)=dW(t)/T(t).$ (154a) Setting $P_{0}=0$ in Eq. (32b), we have
$dW(t)=\left\\{\begin{array}[c]{c}P(t)dV(t)+\mathbf{A}(t)\cdot
d\boldsymbol{\xi}(t)\text{ for }t<t^{\prime}<\tau_{\text{eq}},\\\
A(t)d\xi(t)\text{ for }t^{\prime}<t\leq\tau_{\text{eq}};\end{array}\right.$
(154b) here, we have used the fact that $V(t)$ does not change for
$\tau^{\prime}<t\leq\tau_{\text{eq}}$. Thus,  $\displaystyle\Delta S$
$\displaystyle=\int_{\overline{\mathcal{P}}}\frac{dW(t)+\mathbf{A}(t)\cdot
d\boldsymbol{\xi}(t)}{T(t)}>0,$ $\displaystyle\Delta Q$
$\displaystyle=\int_{\overline{\mathcal{P}}}dW(t)=\Delta W>0;$

the last equation is the fundamental identity in Eq. (22). The irreversible
entropy change $\Delta S$ from EQ macrostate from $\mathcal{A}_{\text{eq}}$ to
$\mathcal{B}_{\text{eq}}$ is the EQ entropy change $\Delta_{\text{i}}S$ is

$\Delta S\equiv S_{\text{fin}}-S_{\text{in}},$ (155)

and can be directly obtained since the EQ entropy $S(E,V)$ is known. The above
analysis is also valid for any arbitrary free expansion process $\mathcal{P}$
and not just $\overline{\mathcal{P}}$ as we have not used any information yet
about $\mathcal{A}_{\text{eq}}$ to $\mathcal{B}_{\text{eq}}$.

For $V_{\text{fin}}=2V_{\text{in}}$, $\Delta_{\text{i}}S=N\ln 2$, 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:

$S(\xi_{\text{V}},\xi_{\text{N}})=N_{\text{L}}\ln(eV_{\text{in}}/N_{\text{L}})+N_{\text{R}}\ln(eV^{\prime}/N_{\text{R}})+f(E_{0}).$

Thus, for arbitrary $\xi_{\text{V}}$ and $\xi_{\text{N}}$, we have
$\Delta_{\text{i}}S(\xi_{\text{V}},\xi_{\text{N}})=S(\xi_{\text{V}},\xi_{\text{N}})-S_{\text{in}}$.
We can determine the two affinities. A simple calculation gives

$\displaystyle A_{\text{V}}/T$ $\displaystyle=\partial
S/\partial\xi_{\text{V}}=1-\xi_{\text{N}}-\xi_{\text{V}},$ $\displaystyle
A_{\text{N}}/T$ $\displaystyle=\partial
S/\partial\xi_{\text{N}}=\ln\frac{(1-\xi_{\text{N}})\xi_{\text{V}}}{(1-\xi_{\text{V}})\xi_{\text{N}}}.$

We see that $A_{\text{V}}$ does not vanish when $V^{\prime}=V_{\text{fin}}$ as
discussed above. It is easy to verify that $\mathbf{A}$ vanish in
$\mathcal{B}_{\text{eq}}$. The pressure of the expanding gas is obtained by
using the derivative $\partial S/\partial V$ as usual. A simple calculation
yields

$\displaystyle\beta P$
$\displaystyle=(1-\xi_{\text{V}})\frac{N_{\text{L}}}{V_{\text{in}}}+\xi_{\text{V}}\frac{N_{\text{R}}}{V^{\prime}}$
(156) $\displaystyle=(1-\xi_{\text{V}})\beta P_{\text{L}}+\xi_{\text{V}}\beta
P_{\text{R}}.$

The last expression for pressure has a close similarity with the Toll-
Narayanaswamy equation (133), which should not be surprising. Before
expansion, we have $\beta P_{\text{in}}=N/V_{\text{in}}$ in
$\mathcal{A}_{\text{eq}}$ and $\beta P_{\text{fin}}=N/V_{\text{fin}}$ in
$\mathcal{B}_{\text{eq}}$ as expected. At EQ in $\mathcal{B}_{\text{eq}}$, the
entropy is given by $S_{\text{fin}}=N\ln(2eV_{\text{in}}/N)$, which gives
$\Delta_{\text{i}}S=N\ln 2$, 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.

###  XIV.2 Quantum Free Expansion

The sudden expansion has been studied Bender ; Gujrati-QuantumHeat ; Gujrati-
JensenInequality  quantum mechanically (without any $\xi_{\text{N}}$) 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 $\xi_{\text{N}}$ to
parallel the study of the classical expansion above but using the $\mu$NEQT.
Thus, we will closely follow the microstates and follow Ref. Gujrati-
JensenInequality  closely.

We make the very simplifying assumptions in the previous section to introduce
$\xi_{\text{N}}$. 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 $t>0$. This is slightly different from what 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 are confined in the initial chamber. We let
$\xi_{\text{N}}=N_{\text{R}}/N$, which gradually increases from
$\xi_{\text{N}}=0$ to $\xi_{\text{N}}=1$. 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 $\mathcal{P}^{\prime}$, $L(t)=L_{\text{fin}}$, but $N_{\text{R}}$
is still not equal to $N$ ($\xi_{\text{N}}\neq 0$). We then follow its
equilibration during $\mathcal{P}^{\prime\prime}$ as the gas come to EQ in the
larger chamber at the end of $\overline{\mathcal{P}}$ when $\xi_{\text{N}}=1$.
Again, there are two internal variables $L$ and $\xi_{\text{N}}$. 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. III) at inverse temperatures
$\beta_{\text{L}}$ and $\beta$, we find that the $N$-particle partition
function is given by

$Z_{N}(\beta_{\text{L}},\beta)=\left[Z(\beta_{\text{L}},L_{\text{in}})\right]^{N(1-\xi_{\text{N}})}\left[Z(\beta,L)\right]^{N\xi_{\text{N}}}$

so that the average energy is
$E_{N}(\beta_{\text{L}},\beta,L_{\text{in}},L,\xi_{\text{N}})=N(1-\xi_{\text{N}})/2\beta_{\text{L}}+N\xi_{\text{N}}/2\beta$.
As this must equal $N/2\beta_{0}$ for all values of $L$ and $\xi_{\text{N}}$,
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 $\Delta E_{k}=E_{k\text{,fin}}-E_{k\text{,in}}$ Gujrati-GeneralizedWork
; Gujrati-JensenInequality .

Below we will show that the quantum calculation here deals with an
irreversible $\overline{\mathcal{P}}$. 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

$P_{k}=-\partial E_{k}/\partial L=2E_{k}/L\neq 0$ (157)

determines the microwork

$\Delta
W_{k}={\displaystyle\int\nolimits_{L_{\text{in}}}^{L_{\text{fin}}}}P_{k}dL>0.$
(158)

It is easy to see that this microwork is precisely equal to $(-\Delta E_{k})$
as expected. It is also evident from Eq. (157) 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 nonnegative and equal: $\Delta Q=\Delta W>0$. 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 occupying the larger box. 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 $\xi_{\text{N}}$,
we have $\Delta
W_{\mathbf{k}}(L,\xi_{\text{N}})=-{\textstyle\sum\nolimits_{i}}\Delta
E_{k_{i}}$, where $i$ runs over the $N_{\text{R}}$ particles. We can compute
the macrowork, which turns out to be $\Delta
W_{N}(\xi_{\text{N}})=N\xi_{\text{N}}\Delta W>0$. The corresponding change in
the free energy is

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

which is consistent with Eq. (84a) for an isolated system for any
$\xi_{\text{N}}$.

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 find that for the isothermal expansion

$\Delta W_{N}=-\Delta\overline{F}_{N}=T_{0}\Delta_{\text{i}}S_{N}>0.$ (159)

after using Eq. (81). The same result is also obtained from the classical
isothermal expansion; see Eq. (84b). All this is in accordance with Theorem 36
in the MNEQT, as expected.

##  XV Discussion and Conclusions

As we noted in Sect. I, thermodynamics is a science of entropy and
temperature. As these macroquantities should uniquely describe the system, we
have required them to SI-quantities in developing the new NEQT, called the
MNEQT, to go beyond the EQ thermodynamics. We will now briefly summarize and
discuss our conclusions form this thermodynamics. We will consider them
separately.

###  XV.1 Unique NEQ $S$ in $\mathfrak{S}_{\mathbf{Z}}$

We first point out the important consequence of the restriction imposed by
quasi-independence discussed in Sects. II and (VI.4). By always dealing with
the SI-entropy $S$, which we have shown to be identical to the statistical
quantity $\mathcal{S}$ in all cases, we can appreciate the concept of quasi-
independence by considering $p_{k}$ that appears in Eq. (47). Considering
$\Sigma$ to consist of two subsystems $\Sigma_{1}$ and $\Sigma_{2}$, which are
in macrostates
$\mathcal{M}_{1}\doteq\left\\{\mathfrak{m}_{k_{1}},p_{k_{1}}\right\\}$ and
$\mathcal{M}_{2}\doteq\left\\{\mathfrak{m}_{k_{2}},p_{k_{2}}\right\\}$. If
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are quasi-independent and form
$\mathcal{M}$ for $\Sigma$, then

$p_{k}\simeq p_{k_{1}}p_{k_{2}}.$

As a consequence, the entropy additivity

$S(\mathbf{X}(t)\mathbf{,}t)\simeq
S_{1}(\mathbf{X}_{1}(t)\mathbf{,}t)+S_{2}(\mathbf{X}_{2}(t)\mathbf{,}t)$

is approximately satisfied. This has a generalization to many subsystems
$\left\\{\Sigma_{i}\right\\}$ given in Eqs. (74a-74c) so that they are _all_
quasi-independent. In terms of the volume $\Delta V_{i}$ of $\Sigma_{i}$ so
that $V={\textstyle\sum\nolimits_{i}}\Delta V_{i}$, the generalization can be
simply written in the form of the entropy additivity requirement over $\Delta
V_{i}$

$S(\mathbf{X}(t)\mathbf{,}t)={\textstyle\sum\nolimits_{i}}S_{i}(\mathbf{X}_{i}(t)\mathbf{,}t),$
(160a) in accordance with quasi-independence. The requirement of quasi-
independence forces the linear size $\Delta l_{i}$ of $\Sigma_{i}$ to be not
less than the correlation length $\lambda_{\text{corr}}$ as discussed in
Sects. II and VI.4; see also Gujrati-II . Thus, there will be no nonlocal
effects (Jou1, ; Hutter, , for example.) to consider in the MNEQT as they are
subsumed within each subsystem. Each subsystem has its own Hamiltonian
$\mathcal{H}$ containing all the information regarding interactions between
its constituent particles and internal variables (see how $\mathcal{H}$ in
Sect. ) so its microstates will contain the effects of all the interactions in
$\left\\{E_{k}\right\\}$.

By a proper choice of $\mathfrak{S}_{\mathbf{Z}}$,
$S(\mathbf{X}(t)\mathbf{,}t)$ can be replaced by a unique state function
$S(\mathbf{Z}(t))$. Similarly, by a proper choice of
$\mathfrak{S}_{\mathbf{Z}_{i}}$, a subspace of $\mathfrak{S}_{\mathbf{Z}}$,
$S_{i}(\mathbf{X}_{i}(t)\mathbf{,}t)$ can be replaced by a state function
$S_{i}(\mathbf{Z}_{i}(t))$ in $\mathfrak{S}_{\mathbf{Z}_{i}}$. By matching the
number of independent variables $n^{\ast}$ on both sides in Eq. (160a) as
discussed in Sect. VI.4, we ensures that $S(\mathbf{Z}(t))$ is uniquely
determined as a sum of $S_{i}(\mathbf{Z}_{i}(t))$ in accordance with Eq.
(74c). By replacing $\mathfrak{S}_{\mathbf{Z}_{i}}$ by
$\mathfrak{S}_{\mathbf{X}},\forall i$, and using Theorem 33 and Corollary 34,
we know that $S(\mathbf{Z}(t))$ uniquely exists in the MNEQT so there is no
freedom to choose any other variables on which $S(\mathbf{Z}(t))$ can depend
on. But the actual choice of $n<<n^{\ast}$ for a given
$\mathcal{M}_{\text{ieq}}$ is determined by the experimental setup. It is this
$n>n_{\text{obs}}$ that is physically relevant for $\mathcal{M}_{\text{ieq}}$
unless we are dealing with a $\mathcal{M}_{\text{eq}}$, where $n_{\text{obs}}$
is the number of independent variables in $\mathbf{X}$. The remaining
$n^{\ast}-n$ internal variables have equilibrated so their affinities vanish.

We turn back to $S(\mathbf{X}(t)\mathbf{,}t)$ and
$S_{i}(\mathbf{X}_{i}(t)\mathbf{,}t)$. If $\Delta l_{i}$ is less than the
correlation length, then Eq. (160a) must be replaced by

$S(\mathbf{X}(t)\mathbf{,}t)={\textstyle\sum\nolimits_{i}}S_{i}(\mathbf{X}_{i}(t)\mathbf{,}t)+S_{\text{corr}}(\mathbf{X}(t)\mathbf{,}t)$
(160b)

due to the correlations present among various $\Sigma_{i}$’s; cf. Eq. (2). In
a continuum NEQT introduced as CNEQT in Sect. VI.3, we are in the limit of
”infinitesimal volume element” over which
$S_{i}(\mathbf{X}_{i}(t)\mathbf{,}t)$ can be expressed as $s(\mathbf{r\mid
x}(\mathbf{r,}t)\mathbf{,}t)d\mathbf{r}$; here, $\mathbf{x}(\mathbf{r,}t)$ is
the local analog of $\mathbf{X}_{i}(t)$ over this volume element. In this
limit, Eq. (160b) reduces to

$S(\mathbf{X}(t)\mathbf{,}t)={\textstyle\int}s(\mathbf{r\mid
x}(\mathbf{r,}t)\mathbf{,}t)d\mathbf{r}+S_{\text{corr}}(\mathbf{X}(t)\mathbf{,}t),$
(160c)

where $S_{\text{corr}}$ cannot be converted to a volume integral as it is a
nonlocal quantity over at least the correlation length. Unfortunately,
$S_{\text{corr}}$ is almost invariably overlooked in the CNEQT Maugin ; Jou0 ,
which allows the function $s(\mathbf{r\mid x}(\mathbf{r,}t)\mathbf{,}t)$ to be
commonly identified as the entropy density. This is most probably a misleading
nomenclature as $S_{\text{corr}}(\mathbf{X}(t)\mathbf{,}t)$ has been
neglected. Even in EQ, the correct entropy density should be precisely
$S(\mathbf{X}(t)\mathbf{,}t)/V(t)$, which is not $s(\mathbf{r\mid
x}(\mathbf{r,}t)\mathbf{,}t)$. With this approximation, the SI-entropy is
replaced by $S_{\text{CT}}(\mathbf{X}(t)\mathbf{,}t)$ given by the integral on
the right side (CT for the CNEQT),

$S(\mathbf{X}(t)\mathbf{,}t)\overset{?}{\simeq}S_{\text{CT}}(\mathbf{X}(t)\mathbf{,}t)={\textstyle\int}s(\mathbf{r\mid
x}(\mathbf{r,}t)\mathbf{,}t)d\mathbf{r;}$ (160d)

the questionmark is because it is hard to estimate the error due to the
neglect of $S_{\text{corr}}$. Thus, the additivity of $s$ in the integral is
not the postulated additivity of $S$ even in EQ thermodynamics. To ensure that
$s$ satisfies the second law, it is postulated that $s$ shares this property
Jou0 . Because of these issues, we do not focus on $s$ in this review as the
volume elements are not usually quasi-independent unless we are at high enough
temperatures so that the correlation lengths become small enough to make them
quasi-independent.

The above limitations also distinguish the MNEQT with all CNEQT theories,
which fall under the category of the M̊NEQT. Here, we will briefly comment on
two successful theories.

The first one is the extended irreversible thermodynamics (ENEQT) Jou0 , a
well-known CNEQT, which also neglects $S_{\text{corr}}$ but treats the
corresponding entropy density $s_{\text{ET }}$(ET for the ENEQT) as a state
function involving various dissipative fluxes such as the heat flux. As said
above, one needs to be careful to incorporate nonlocal effects (Jou1, ;
Hutter, , for example.) in the CNEQT. In addition, the total entropy
$S_{\text{ET}}$ (Jou0, , see Eq. (5.66) and the discussion thereafter) is also
a state function involving the same fluxes for $\Sigma$, which violates
Corollary 34 about requiring a larger state space relative to $s_{\text{ET}}$:
Both $S_{\text{ET}}$ and $s_{\text{ET}}$ cannot have the same state space for
the additivity of entropy to be an identity; see also Remark 43. As the fluxes
determine MI-macroquantities $d_{\text{e}}Q$ and $d_{\text{e}}W$,
$S_{\text{ET}}$ is not a SI-entropy as $S$ is in the MNEQT.

The second one is the MNET Rubi-Vilar-Reguera  that is based on the idea of
_internal_ dof (dof${}_{\text{in}}$) proposed by Prigogine and Mazur
Prigogine2  for a $\Sigma$ in contact with a $\widetilde{\Sigma}$. The authors
provide a very good comparison of the MNET with other important theories to
which we direct the reader. Here, we only compare it with the MNEQT. The
emphasis in the MNET is to study slow relaxation in $\Sigma$ (cf. Sect. XI)
caused by the dof, that we denote here by dof${}_{\text{slow}}$ or by
observables $\mathbf{X}_{\text{slow}}$, and the corresponding part of $\Sigma$
by $\Sigma_{\text{slow}}$; the remainder of the system is denoted by
$\Sigma_{\text{fast}}$ with observables $\mathbf{X}_{\text{fast}}$. In
addition to $\widetilde{\Sigma}$, $\Sigma_{\text{slow}}$ is also allowed to
interact with another work medium, which we denote by
$\widetilde{\Sigma}^{\prime\text{(w)}}$ with an extra prime, with which it
exchanges macrowork only; see (Landau, , Sect. 20). This makes As is well-
known from the Tool-Narayanaswamy equation, see Eq. (133), and other works
Gujrati-II ; Langer ; Nagel ; Liu , $\Sigma_{\text{slow}}$ and
$\Sigma_{\text{fast}}$ usually have different temperatures, see Eq. (133),
pressures, see for example Eq. (132), etc. and they need not be equal to those
of $\widetilde{\Sigma}$. This is not considered in the MNET, where it is
assumed that $T=T_{0},P=P_{0}$, etc. so $\Sigma$ is assumed to be in EQ with
$\widetilde{\Sigma}$, so there cannot be any internal exchanges between
$\Sigma_{\text{slow}}$ and $\Sigma_{\text{fast}}$. The main focus in the MNET
is only on $\Sigma_{\text{slow}}$ and not the entire $\Sigma$. This makes
$\Sigma$ as the realization $\Sigma_{\text{C}}$; see Sect. X. The entropy in
the MNET in our discrete notation is given by

$S_{\text{MNET}}(\mathbf{X},t)=S_{\text{eq}}(\mathbf{X})-H(\mathbf{X}_{\text{slow}},t),$
(161)

where

$H(\mathbf{X}_{\text{slow}},t)\doteq{\textstyle\sum\limits_{k}}P_{k}(t)\ln\frac{P_{k}(t)}{P_{k}^{\text{eq}}}$

is the net contribution from $\mathbf{X}_{\text{slow}}$ (Rubi-Vilar-Reguera, ,
Eq. (7)); here, $P_{k}(t)$ is the probability of
$\mathfrak{m}_{k}^{\text{c}}$, the microstate in the _internal configurational
space_ (c) formed by dof${}_{\text{slow}}$ Prigogine2 , with
$\mathbf{X}_{\text{slow}k}$ denoting its configuration. As
$\Sigma_{\text{slow}}$ equilibrates with $\widetilde{\Sigma}^{\prime}$,
$P_{k}(t)\rightarrow P_{k}^{\text{eq}}$ so that $H(t)\rightarrow 0$.
Consequently, $S_{\text{MNET}}(\mathbf{X},t)\rightarrow
S_{\text{eq}}(\mathbf{X})$ so that
$S_{\text{MNET}}(\mathbf{X},t)-S_{\text{eq}}(\mathbf{X})=-H(\mathbf{X}_{\text{slow}},t)\
$is the contribution from the NEQ dof${}_{\text{slow}}$ vanKampen . The
presence of $P_{k}^{\text{eq}}$, which surely depends on the conjugate fields
of the medium $\widetilde{\Sigma}_{\text{slow}}$ controlling
$\mathbf{X}_{\text{slow}}$, makes $S_{\text{MNET}}(\mathbf{X},t)$ an MI-
quantity. Thus, it is different from our SI-entropy $S(\mathbf{X},t)$.
Moreover, as $\Sigma_{\text{slow}}$ interacts with
$\widetilde{\Sigma}^{\prime\text{(w)}}$, there is an exchange work
$\Delta\widetilde{W}^{\prime}=-\Delta_{\text{e}}W_{\text{slow}}$ done by
$\widetilde{\Sigma}^{\prime\text{(w)}}$ Landau . As $\mathbf{X}_{\text{slow}}$
in the MNET is controlled externally, it does not represent an internal
variable in the sense used in the MNEQT, which explains the use of
$\mathbf{X}_{\text{slow}}$ and not $\boldsymbol{\xi}$ to represent
dof${}_{\text{slow}}$. This is also consistent with Conclusion 51 since there
is no need to consider any internal variable for the realization
$\Sigma_{\text{C}}$. This is further clarified in the next paragraph. It is
easy to see that $\Delta_{\text{e}}W_{\text{slow}}$ satisfies Eq. (23b):

$\Delta E=T_{0}\Delta_{\text{e}}S-P_{0}\Delta
V-\Delta_{\text{e}}W_{\text{slow}};$ (162)

see (Landau, , see the derivation leading to Eq. (20.1)). Thus, MNET belongs
to the M̊NEQT as pointed out above. A configuration temperature for
$\mathfrak{m}_{k}^{\text{c}}$ is also introduced in the MNET by using
$s_{\text{c}k}=-\ln P_{k}(t)$, which is not considered in the MNEQT, where
only a global thermodynamic temperature is defined.

As the examples in Sect. X.3 have revealed, we can treat $\Sigma$ either as
$\Sigma_{\text{B}}$ or $\Sigma_{\text{C}}$. We need internal variables to
specify $\Sigma_{\text{B}}$ that help to describe whatever is going on within
$\Sigma$ without knowing these processes. While we do not need internal
variables to specify $\Sigma_{\text{C}}$, we need to know internal processes
such as the internal transfer
$dE_{\text{in}}(t)=d_{\text{e}}Q_{\text{in}}(t)$. Both realizations are
equivalent in the MNEQT. As the entropy is a unique function in
$\mathfrak{S}_{\mathbf{Z}}$, there is no room for any extra dependence such as
external fluxes in either realization; see Theorem 33. The internal fluxes
such as $dE_{\text{in}}(t)$ are needed for $\Sigma_{\text{C}}$, but they are
not controlled by the medium (they are present even if $\Sigma$ is isolated).
Thus, the MNEQT always deals with a SI-entropy.

Thus, the two entropies, $S_{\text{ET}}$ and $S$, are very distinct in many
ways, and cannot be compared as their predictions will be very different.

We now turn to the significance of the new NEQT (MNEQT) in the enlarged state
space, which is a SI-thermodynamics. As macroheat and macrowork are two
independent quantities in the MNEQT, it is clear that the notion of
temperature can be understood by merely focusing of the relationship between
$dQ$ and $dS$ for any arbitrary process; $dW$ plays no role in it. This is
what makes the MNEQT a very useful thermodynamic approach. It should be
stressed that the generalized macrowork $dW$ (the generalized macroheat $dQ$)
is not the same as the exchanged macrowork $d_{\text{e}}W$ (exchanged
macroheat $d_{\text{e}}Q$) with the medium unless $d_{\text{i}}W=0$
($d_{\text{i}}Q=0$), i.e., unless there is no _internal irreversibility_
caused by internal processes.

Thus, any deviation of $dW$ from __ $d_{\text{e}}W$ or $dQ$ from
$d_{\text{e}}Q$ in a process is the result of irreversibility due to internal
processes alone. Indeed, $d_{\text{i}}W$ is the macrowork done internally by
the system against all _dissipative forces within the system_ , see Eq. (82),
which explains why $d_{\text{i}}W$ is a measure of dissipative irreversibility
(Definition 38) within the system. In a similar manner, $d_{\text{i}}Q$ is the
macroheat generated internally by the system, which from
$d_{\text{i}}Q=d_{\text{i}}W$ is also due to all dissipative forces within the
system. It must be emphasized that irreversible macroheat transfer due to
temperature difference between a system and a medium does not affect
$d_{\text{i}}W=d_{\text{i}}Q$; see Eq. (86). On the other hand, irreversible
macroheat transfer between different _internal_ parts of $\Sigma$ will be part
of $d_{\text{i}}W=d_{\text{i}}Q$ as seen from the discussion of
$\Sigma_{\text{C}}$ in Sect. X.

The use of SI-quantities to specify a macrostate and microstates of a system
allows us to determine a statistical definition of the entropy of any
$\mathcal{M}_{\text{arb}}$ in Sect. IV, which is based on the ideas of
Boltzmann. Using the flatness hypothesis, see Remark 21, known to be valid for
macroscopic systems, we provide a simple proof of the second law in Sect.
IV.3; see also (Gujrati-Symmetry, , Theorem 4).

###  XV.2 Unique NEQ $T$ in $\mathfrak{S}_{\mathbf{Z}}$

As $dQ$ and $dS$ are SI-macroquantities, their extensivity requires a linear
relation between them for any $\mathcal{M}_{\text{arb}}$ as discussed in Sect.
V.2; see Eq. (62). The proportionality parameter is identified as the
temperature $T$ in Eq. (104). With this extension to deal with
$\mathcal{M}_{\text{nieq}}$ in
$\mathfrak{S}_{\mathbf{Z}^{\prime}}\supset\mathfrak{S}_{\mathbf{Z}}$, the
definition is applicable to any $\mathcal{M}_{\text{arb}}$. Thus, there is no
need to differentiate between $T_{\text{arb}}$ and $T$ in the MNEQT as said
earlier. The same definition also applies to an isolated system in an
arbitrary macrostate. Determining other fields related to $dW$ such as the
pressure do not pose any new complications in the new approach as they are
mechanical in nature as discussed in Sect. VI.2. We have thus proposed a novel
approach to define the unique temperature (see Theorem 33) that is applicable
to any $\mathcal{M}_{\text{arb}}$ by selecting the particular
$\mathfrak{S}_{\mathbf{Z}^{\prime}}$ where $\mathcal{M}_{\text{nieq}}$ is
convert to $\mathcal{M}_{\text{ieq}}$. Then the changes in
$\left\\{dp_{k}\right\\}$ and $\left\\{E_{k}\right\\}$ identify $dS,dQ$ and
$dW$ for given $T$ and $\mathbf{W}$ in $\mathfrak{S}_{\mathbf{Z}^{\prime}}$.
All these SI-macroquantities have the same values also in
$\mathfrak{S}_{\mathbf{Z}}$ for $\mathcal{M}_{\text{nieq}}$. All of this
strongly supports Proposition 47 as the fundamental axiom of the MNEQT that
can explain the behavior of any $\mathcal{M}_{\text{arb}}$.

It should be pointed out that the statistical definition of temperature in
Eqs. (5a) or (81) is not limited to extensive systems only. The discussion and
the conclusions are also valid for systems for which $dQ$ and $dS$ scale the
same way with $N$. We have only considered a linear scaling between the two
SI-macroquantities in this work. It should be pointed out that our concept of
temperature has some similarity with the idea of a contact temperature for a
system in thermal contact with a medium. The latter is introduced Muschik ;
Muschik-2016 ; Muschik-2020  by the inequality in Eq. (88) for a system in
thermal contact with a medium (but not when the system is isolated). We,
instead, define the temperature as an equality in Eq. (5a) for any arbitrary
macrostate, which works even for an isolated system. Eq. (88) is a consequence
of our definition.

We have seen that this definition of $T$ satisfies the four requirements, see
Criterion 32, listed in Sect. VI. This thus solves the dreams of Planck and
Landau Landau0 ; Planck . For example, we need to ensure that _macroheat_
$d_{\text{e}}Q$_, if it is transferred at different temperatures, always flows
from hot to cold_. Indeed, this is a fundamental requirement for a consistent
notion of temperature due to the second law; see Criterion C3. To the best of
our knowledge, this question has not been answered satisfactorily Muschik ;
Keizer ; Morriss ; Jou ; Hoover ; Ruelle  for an arbitrary nonequilibrium
macrostate. The question is not purely academic as it arises in various
contexts of current interest in applying nonequilibrium thermodynamics to
various fields such as the Szilard engine Marathe ; Zurek ; Kim , Jarzynski
process Jarzynski , stochastic thermodynamics Seifert , Maxwell’s demon Wiener
; Brillouin , thermogalvanic cells, corrosion, chemical reactions, biological
systems Hunt ; Horn ; Forland , etc. to name a few. Our approach thus finally
_solves_ the long-standing unsolved problem of defining the temperature for an
arbitrary macrostate in a consistent way that satisfy the stringent criteria
C1-C4 as proven in Theorem 40. Thus, $T$ must be treated as a genuine unique
temperature of the system in any macrostate $\mathcal{M}$.

Our definition of the temperature in Eqs. (5a) and (81) introduces $T(t)$ as a
global quantity, see C4 in Criterion 32, for the entire system and should not
be confused as a local quantity, which varies from region to region within the
system. This is true even if the system is inhomogeneous. Recall that we have
not imposed any requirement for the system to be homogeneous in our discussion
in Sec. VI. One may wonder if it makes any sense to call $T(t)$ the
temperature of the system even if it is inhomogeneous. It is possible to think
of an inhomogeneous system to be composed of a number of homogeneous
subsystems $\Sigma_{1},\Sigma_{2},\cdots$, each macroscopic in its own right.
In that case, we can assign a temperature $T_{1}(t),T_{2}(t),\cdots$ to
$\Sigma_{1},\Sigma_{2},\cdots$, respectively. It is then possible to relate
$T(t)$ to $T_{1}(t),T_{2}(t),\cdots$. We have explicitly shown this here by
considering only two subsystems in Sect. X when they are of identical sizes,
and Sect. XI when they are of different sizes, and treating the system as
$\Sigma_{\text{B}}$; see Eqs. (120a) and (133), respectively. For example, we
can divide $\Sigma$ into four subsystems $\Sigma_{1},\Sigma_{2},\Sigma_{3}$,
and $\Sigma_{4}$ of equal volumes and numbers of particles, but of different
energies. We can assume them in their own EQ macrostate
$\mathcal{M}_{i}(E_{i},V/4,N/4)$ at temperature $T_{i}$. Then, we will obtain
for $\Sigma=\Sigma_{\text{B}}$

$\beta(t)=[\beta_{1}(t)+\beta_{2}(t)+\beta_{3}(t)+\beta_{4}(t)]/4,$

which will require three internal variables as shown in Eq. (37). It is easy
to generalize the above relation to many subsystems and allowing the
possibility of different sizes. We can also allow for volumes to be different
for different subsystems as was done in deriving Eq. (132).

The possibility to study the formation of internal structures in $\Sigma_{i}$
in a NEQ $\Sigma$ should prove very useful to understand what drives their
formation. A very simple example of this is the pattern formation of Rayleigh-
Bérnard cells and their competition Hohenberg  in a fluid system. This pattern
formation has received a lot of attention recently (Chatterjee, , for
example), where stable cells are studied in nonturbulent convection in steady
state. It is found that each cell can be described in one of its EQ macrostate
to a very good approximation with its own temperature $T_{i}$. What our
approach shows is that the stable convection here can also be described by a
thermodynamic constant (steady) temperature $T$ associated with the steady
macrostate of the entire fluid.

Having a global temperature for an inhomogeneous system does not mean that if
we insert a thermometer in it anywhere, we will measure $T$. This is because
the act of ”inserting” a thermometer amounts to looking at the ”internal”
structure of the system, so we will be probing it as $\Sigma_{\text{C}}$.
Thus, if we insert it in $\Sigma_{1}$, we will record $T_{1}$; and if we
insert it in $\Sigma_{2}$, we will record $T_{2}$, and so on. This should not
be a surprise. We refer the reader to an interesting discussion of this issue
in Muschik-2016 .

As far as fields such as the pressure that are associated with $dW$ are
concerned, they do not pose the same kind of problem as they are purely
mechanical. All one needs to do is to take their instantaneous averages over
microstate probabilities for any arbitrary macrostate; see, for example, Eq.
(64a) involving such an average. This is possible because $\mathbf{W}$ is a
parameter, which makes $\mathbf{F}_{\text{w}k}$ fluctuating quantities over
$\mathfrak{m}_{k}$. This cannot be done for the temperature as $E$ is not a
parameter in the Hamiltonian. In this sense, we are considering a NEQ version
of the canonical ensemble in the MNEQT, which makes $E_{k}$ fluctuating over
$\mathfrak{m}_{k}$. Thus, $T$ plays the role of a ”parameter.” For this
reason, there is no way to define a temperature $T_{k}$ for $\mathfrak{m}_{k}$
and then take its average. What we can do in the MNEQT is to use the
temperature of various subsystems to obtain $T$ as is done in Eqs. (120a) or
(133).

We have shown that the definition of the irreversible macrowork
$d_{\text{i}}W$ is always _nonnegative_ as required by the second law; see Eq.
(77c). Various consequences of the second law are discussed in Sect. VI.5. We
have shown that, once a model for a system is given, we can identify the
required number and nature of internal variables as a computational scheme in
Sects. III and X, and later sections in the second half of the review. These
applications provide a clear strategy, once a model has been created, for
computation for an arbitrary thermodynamic process and should prove useful in
the field.

We have mostly alluded to $\mathcal{M}_{\text{ieq}}$’s above to highlight the
importance of internal variables in $\mathfrak{S}_{\mathbf{Z}}$, and to
$\mathcal{M}_{\text{nieq}}$’s for memory-effects with respect to
$\mathcal{M}_{\text{ieq}}$’s in $\mathfrak{S}_{\mathbf{Z}}$. In the absence of
a reliable model, finding $\mathfrak{S}_{\mathbf{Z}}$ in many cases may not be
easy to do. Compared to this, the identification of the state space
$\mathfrak{S}_{\mathbf{X}}$ is almost trivial based on the experimental setup.
Therefore, it is much more convenient to work with
$\mathfrak{S}_{\mathbf{X}}$, with respect to which all NEQ states possess
memory. Thus, _the novel approach we develop here is extremely useful as it
does not require knowing the internal variables_ as discussed in Sect. VIII.3.
However, for completeness, we have developed the MNEQT in
$\mathfrak{S}_{\mathbf{Z}}$, which can be easily adapted to
$\mathfrak{S}_{\mathbf{X}}$ by the procedure outlined in Remark 46.

It should be stressed, as noted in Remarks 25 and 50 that both $dQ$ and $dS$
exist, and so does their relation in Eq. (5a), regardless of the speed of the
arbitrary process $\mathcal{P}_{\text{arb}}$. This makes the Clausius equality
extremely important and useful as there is no restriction on its validity. It
is a genuine equality even in the presence of irreversibility without any
restriction on the process. This should be contrasted with the conventional
form of Clausius’s inequality in Eq. (94); the equality here holds only in the
absence of irreversibility.

The existence of a unique $T$ also appears in the microstate probabilities,
see Sect. VIII.2 that can be used to determine various fluctuations of
interest. These probabilities for $\mathcal{M}_{\text{ieq}}$ also give a
generalization of the EQ partition function to a NEQ partition function in Eq.
(102a). Because of the space limitation, we did not cover its consequences.

###  XV.3 Applications

We now come to the various applications of the MNEQT in the later half of the
review. The main lesson here is that several applications cannot be carried
out in the M̊NEQT. Apart from the many applications in Sect. X that we have
already discussed above, we have applied it to glasses when we derive the
famous Tool-Narayanaswamy equation in Sect. XI. It is a phenomenological
equation for which we provide a theoretical justification within the MNEQT. We
study an irreversible Carnot cycle in Sect. XII and derive its efficiency in
terms of the entropy generation $\Delta_{\text{i}}S$ and show how it differs
from the that of a reversible Carnot cycle. We also show how to compute
$\Delta_{\text{i}}S$ for a simple case in which each segment is irreversible
but between EQ macrostates; see Eq. (138).

The next important application is about friction and the Brownian motion in
Sect. XIII. By considering the relative motion between $\Sigma$ and
$\widetilde{\Sigma}$, we theoretically predict the well-know empirical fact
that friction is caused by the relative motion. We apply the approach to a
system of piston in a cylinder, a moving particle-spring system in a fluid,
and just a particle fluid system.

The last application is on free expansion in Sect. XIV. Here, we consider
classical and quantum expansion. In both cases, we make a simple model of the
process and show how it can used to determine $\Delta_{\text{i}}S$ between not
only two $\mathcal{M}_{\text{eq}}$ macrostates but also between two
$\mathcal{M}_{\text{nieq}}$; the latter cannot be determined in the M̊NEQT.

###  XV.4 Summary

To summarize, we have given a detailed review of the MNEQT in an extended
state space that was initiated a while back Gujrati-II ; Gujrati-Heat-Work ;
Gujrati-Entropy2 . Its main attraction is the variety of new applications,
many of which cannot be investigated in the M̊NEQT in which internal variables
play no direct role. The approach is applicable to a system in any arbitrary
macrostate $\mathcal{M}_{\text{arb}}$ and is used to provide a unique but very
sensible definition of the temperature, which satisfies all of its important
requirements. The useful aspect of the statistical approach needed for the
MNEQT is that it provides a _unique_ definition of generalized macroheat and
macrowork $dQ$ and $dW$, respectively, that are independent contributions in
the generalized first law in Eq. (23a); both quantities are system intrinsic
and obey the conventional partitioning in Eq. (4) valid for any process. These
macroquantities differ from the exchange macroheat and macrowork
$d_{\text{e}}Q$ and $d_{\text{e}}W$, respectively. Therefore, the MNEQT
directly considers the irreversible components $d_{\text{i}}Q$ and
$d_{\text{i}}W$ that originate from all _internal_ dissipation within the
system and satisfy an important identity $d_{\text{i}}Q\equiv
d_{\text{i}}W>0$, see Corollary 35, for any arbitrary irreversible process.
The irreversible macroquantities vanish for a reversible process. The
identification of a global and unique temperature $T$ is the most significant
aspect of the MNEQT in that it allows us to deal with $\Sigma$ as a blackbox
so that we do not need to know its interior. This requires a certain number of
internal variables, which explains the extended state space. We similarly
define other fields like the pressure, etc. statistically in terms of
generalized ”mechanical” forces; these also include generalized forces for
internal variables. All these definitions are instantaneous and are not
affected by how slow or fast any arbitrary process is. The latter only
determines the time window of relaxations of the internal processes, and the
choice of the state space. We believe that our novel approach provides a
first-ever definition of the temperature, pressure, etc. and of $dQ$ and $dW$
for any arbitrary macrostate, whether the system is isolated on in a medium.
Our approach is also valid to investigate nonequilibrium macrostates with
respect to $\mathfrak{S}_{\mathbf{X}}$, which brings memory effects in the
investigation. Thus, the approach is applicable in a wide variety of
situations, and fulfils Planck’s dream.

## References

  * (1) Th. De Donder and P.V. Rysselberghe, Thermodynamic Theory of Affinity: A Book of Principles. Oxford, England: Oxford University Press (1936).
  * (2) I. Prigogine, Thermodynamics of Irreversible Processes, Wiley-Interscience, New York (1971).
  * (3) S.R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, First Edition, Dover, New York (1984).
  * (4) B.G. Eu, Kinetic Theory and Irreversible Thermodynamics, John Wiley, New York (1992).
  * (5) G.D.C. Kuiken, Thermodynamics of Irreversible Processes, John Wiley, Chichester, (1994).
  * (6) H.C. Ottinger, Beyond Equilibrium Thermodynamics, Wiley, Hoboken (2005).
  * (7) S. Kjelstrum and D. Bedeaux, Nonequilibrium Thermodynamics of Heterogeneous Systems, World-Scientific, Singapore (2008).
  * (8) D.J. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Second edition, Cambridge University Press, Cambridge (2008).
  * (9) E. Fermi, Thermodynamics, Dover, New York (1956).
  * (10) F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, Inc. New York (1965).
  * (11) L.C. Woods, The Thermodynamics of Fluids Systems, Oxford University Press, Oxford (1975).
  * (12) J. Kestin, A course in Thermodynamics, vols. 1 & 2, McGraw-Hill Book Company, New York (1979).
  * (13) J.R. Waldram, The Theory of Thermodynamics, Cambridge University, Cambridge (1985).
  * (14) L.D. Landau, E.M. Lifshitz, Statistical Physics, Vol. 1, Third Edition, Pergamon Press, Oxford (1986).
  * (15) R. Balian, From Microphysics to Macrophysics, Vol. 1, Springer-Verlag, Berlin (1991).
  * (16) D. Kondepudi and I. Prigogine, Modern Thermodynamics, John Wiley and Sons, West Sussex (1998).
  * (17) L. Boltzmann, Lectures on Gas Theory, University of California Press: Berkeley, CA, USA (1964); see also, L. Boltzmann, Wien Ber. 76, 373 (1877).
  * (18) J.W. Gibbs, Elementary Principles in Statistical Mechanics, Scribner’s Sons, N.Y. (1902).
  * (19) D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics, Second Ed, Springer, Berlin (1996).
  * (20) W. Muschik, Forsch Ingenieurwes, 71: 149 (2007).
  * (21) W. Schottky, Thermodynamik. Springer, Berlin (1929).
  * (22) W. Muschik, arxiv:2010.14241v1.
  * (23) Y. Oono and M. Paniconi, Prog.Theor. Phys. Suppl. 130, 29 (1998).
  * (24) A. Bejan, Applied Engineering Thermodynamics, 3rd ed. New York, N.Y., USA, John Wiley (2006); see also, A. Bejan, J. Appl. Phys. 79, 1191 (1996).
  * (25) S-i Sasa and H. Tasaki, J. Stat. Phys. 125, 125 (2006).
  * (26) By definition, an observable is an extensive variable that can be controlled from outside the system such as a medium with which the system interacts. In contrast, an internal variable, to be introduced later, is an extensive variable that cannot be controlled from outside as it is generated internally within the system.
  * (27) We are assuming that there is only one species of stable particles, whose number $N$ is an observable, and is held fixed to fix the size of $\Sigma$. We can list $N$ in $\mathbf{X}$ if we keep another observable such as $V$ fixed to fix the size of the system. Here, we will keep $N$ fixed for the size. If there are several species $k=1,2,\cdots,r$ of particles that undergo $l$ distinct chemical reactions among themselves, then the individual numbers $N_{k},k\in\left\\{1,2,\cdots,r\right\\}$ of the species are not constant, only their total $N$ remains constant. In this case, we need distinct $l^{\prime}\doteq l-1$ extents of reaction deGroot ; Prigogine  as internal variables in $\mathbf{Z}$ as has been discussed later. If the species do not undergo chemical reactions among themselves, then $N_{k}$’s are individually observables. In this case, we can choose $l^{\prime}$ independent numbers that are contained in $\mathbf{X}$. In this review, we only consider a single species for simplicity.
  * (28) J. Keizer, Statistical thermodynamics of nonequilibrium processes, Springer-Verlag, New York (1987).
  * (29) R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Springer Verlag, Berlin (1992).
  * (30) Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer, New York (2010).
  * (31) W.T. Coffey and Y.P. Kalmykov, The Langevin equation, 4th ed., World Scientific, Singapore (2017).
  * (32) P.D. Gujrati, Phys. Rev. E 102, 012140 (2020).
  * (33) K. Sekimoto, J. Phys. Soc. Japan, 66, 1234 (1997).
  * (34) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997); C.R. Physique 8, 495 (2007).
  * (35) U. Seifert, Eur. Phys. J. B 64, 423 (2008).
  * (36) G.N. Bochkov and Yu.E. Kuzovlev, Sov. Phys. JETP 45, 125 (1977); ibid 49, 543 (1979).
  * (37) B.D. Coleman, J. Chem. Phys. 47, 597 (1967).
  * (38) G.A. Maugin, The Thermodynamics of Nonlinear Irreversible Behaviors,World Scientific,Singapore (1999).
  * (39) P.D. Gujrati, Phys. Rev. E 85, 041128 (2012); P.D. Gujrati, arXiv:1101.0438. It should be remarked that only macroheat exchange is considered in Sec. IVB so $dQ$ there stands for the exchange macroheat; compare with Definition 16.
  * (40) E. Bouchbinder and J.S. Langer, Phys. Rev. E 80, 031131 (2009); ibid 80, 031132 (2009).
  * (41) V.N. Pokrovskii, ISRN Thermodynamics, Article 906136 (2013); https://doi.org/10.1155/2013/906136.
  * (42) P.D. Gujrati, Entropy 20, 149 (2018).
  * (43) The Glass Transition and the Nature of the Glassy State, M. Goldstein and R. Simha, ed., N.Y. Academy of Sciences, New York (1976).
  * (44) R.O. Davies and G.O. Jones, Proc. R. Soc. (Lond.) A217, 26 (1953); Adv. Phys. 2, 370 (1953).
  * (45) I.S. Gutzow and J.W.P. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization, Second Edition, Springer, Berlin (2013).
  * (46) S.V. Nemilov, Thermodynamic and Kinetic Aspects of the Vitreous State, CRC Press, Boca Raton (2018).
  * (47) I. Prigogine, Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York (1967).
  * (48) P.D. Gujrati, arXiv:1304.3768.
  * (49) P.D. Gujrati, Entropy, 17, 710 (2015).
  * (50) P.D. Gujrati, Symmetry 2, 1201 (2010) and references therein.
  * (51) P.D. Gujrati, Phys. Rev. E 81, 051130 (2010); P.D. Gujrati, arXiv:0910.0026. It should be warned that $dQ$ in Eqs. (15)-(16) there stands for the exchange macroheat; recall Definition 16.
  * (52) J.M.G. Vilar and J.M. Rubi, Proc. Natl. Acad. Sci. USA 98, 11081 (2001); https://doi.org/10.1073/pnas.191360398.
  * (53) C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948), where a information entropy is derived that has identical mathematical form.
  * (54) See Eq. (7.9) in Landau , which shows how this entropy formulation emerges in statistical physics. It is applicable to both EQ and NEQ macrostates as is clear from Sec. 40 (see Eq. (40.7) in particular) dealing with NEQ ideal gas.
  * (55) P.D. Gujrati, arXiv:0908.1075.
  * (56) R.C. Tolman, The Principles of Statistical Mechanics,Oxford University, London (1959).
  * (57) S.A. Rice and P. Gray, The Statistical Mechanics of Simple Liquids, Interscience Publishers, New York (1965).
  * (58) E.T. Jaynes, Am. J. Phys. 33, 391 (1965).
  * (59) G.E. Crooks, Phys. Rev. E 60, 2721 (1999).
  * (60) L.P. Pitaevskii, Physics-Uspekhi 54, 625 (2011).
  * (61) K. Sekimoto, Stochastic Energetics, Springer, Berlin (2010).
  * (62) H. Spohn and J.L. Lebowitz, Adv. Chem. Phys. 38, 109 (1978).
  * (63) R. Alicki, J. Phys. A 12, L103 (1979).
  * (64) K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009).
  * (65) U. Seifert, Rep. Prog. Phys. 75, 126001, (2012).
  * (66) C.Van den Broeck and M. Esposito, Physica A 418, 6 (2015).
  * (67) E.A. Gislason and N.C. Craig, J. Chem. Educ. 84, 499 (2007).
  * (68) G.L. Bertrand, J. Chem. Educ. 82, 874-877 (2005).
  * (69) R.P. Bauman, J. Chem. Educ. 41, 102-104, 675, 676-677 (1964).
  * (70) D. Kivelson and I Oppenheim, J. Chem. Educ. 43, 233-235 (1966).
  * (71) (a) Th. M. Nieuwenhuizen, Phys. Rev. Lett. 80, 5580 (1998); (b) A.E. Allahverdyan and Th. M. Nieuwenhuizen, Phys. Rev. E 62, 845 (2000).
  * (72) E.G.D. Cohen and D. Mauzerall, J. Stat. Mech. P07006 (2004); Mol. Phys. 103, 21 (2005).
  * (73) C. Jarzynski, J. Stat. Mech. P09005 (2004).
  * (74) J. Sung, arXiv:cond-mat/0506214v4.
  * (75) D.H.E. Gross, arXiv:cond-mat/0508721v1.
  * (76) C. Jarzynski, arXiv:cond-mat/0509344v1.
  * (77) L. Peliti, J. Stat. Mech. P05002 (2008).
  * (78) J.M.G. Vilar and J.M. Rubi, Phys. Rev. Lett. 101, 020601 (2008).
  * (79) J. Horowitz and C. Jarzynski, Phys. Rev. Lett. 101, 098901 (2008).
  * (80) J.M.G. Vilar and J.M. Rubi, Phys. Rev. Lett. 101, 098902 (2008).
  * (81) L. Peliti, Phys. Rev. Lett. 100, 098903 (2008).
  * (82) J.M.G. Vilar and J.M. Rubi, Phys. Rev. Lett. 100, 098904 (2008).
  * (83) D.B. Amotz and J.M. Honig, J.Chem. Phys. 118, 5932 (2003); D.B. Amotz and J.M. Honig, Phys. Rev. Lett. 96, 020602 (2006); see also, J.M. Honig, Thermodynamics, 4${}^{\text{th}}$ edition, Academic Press, Oxford (2014).
  * (84) J.P.S. Bizarro, Phys. Rev. E 78, 021137 (2008).
  * (85) P.D. Gujrati, Phys. Rev. E 85, 041129 (2012); P.D. Gujrati, arXiv:1101.0431.
  * (86) P.D. Gujrati, arXiv:1105.5549.
  * (87) P.D. Gujrati, arXiv:1206.0702.
  * (88) D. Ruelle in Boltzmann’s Legacy, edited by G. Gallavotti, W. L. Reiter, and J. Yngvason, European Mathematical Society, Zürich, Switzerland (2008).
  * (89) S.F. Edwards and D.V. Grinev, Adv. Phys. 51, 1669 (2002).
  * (90) J.D. Bekenstein, Phys. Rev. D7,2333(1973).
  * (91) S.W. Hawking, Commun. Math. Phys. 43,199(1975).
  * (92) M. Planck in Festschrift Ludwig Boltzmann, p. 113, Barth, Leipzig (1904).
  * (93) L.D. Landau, Zh. Eksp.Teor. Fiz. 7,203 (1937); Collected papers of L.D. Landau, p. 169, ed. D. ter Haar, Gordon and Breach, New York (1965).
  * (94) J. Keizer, J. Chem. Phys. 82, 2751 (1985).
  * (95) W. Muschik, Aspects of non-equilibrium thermodynamics, World Scientific, Singapore (1990); see also W. Muschik and G. Brunk, Int. J. Engng Sci., 15, 377 (1977), and W. Muschik, C. Papenfuss and H. Ehrentraut, J. Non-Newtonian Fluid Mech., 96, 255 (2001).
  * (96) G.P. Morris and L. Rondoni, Phys. Rev. E 59, R5 (1999).
  * (97) J. Casas-Vázquez and D. Jou, Rep. Prog. Phys. 66, 1937 (2003).
  * (98) W.G. Hoover and C.G. Hoover, Phys. Rev. E 77, 041104 (2008).
  * (99) N.F. Ramsey, Phys. Rev. 103, 20 (1956) and references therein.
  * (100) B.D. Coleman, Arch. Rat. Mech. Anal. 17, 1 (1964).
  * (101) U. Lucia and G. Grisolia, Materials 14, 2024 (2021).
  * (102) P.D. Gujrati, arXiv:1702.00455.
  * (103) B.C. Eu and L.S. Garcia-Colin, Phys. Rev. E 54, 2501 (1996).
  * (104) I. Boyko and P.D. Gujrati, to be published.
  * (105) P.D. Gujrati and Tyler Johnson, to be published.
  * (106) P.D. Gujrati, arXiv:1512.08744.
  * (107) P.D. Gujrati, Phys. Lett. A 384, 126460 (2020).
  * (108) F. Wu, L. Chen, S. Wu, F. Sun, and C. Wu, J. Chem. Phys. 124, 214702 (2006).
  * (109) C.M. Bender, D.C. Brody and B.K. Meister, J. Phys. A 33, 4427 (2000), and Proc. Roy. Soc. A 461, 733 (2005).
  * (110) S.W. Doescher and M.H. Rice, Am. J. Phys. 37, 1246 (1969).
  * (111) D.W. Schlitt and C. Stutz, Am. J. Phys. 38, 70 (1970); C. Stutz and D.W. Schlitt, Phys. Rev. A 2, 897 (1970).
  * (112) L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Third Edition, Pergamon Press, Oxford (1977).
  * (113) L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford (1982).
  * (114) J. Casas-Vazquez, M. Criado-Sancho, and D. Jou, Physica A, 311, 353 (2002).
  * (115) M. Hutter and J.M. Brader, J. Chem. Phys. 130, 214908 (2009).
  * (116) D. Reguera, J.M. Rubi, and J.M.G. Vilar, J. Phys. Chem. B 109, 21502 (2005).
  * (117) I. Prigogine and P. Mazur, Physica 19, 241 (1953).
  * (118) I.K. Ono, C.S. O’Hern, D.J. Durian, S.A. Langer, A.J. Liu and S.R. Nagel, Phys. Rev. Lett. 89, 095703 (2002).