Correcting the Mistaken Identification of Nonequilibrium Microscopic Work
P. D. Gujrati

UATP/1804


pdg@uakron.edu
Department of Physics, Department of Polymer Science, The University of Akron,
Akron, OH 44325


Nonequilibrium work-Hamiltonian connection at the level of a microstate plays
a central role in diverse modern branches of statistical thermodynamics
(fluctuation theorems, quantum thermodynamics, stochastic thermodynamics,
etc.). The energy change dE_k for the kth microstate is always
but erroneously equated with the external work dW_k
done on the microstate, whereas the correct identification is
dE_k=-dW_k, dW_k being the generalized work done by the
microstate; the average of d_iW_k=dW_k+dW_k
represents the dissipated work. We show that d_iW_k is ubiquitous
even in a purely mechanical system, let alone a nonequilibrium system. As
d_iW_k is not included in dW_k, the current
identification dE_k=dW_k does not account for
irreversibility such as in the Jarzynski equality (JE). Using dW_k to
account for irreversibility, we obtain a new work relation that works even for
free expansion, where the JE fails. In the new work relation, Δ
E_k=-Δ W_k depends only on the energies of the initial and final
states and not on the actual process. This makes the new relation very
different from the JE. Thus, the correction has far-reaching consequences and
requires reassessment of current applications in theoretical physics.



Correcting the Mistaken Identification of Nonequilibrium Microscopic Work
    P.D. Gujrati
    January 31, 2017
=========================================================================






§ INTRODUCTION




 §.§ Motivation


In this work, we are mainly interested in an interacting system Σ in
the presence of a medium Σ as shown in Fig.
<ref>; the latter will be taken to be a collection of two parts: a
work source Σ_w and a heat source Σ_h, both of which can interact with the system Σ
directly but not with each other. We will continue to use Σ
to refer to both of them together. The collection Σ_0=Σ∪Σ forms an isolated system, which we assume to be
stationary, with all of of its observables such as its energy E_0, its
volume V_0, its number of particles (we assume a single species) N_0
constant in time, even if there may be internal changes going on when its
parts are out of equilibrium. These internal changes require some internal
variables for their description <cit.>. In the
following, we will assume Σ to be always in equilibrium
(which requires it to be extremely large compared to Σ) so that any
irreversibility going on within Σ_0 is ascribed to Σ alone,
and is caused by processes such as dissipation due to viscosity, internal
inhomogeneities, etc. that are internal to the system. Moreover, we assume
additivity of volume, a weak interaction between, and quasi-independence of,
Σ and Σ; the last two conditions ensure that the
energies and entropies are additive
<cit.> but
also impose some restriction on the size of Σ in that it cannot be too
small. In particular, the size should be at least as big as the correlation
length for quasi-independence as discussed earlier
<cit.>. In this work, we will
assume that all required conditions necessary for the above-mentioned
additivity are met.

All quantities pertaining to Σ carry no suffix; those for
Σ carry a tilde, while those of Σ_0 carry a suffix
0. As Σ is always in equilibrium, all its fields have the
same value as for Σ_0 so that its temperature, pressure, etc. are
shown as T_0,P_0, etc. as seen in Fig. <ref>. To simplify our
discussion, we are going to mostly restrict our discussion to two parameters
in the Hamiltonian; however, later in Secs. <ref>
-<ref>, we will consider a general set 𝐖 of work
parameters. In equilibrium (EQ), the state of Σ is described by two
observables E and V of which V appears as a parameter in its
Hamiltonian ℋ(𝐳,V); here, 𝐳 is the collection
of coordinates and momenta of the N particles. Throughout this work, N is
kept fixed so we will not exhibit it unless necessary. We can manipulate V
from the outside through Σ_w, which exerts an
external "force" such as the pressure to do some "work." For this work, V is
a symbolic representation of an externally controlled parameter, which is
commonly denoted by λ in the current literature and whose nature is
determined by the experimental setup. Thus, V here should be considered a
parameter in a wider sense than just the volume and P its conjugate force;
see Fig. (<ref>).


At the microscopic level, the state of the system is
specified by the set of microstates {𝔪_k}, their energy set {  E_k} and their probability
set {  p_k}. For the same set {𝔪
_k,E_k}, different choices of {  p_k}
describes different states, one of which corresponding to {
p_keq} uniquely specifies an EQ state; all other states are
called nonequilibrium (NEQ) states.

















The EQ entropy is a state function S(E,V). Away from EQ, we also need an
additional set of independent extensive internal variables to specify the
state <cit.>; we take a single variable ξ for our
study to simplify the presentation <cit.>. In this case,
V and ξ appear as parameters in the Hamiltonian ℋ
(𝐳,V,ξ). However, the discussion can be readily extended to many
internal variables. The force conjugate to ξ is known as the
affinity and conventionally denoted by A
<cit.>. A NEQ state for which its entropy is a state
function S(E,V,ξ) is said to be an internal equilibrium (IEQ) state
<cit.>. We will only consider IEQ states in this work for
simplicity. It is well known that the energy E=ℋ_V,ξ does not
change at fixed V and ξ as 𝐳 evolves in accordance with
Hamilton's equations of motion. Therefore, 𝐳 plays no role in the
performance of work so we will usually not exhibit it unless clarity is needed.

Focus on the Hamiltonian ℋ(V,ξ) provides us with the advantage
of obtaining a microscopic description of NEQ thermodynamics, where a central
role is played by the microscopic work-Hamiltonian relation at the level of
microstates {𝔪_k} of a NEQ system
Σ. The work relation is a key ingredient in developing a microscopic
NEQ statistical thermodynamics, where the emphasis is in the application of
the first law in diverse branches including but not limited to
nonequilibrium work theorems,
<cit.> stochastic
thermodynamics, <cit.> and quantum
thermodynamics, <cit.> to name a few. Once microscopic work
is identified, microscopic heat can also be identified by invoking the first
law. This makes identifying work of primary importance in NEQ thermodynamics.
Unfortunately, this endeavor has given rise to a controversy about the actual
meaning of NEQ work, which apparently is far from settled
<cit.>
.



 §.§ Mistaken Identity


The above controversy is distinct from the confusion about the meaning of work
and heat in classical nonequilibrium thermodynamics <cit.>
involving a system-intrinsic (SI) or medium-intrinsic
(MI) description, the latter only recently having been clarified
<cit.>
; see Sec. <ref> for more details and clarification.


By a SI-description (MI-description), we mean a description
that uses quantities associated with Σ only (with Σ
only); see for example Eqs.(<ref>) for dW and (<ref>) for
dW.


The SI-work dW is the work done by the system and the MI-work dW is the work done by the medium. At the level of microstates, one is faced
with the fact that both of these works dW_k and dW_k cannot be related to the change dE_k in the microstate energy of
𝔪_k. Indeed, as pointed out recently
<cit.>, the identification of microscopic work has
proven erroneous in that the external work dW_k, an MI
quantity for 𝔪_k, is identified with the SI energy change
dE_k for 𝔪_k, whereas in a NEQ state, the thermodynamics of
the medium and the system are very different. As pointed above, it is Σ that experiences dissipation <cit.>, which is partly
responsible for the irreversible entropy generation d_iS in the
system. This dissipation is not captured by dW_k in its
current formulation as we will see in this work. This is related to a
surprising fact hitherto unappreciated that at the microscopic level,
dW_k and dE_k always differ in a thermodynamic system,
even when dealing with a reversible process as shown by a thermodynamic
example in Sec. <ref>. In fact, they differ
even for a purely mechanical system as shown by the mechanical example in Sec.
<ref>.

Jarzynski Equality: As we will show here (this has been briefly
reported earlier <cit.>), not recognizing this subtle
fact has given rise to the "mistaken identity"

    dW_k?=dE_k,

between an MI-work and SI-energy change in the current literature; see for
example Jarzynski <cit.> who exploits the mistaken identity to prove
by now the famous Jarzynski equality (JE) discussed below in Sec.
<ref>.

Following Prigogine <cit.>, see also Fig. (<ref>) and
Sec. <ref> for clarification, we partition

    dE_k=d_eE_k+d_iE_k,

in which d_eE_k≡ dW_k represents the change
caused by the exchange with Σ through external work and
d_iE_k represents the change caused by internal processes going
on within Σ. As the work dW_k is the work done by 𝔪
_k at the expense of its energy loss, we have dW_k=-dE_k as we will
establish here. We will show by examples and arguments that d_i
E_k≠0 is ubiquitous not only in purely mechanical systems but
also in a reversible process and is necessary for dissipation. The main
thesis of this work to be established here is the following


It is the contribution d_iE_k that is
necessary but not sufficient to describe dissipation but is not captured by
dW_k=d_eE_k. Therefore, assuming dW_k=dE_k is to ignore dissipation completely regardless of the time
dependence of the work protocol. The SI-work relation dW_k=-dE_k, which
will be established here, contains dissipation and will provide the required
NEQ work relation.


The same conclusion also applies to the accumulation Δ_i
E_k≠0 along a trajectory, which is the path γ taken by the kth
microstate 𝔪_k during a NEQ process 𝒫_0 over a
time interval (  0,τ). The energy changes dE_k
,d_eE_k and d_iE_k occur during a segment of
𝒫_0 between t and t+dt, while Δ E_k,Δ
_eE_k and Δ_iE_k are cumulative changes over the
entire trajectory γ; see Sec.<ref> for
more details.

Quantum and Stochastic Thermodynamics: In a seminal paper laying down
the foundation of quantum thermodynamics, Alicki <cit.> identifies the average work dW, the average of
dW_k, done by external forces with the average of dE_k in
accordance with Eq. (<ref>). In a paper of similar importance
for stochastic thermodynamics, Sekimoto <cit.> also identifies the change dE_k in the energy E_k by
work variables with the work dW_k done by the external system.

What is Accomplished: The recent report
<cit.> was very compact. This work is designed to
elaborate on what was reported, to provide the missing details and more.
Throughout the work, we include an internal varaiable as a work variable. The
main emphasis here will be to demonstrate the ubiquitous nature of
d_iE_k, whose existence has not been previously appreciated. Not
recognizing its existence has resulted in the above mistaken identification.
It is the force imbalance between the internal and external forces that
generates d_iE_k and is very common in almost processes, even a
reversible one as we will demonstrate. This is the most important outcome of
the our approach, which emphasizes the importance of SI-quantities (such as
dE_k=-dW_k) that are very different from the MI-quantities (such as
d_eE_k=dW_k) in any process, even a reversible
one. The use of generalized work as isentropic change dW=-dE_S allows us
to calculate microscopic work dW_k, which changes dE_kbut not
p_k. On the other hand, the generalized heat dQ_k does not change
dE_k but changes p_k. This proves to be a great simplification and
allows us to treat dW_k and dQ_k as purely a mechanical and a
stochastic concept, respectively. In addition, as 𝔪_k uniquely
determines E_k for a fixed work set 𝐖, we find that dE_k is
uniquely determined by d𝐖 and not by the nature of the work
process. This provides a simplification in evaluating the cumulative change
Δ W_k and allows us to prove a new work theorem, both for the
exclusive and the inclusive Hamiltonian. This work theorem is very general and
we apply it successfully to the free expansion.



 §.§ Layout


The layout of the paper is the following. In the next section, we introduce
the extension of Prigogine's notation as it may be unfamiliar to many readers.
In Sec. <ref>, three important concepts of NEQ quantities that
will play a central role in this work: the concepts of SI- and MI-quantities,
and that of microwork. In Sec. <ref>, we give
two alternate versions of the first law using the MI- and SI-quantities, and
introduce thermodynamic forces and their role in dissipation that does not
seem to have been appreciated in the field. A new concept of force imbalance
is also introduced that plays an important role in microscopic thermodynamics
but has been hitherto overlooked. It is one of the central new quantities that
is ubiquitous and needs to be incorporated in any process, EQ or NEQ. In Sec.
<ref>, we introduce a new approach to investigate
NEQ thermodynamics at the microscopic level by using microstates. Here, we
introduce a different way of looking at work and heat, which we call
generalized work and generalized heat, that has only recently been developed
but proves very useful for a microscopic understanding of statistical
thermodynamics. The generalized work is isentropic and the generalized heat
satisfies the Clausius equality dQ=TdS even for an irreversible process.
This makes generalized work a purely mechanical quantity. The stochasicity
arises from heat but not from work. This results in a clear division between
work and heat, making them independent quantities as we explain here. In the
next section, we introduce exclusive and inclusive Hamiltonians. It is the
latter Hamiltonian that is used by Jarzynski, but we find basically no formal
difference between the two. In Sec. <ref>, we consider two
different models, a mechanical and a thermodynamic, to explain various
different kinds of NEQ work and heat. In Sec. <ref>, we show how
internal variables emerge in a system's Hamiltonian. We particularly pay
attention to the relative velocity that induces a force that is dissipative,
i.e., frictional and contributes to the dissipative work. In Sec.
<ref>, we finally develop our theoretical framework for NEQ
statistical thermodynamics and introduce statistical formulation of various
quantities. In Sec. <ref>, we propose a new work theorem both
for an exclusive and an inclusive Hamiltonian, which differs from the
Jarzynski formulation. The latter is well known for its failure for free
expansion. Therefore, we apply the new theorem to free expansion and show that
it holds in Sec. <ref>. Indeed, we argue that the new
theorem works for any arbitrary process and not just the free expansion. In
the last section, we give a brief summary of our results.



§ GENERALIZATION OF PRIGOGINE'S PARTITION


Any extensive SI quantity X(t) of Σ, see Fig. <ref>, can
undergo two distinct kinds of changes in time: one due to the exchange with
the medium and another one due to internal processes. Following modern
notation <cit.>, exchanges of X (we will suppress the time
argument unless clarity is needed) with the medium and changes within the
system carry the suffix e and i, respectively:

    dX≡ d_eX+d_iX;

here

    dX(t)≐ X(t+dt)-X(t).

For the medium, we must replace X by X so that dX(t)=X(t+dt)-X(t). For Σ_0, we must use
X_0 so that dX_0(t)≐ X_0(t+dt)-X_0(t). We will assume
additivity of X for Σ_0:
    X_0=X(t)+X(t).

For this to hold, we need to assume that Σ and Σ
interact so weakly that their interactions can be neglected; recall
that E is one of the possible X. As there is no irreversibility within
Σ , we must have d_iX(t)=0 for any
medium quantity X(t) and

    d_eX≐-dX=-d_eX.

It follows from additivity for Σ_0 that

    dX_0≡ dX+dX=d_iX.

This means that any irreversibility in Σ_0 is ascribed to Σ,
and not to Σ, as noted above. In a reversible change,
d_iX≡0. The additivity of entropy

    S_0(t)=S(t)+S(t)

requires Σ and Σ  to be quasi-independent as said
earlier so that

    dS_0=dS+dS.



As an example of the Prigogine's notation, the entropy change dS is written
as

    dS≡ d_eS+d_iS

for Σ; here,

    d_eS=-d_eS

is the entropy exchange with the medium and

    d_iS≥0

is irreversible entropy generation due to internal processes within
Σ and is nonnegative in accordance with the second law. It follows from
Eq. (<ref>) that d_iS is also the
entropy change dS_0 of Σ_0. For the energy change dE, we
write

    dE≡ d_eE+d_iE,

compare with Eq. (<ref>), except that

    d_iE≡0,

since no internal process, even chemical reaction, can change the energy of
the system. [The surprising fact is that d_iE_k≠0, as we will
establish below; see Eq. (<ref>).] Similarly, if
dW and dQ represent the SI work done by and the heat change of the system,
then

    dW≡ d_eW+d_iW,dQ≡ d_eQ+d_iQ.

Here, d_eW and d_eQ are the work exchange and
heat exchange with the medium, respectively, and d_iW≡
d_iW_0and d_iQ≡ d_iQ_0 are
irreversible work done and heat generation due to internal processes in
Σ. For an isolated system such as Σ_0, the exchange quantity
vanishes so that dX_0=d_iX_0. In particular,

    dS_0=d_iS_0;dE_0=d_iE_0≡0,dW_0=d_i
    W_0;dQ_0=d_iQ_0.

Thus, a description in terms of SI quantity alone can describe an isolated and
an open system, whether in equilibrium or not. This makes the SI description
highly desirable.

The use of exchange quantities results in an MI-description for Σ. No
internal variables are required to determine the exchange quantities since
their affinities for Σ vanish <cit.>,
which is always assumed to be in equilibrium. The SI-description always refers
to SI quantities (dW,dQ, etc.). The SI quantities may include internal
variables as their affinities need not vanish for Σ; in many cases,
they may not be readily measurable or even identifiable and require care in
interpreting results <cit.>. Therefore, the use of exchange quantities
(MI) is quite widespread. Despite this, the SI-description, which we call the
internal approach, is more appropriate to study nonequilibrium
processes at the microscopic level, even if we do not use ξ, since
ℋ, itself an SI quantity, plays a central role. In contrast, the
MI-description is called the external approach here.

Intuitive Meaning: Intuitively, dW denotes the work done by the
system, a part d_eW (=P_0dV) of which is transferred to
Σ_w through exchange and d_iW is spent
to overcome internal dissipative forces. Of dQ, a part d_e
Q is transferred from Σ_h through exchange and
d_iQ=d_iW is generated by internal dissipative forces.

The modern notation requires three different operators d,d_e and
d_i having the additive property

    d≡ d_e+d_i,

acting on any extensive quantity. They are linear differential operators,
which satisfy

    d_α(aX+bX^')       =ad_αX+bd_αX^',
    
    d_α(XX^')       =Xd_αX^'+(d_αX)X^';

here d_α stands for d,d_e and d_i, X and
X^' are two extensive quantities, and a and b are two pure
numbers. These operators also work on microstate quantities as well and will
prove very useful. We have already seen its use for E_k in Eq.
(<ref>).

Treating microstate probabilities {  p_k} as fractions
𝒩_k/𝒩 of the number of replicas in the kth
microstate out of a large but fixed number 𝒩 of replicas of the
same system, we can also apply d_α on p_k with the result

    dp_k=d_ep_k+d_ip_k,

that will be very useful later.



§ NEW CONCEPTS




 §.§ SI- and MI-works


In a NEQ state, the thermodynamics of the medium and the system are very
different. As pointed above, it is Σ that experiences dissipation
<cit.>, which is partly responsible for the irreversible
entropy generation d_iS in the system as we will see in Eq.
(<ref>). To clarify the difference
between the SI-description and the MI-description, we consider the
pressure-volume work and the work done by some internal variable ξ as an
example; see the gas confined in a cylinder with one end closed and a movable
piston on the other end in Fig. <ref>(a). The internal
variable may describe the nonuniformity of the gas in an irreversible process.
It should be recalled that a system in EQ must be uniform
<cit.> so any nonuniformity will result in a NEQ state. The
work in the two descriptions is the

    SI-work dW=PdV+Adξ

done by Σ (SI) or the

    MI-work dW=-P_0dV

done by Σ (MI) in terms of the instantaneous
pressure P of Σ or P_0of Σ, and their volume
change dV or -dV, respectively, where we have used the additivity of the
volume V_0=V+V and the fact that dV=-dV to
ensure a constant V_0. Notice that dW does not contain any
contribution from the internal variable since A_0=0
<cit.>. It represents the MI-work done on the system by
Σ and its negative denotes the exchange work between
Σ and Σ as the MI-work done by Σ
against Σ; we denote it by

    d_eW=-dW=P_0dV.




















Let dQ denote the heat added to the medium Σ,
here by the system. Then, d_eQ=-dQ is the heat
transferred (exchanged) to Σ by Σ and d_e
W=-dW is the work transferred (exchanged) by Σ to
Σ. Both d_eQ and d_eW refer to energy
flows across the boundary between Σ and Σ (see the
exchange d_eX in Fig. <ref>), and are determined by the
process of transfer that depends on the combined properties of Σ and
Σ. They are customarily used to express the first law for
Σ

    dE=d_eQ-d_eW

Our sign convention is that d_eQ is positive when it is added to
Σ, and d_eW is positive when it is transferred to
Σ. Once, work has been identified, the use of the first law
allows us to uniquely determine the heat.



 §.§ Microwork


The above macroscopic concepts of work represent thermodynamic averages (this
will become more clear later) <cit.>. This allows us
to identify the microscopic analogs dW_k and dW_k of dW
and dW, respectively, at the level of kth microstates
𝔪_k. The thermodynamic average connection, see also Sec.
<ref>, is the following:

    dW=⟨ dW⟩≐
    
    
    ∑_k
    p_kdW_k,dW=⟨ dW⟩≐
    
    
    ∑_k
    p_kdW_k,

which also introduces thermodynamic averaging ⟨·⟩ with respect to the instantaneous microstate probabilities {
p_k}. For the energy change dE_k, we have its average given by

    dE_S=⟨ dE⟩≐
    
    
    ∑_k
    p_kdE_k;

the reason for the special symbol becomes once we recall the definition of
entropy S:

    S=-⟨ln p⟩≐-
    
    
    ∑_k
    p_kln p_k.

As the above three averages are at fixed p_k's, they are evaluated at
fixed entropy. This is shown by the suffix S in dE_S. This quantity will
turn out to have an important role in our theory.

We will refer to dW_k in the following as microstate work or
microwork in short (but not dW_k). One can then
determine the accumulated work Δ W_k,ΔW_k and
Δ E_k along some path γ followed by the microstate in some
process 𝒫_0 by simply integrating dW_k but without the
probabilities:

    Δ W_k≐
    
    
    ∫_γ
    dW_k,ΔW_k≐
    
    
    ∫_γ
    dW_k,Δ E_k≐
    
    
    ∫_γ
    dE_k.

We observe that dW_k,dW_k and dE_k are defined
regardless of the probability p_k. Because of this, we wish to emphasize
the following important point, which we present as a


The cumulative quantities Δ W_k
,ΔW_k or Δ E_k exist or are defined regardless of
p_k. Thus, they exist or defined even if p_k=0.


This condition will prove useful when we consider free expansion of a
classical gas in Sec. <ref>.

Traditional formulation of NEQ statistical mechanics and thermodynamics
<cit.> takes a mechanical approach in which Σ follows its
classical or quantum mechanical evolution dictated by its SI-Hamiltonian
ℋ(V,ξ) <cit.>; the interaction with
Σ is usually treated as a very weak stochastic
perturbation. However, investigating microwork (SI approach) is very
convenient as p_k's do not change. Unfortunately, this description has
been overlooked by the current practitioners in the field who have
consistently used a MI-description for work by using the external approach but
then mistakenly confuse it with its SI-analog; see Eq. (<ref>).



 §.§ Jarzynski Process


As ΔW_k is not affected by the internal state (P,A, and
{  p_k}) of Σ, it is a choice candidate for studying
work fluctuations and for the Jarzynski process. By setting the MI-works
dW_k,ΔW_k equal to the SI-changes
dE_k,Δ E_k, respectively,

    dW_k=dE_k,ΔW_k=Δ E_k,

[compare with Eq. (<ref>)] Jarzynski developed the following
famous nonequilibrium work relation, the JE, using a very clever averaging
⟨·⟩ _0

    ⟨ e^-β_0ΔW^'⟩
    _0≐
    
    
    ∑_k
    p_k0e^-β_0ΔW^'_k=e^-β_0Δ
    F^'

for a Jarzynski process (we use a prime now reserved for the inclusive
approach discussed in Sec. <ref>). In Eq.
(<ref>), where an externally driven nonequilibrium process
𝒫_0 between two equilibrium states, the initial state
_eq and the final state _eq, at the
same inverse temperature β_0 is considered, ⟨·⟩ _0 refers to a special averaging with respect to the
canonical probabilities of the initial state _eq:

    p_k0^'≐ e^-β_0E_k^'/Z_in^'(β_0);

here, the suffix 0 refers to the initial state _eq,
Z_in^'(β_0) the initial equilibrium partition function
for the system, ΔW^'_k the cumulative work done on
𝔪_k during 𝒫_0, and Δ F^' the
change in the free energy over 𝒫_0
<cit.>. However, the average ⟨·⟩ _0 in Eq. (<ref>) does not represent a
thermodynamic average [compare with Eq. (<ref>)] over
𝒫_0 as recently pointed out <cit.>.
The inverse temperature along 𝒫_0 may not always exist or may be
different than β_0 due to irreversibility <cit.>. The
process 𝒫_0 consists of two stages: in the driving stage
𝒫, external work is done (ΔW_k^'≠
0), and the reequilibration stage 𝒫
≐𝒫_0\𝒫 (𝒫_0=𝒫∪𝒫) in which Σ is in thermal contact with the
heat bath Σ_h at β_0, during which no work
is done (ΔW_k^'≡0); Σ_h may or may not be present during 𝒫. The equality is
not obeyed for free expansion of a classical gas for which ΔW^'_k≡0 but Δ F^'≠0 as previously
observed <cit.>.



§ THE FIRST AND SECOND LAWS, THERMODYNAMIC FORCES AND
DISSIPATION




 §.§ Gibbs Fundamental Relation


For the IEQ entropy S(E,V,ξ), the Gibbs fundamental relation <cit.>
is given by

    dS=(dE+PdV+Adξ)/T,

where

    1/T=∂ S/∂ E,P=T∂ S/∂ V,A=T∂ S/∂ξ

are the inverse temperature, pressure and the affinity of the system that
represent its fields. As S is a SI-quantity, the above fields are also
SI-quantities. In EQ, the affinity vanishes, which means that S no longer
depends on ξ. In other words, ξ is no longer independent of E and
V in EQ so we can write it as ξ_eq≡ξ(E,V). Otherwise,
ξ is an independent variable in NEQ states. We can rewrite Eq.
(<ref>) as

    dE=TdS-PdV-Adξ.

We see that the first term on the right side of the equation represents the
change in the energy due to the entropy, while the last two terms represent
the isentropic change dE_S in the energy; see Eq.
(<ref>). These changes represent SI-changes in E and play an
important role in our approach. In particular, as V and ξ are work
variables that we collectively denote by 𝐖, we see from Eq.
(<ref>) that .  dE| _S is nothing but the SI-work
dW. Thus,

    TdS=dE-dE_S≡ dE_𝐖,

where dE_𝐖 represents the change dE at fixed 𝐖. For
brevity, we will call dE_𝐖 the isometric change in which
all work variables are held fixed. Thus, we arrive at the following conclusion:


 The change dE consists of two
independent contributions- an isentropic change dW=dE_S, and an isometric
change TdS=dE_𝐖.


The conclusion allows us to express the first law in an alternative form using
SI-quantities as we discuss below.



 §.§ The First Law


Using the exchange work d_eW=-dW=P_0dV, the (MI)
work done by Σ against Σ, and the work
partition

    dW=d_eW+d_iW,

we identify d_iW. We also introduce the exchange heat d_e
Q=-dQ=T_0d_eS between Σ and
Σ, which represents a MI-quantity. Defining

    d_iQ≐ d_iW

to emphasize the well-known fact that internal work can also be treated as
internal heat, we introduce

    dQ=d_eQ+d_iQ

so that the conventional MI-form of the first law in Eq. (<ref>)
can also be written in an SI-form

    dE=dQ-dW.

From now on, we will refer to dW and dQ (d_eW and d_e
Q) as generalized work and generalized heat (exchange work and
exchange heat), respectively, so that no confusion can arise.

If we compare this form with Eq. (<ref>), we recognize that the
last two terms there is dW above. Hence, the first term TdS there is
nothing but dQ introduced above. This provides us with an important
consequence of our approach
<cit.>
in which

    dQ=TdS;

compare with Eq. (<ref>). In other words, dQ is nothing but
the isometric change .  dE| _𝐖 in E: it is the
change when no work is done (dW=0). The relation dQ=TdS is very
interesting in that it not only turns the well-known Clausius inequality
d_eQ=T_0d_eS≤ T_0dS into an equality but also
makes the generalized work dW in Eq. (<ref>) isentropic, whereas
d_eW=-dW is not. This aspect of dW will prove useful
later. Moreover, Eq. (<ref>) allows us to uniquely identify
generalized heat and work; this work is always isentropic and this heat is
always isometric. On the other hand, the MI-heat and the MI- work suffer from
ambiguity; see, for example, Kestin <cit.>. We summaries these
important observations here as


 The generalized work dW is isentropic
change in the energy, while the generalized heat dQ=TdS is the isometric
change in the energy.


It is important to draw attention to the following important fact. We first
recognize that the first law in Eq. (<ref>) really refers to the
change in the energy caused by exchange quantities. Therefore, dE on the
left truly represents d_eE; see Eq. (<ref>).
Accordingly, we write Eq. (<ref>) as

    d_eE=d_eQ-d_eW.

Subtracting this equation from Eq. (<ref>), we obtain the identity

    d_iE=d_iQ-d_iW≡0,

where we have used Eq. (<ref>). The above equation is nothing but the
identity in Eq. (<ref>). However, the analysis also demonstrates
the important fact that the first law can be applied either to the exchange
process or to the interior process. In the last formulation, it is also
applicable to an isolated system.

The analog of d_iW between two nearby equilibrium states is
well known in classical thermodynamics and is usually called the
dissipated work; see for example, p. 12 in Woods <cit.>. The
dissipated work (P-P_0)dV is the difference between the generalized work
dW=PdV and the reversible work dW_rev=P_0dV done by Σ between the same two equilibrium states and is always nonnegative in
accordance with Theorem <ref> that will be
proved later in its general form (which refers to d_iW between any
two nearby arbitrary states that need not necessarily be equilibrium
states). If they are not, then the procedure described by Wood  to determine
dissipated work cannot work as there is no reversible path connecting two
nonequilibrium states. This makes our approach for introducing d_i
W=dW-d_eW much more general.



 §.§ The Second Law


We rewrite dE in Eq. (<ref>) as

    dW=TdS-dE=-dF+(T-T_0)dS,

where

    F=E-T_0S

is the Helmholtz free energy; note the presence of T_0 and not T in F.
As (T-T_0)dS≤0 established later, see Eq. (<ref>
), we conclude

    dW≤-dF,

as a consequence of the second law. It is also easy to see from this that

    dW≥ dF+d_iW≥ dF,

where we have used the fact that d_iW≥0, which is again a
consequence of the second law as shown in Theorem
<ref>. Similar inequalities also hold in the
inclusive approach:

    dW^'≤-dF^',dW^'≥ dF^'.





 §.§ Dissipation and Thermodynamic Forces


We first turn to the relationship of dissipation with the entropy generation
d_iS for Σ. The general result for the present case is

    Td_iS=(T_0-T)/T_0d_eQ+(P-P_0)dV+Adξ≥0;

see, for example, Ref. <cit.>. In accordance with the second law in
Eq. (<ref>), each term on the right of the first equation must be
nonnegative. The first term is due to heat exchange d_eQ with
Σ at different temperatures, the second term is the
irreversible work due to pressure imbalance and the third term is the
irreversible work due to ξ. It is customary to think of A as the
instantaneous "force" associated with the "displacement" dξ. As the
affinity A_0=0 for Σ, A-A_0≡ A also denotes
the affinity imbalance. Therefore, the last two terms above denote dissipated
work, which is customarily called dissipation; the first term is not
considered part of it. Therefore, in this work, we will use dissipation to
denote the dissipated work.

It is clear that the root cause of dissipation is a "force imbalance"
P(t)-P_0,A(t)-A_0≡ A(t), etc.
<cit.>
between the external and the internal forces performing work, giving rise to
an internal work d_iW due to all kinds of force imbalances, which
is not captured by using the MI-work dW unless we recognize that
there must be some nonvanishing force imbalance to cause dissipation.
The irreversible or dissipated work is <cit.>

    d_iW=(P-P_0)dV+Adξ≥0,

which is generated within Σ; we give a general proof of this result
(see Eq. (<ref>)) in Theorem
<ref>. The pressure and affinity imbalance
P-P_0and A-A_0≡ A are commonly known as thermodynamic
forces driving the system towards equilibrium.

If we include the relative velocity between the subsystem Σ_p
formed by the piston and the subsystem Σ_gc of the gas and the
cylinder (Σ=Σ_p∪Σ_gc), we must account
for <cit.> an additional term -𝐕·d𝐏
_p in d_iW due to the relative velocity 𝐕:

    d_iW=(P-P_0)dV-𝐕·d𝐏_p+Adξ.

This is reviewed in Sec. <ref>. We will come back to this term
later when we consider the motion of a particle attached to a spring; see Fig.
<ref>(b).

The irreversible work is present even if there is no temperature difference
such as in an isothermal process as long as there exists some nonzero
thermodynamic force. The resulting irreversible entropy generation is then
given by

    Td_iS=d_iW≥0.

We summarize this as a conclusion <cit.>:


To have dissipation, it
is necessary and sufficient to have a nonzero thermodynamic force. In its
absence, there can be no dissipation regardless of the time-dependence of the
work process.




§ MICROSTATE THERMODYNAMICS


We have applied the internal approach microscopically to the set {𝔪_k} of microstates
<cit.> to obtain a
microscopic representation of (generalized) work and heat in terms of
the set of microstate probabilities {  p_k} and other
SI-quantities. We expand on this approach here and exploit it. As we will be
dealing with microstates, we will mostly use their energy set {
E_k} instead of ℋ.



 §.§ General Discussion


As ℋ depends on V and ξ as parameters, E_k also depends
on them as parameters so we write it as E_k(V,ξ). Then the thermodynamic
energy is given by

    E=⟨ E⟩≐
    
    
    ∑_k
    p_kE_k(V,ξ).

The dependence of p_k is not important. Now, we have

    dE=d⟨ E⟩ =
    
    
    ∑_k
    p_kdE_k+
    
    
    ∑_k
    E_kdp_k;

here,

    dE_k≐(∂ E_k/∂ V)dV+(∂ E_k/∂ξ)dξ.

As {  p_k} is not changed in the first sum, it is evaluated
at fixed entropy of Σ <cit.>
so this isentropic sum must be identified with -dW:

    dW=-⟨ dE⟩≐-
    
    
    ∑_k
    p_kdE_k;

the summand must denote dW_k; compare with Eq. (<ref>):

    dW_k=-dE_k.

It follows from Eq. (<ref>) that

    d_eW_k=-d_eE_k,d_iW_k=-d_iE_k.

We also find that dW only changes E_k's but not p_k's. The second
sum in dE must be identified with the generalized heat dQ

    dQ≐
    
    
    ∑_k
    E_kdp_k=TdS,

see Eq. (<ref>). It is possible to express dQ also as an average
by introducing Gibbs index of probability <cit.>

    η_k≐ln p_k.

We have

    dQ=⟨ dQ⟩ =⟨ Edη⟩≐
    
    
    ∑_k
    p_kE_kdη_k,

so that we can formally introduce a quantity

    dQ_k≐ E_kdη_k

for 𝔪_k, which we will refer to as microstate heat or
microheat in short. It also changes the probability at fixed E_k.
Thus, both dQ and dQ_k are stochastic quantities. We should emphasize
that our concept of microstate heat is very different from the concept of heat
currently used in the literature.

We also observe that the generalized heat dQ and dQ_k only changes
p_k's, but not E_k's. Therefore, the following aspects of the
generalized quantities will be central in our discussion later, which we
present as two conclusions:


The microwork dW_k changes E_k without
changing p_k. Thus, a purely mechanical approach can be used for
microwork. The effect of microheat is to change p_k but not E_k so it
is microheat that makes a thermodynamic process stochastic by changing p_k.



While the microheat dQ_k doe not change
E_k, it does contribute to the energy change dE through dQ=


∑_k
E_kdp_k as p_k's change.


The following conclusion also follows from the above general discussion.


Comparing d⟨ E⟩ in Eq. (<ref>)
with ⟨ dE⟩ =-dW given above, we conclude that

    d⟨ E⟩≠⟨ dE⟩ .

The difference is dQ above.




 §.§ Microscopic Force Imbalance


The force imbalance necessary for irreversibility has its root in a similar
imbalance at the microstate level <cit.>. We only have
to recall Eq. (<ref>) and to recognize that

    P=
    
    
    ∑_k
    p_kP_k,A=
    
    
    ∑_k
    p_kA_k,

where P_k≐-∂ E_k/∂ V,A_k≐-∂
E_k/∂ξ are the instantaneous pressure and affinity associated with
𝔪_k at that time t. (We can similarly introduce
𝐕_k if we are interested in it.) It follows from Eq.
(<ref>) that dE_k=-dW_k. Using Eqs. (<ref>)
and (<ref>), along with (<ref>), (<ref>),
(<ref>) and (<ref>), we conclude that



    dW_k     =-dE_k=P_kdV+A_kdξ,
    
    dW_k     =-d_eW_k=d_eE_k=-P_0
    dV,
    
    d_iW_k     =-d_iE_k=(P_k-P_0)dV+A_kdξ.

The important point to note is that the force imbalances P_k-P_0 and
A_k determine the internal changes d_iE_k or d_i
W_k for 𝔪_k. On the other hand, dE_k is determined by
the SI-fields P_k and A_k, while d_eE_k is determined by
the MI-field P_0. From now on, we will reserve the use of force imbalance
to denote it only at the microstate level. At this level, it appears as a
mechanical force imbalance. In contrast, we will refer to macroscopic force
imbalance from now on as thermodynamic forces. We will in the following see
that there are reasons to make a clear distinction between the two; compare
Conclusion <ref> with Conclusion
<ref>.

The following comment for various microscopic works in Eq.
(<ref>) should be obvious: the average of Eq.
(<ref>) gives Eq. (<ref>), the average of Eq.
(<ref>) yields Eq. (<ref>), and the average
of Eq. (<ref>) gives Eq. (<ref>).
This helps us extend Conclusion
<ref> for thermodynamic forces
to a new conclusion valid only for mechanical force imbalances P_k-P_0
and A_k to accommodate the possibility when they result in


    ∑_k
    p_k(P_k-P_0)=0,
    
    
    ∑_k
    p_kA_k=0,

even if P_k-P_0 and A_k are nonzero. In this case, the system is in
EQ despite the fact that the force imbalance is present. Indeed, we will see
later that the presence of force imbalance is a ubiquitous phenomenon and must
be incorporated even if they result in zero thermodynamic forces. The new
conclusion is the following:


To have dissipation, it is
necessary but not sufficient to have a nonzero mechanical force imbalance.
Even in its presence, there may be no dissipation if Eq. (<ref>)
is satisfied. Nevertheless, it must be incorporated for a consistent theory.




 §.§ Accumulation of thermodynamic work along a Trajectory


Consider a path γ taken by the kth microstate 𝔪_k
during a NEQ process 𝒫_0 and dW_k and dW_k
during t and t+dt along γ. The net works Δ W_k and
ΔW_k is the integral over γ as shown in Eq.
(<ref>). The thermodynamic works are averages ⟨·⟩ over {  p_k} (not to be confused with
⟨·⟩ _0 in Eq. (<ref>)):

    dW       =⟨ dW⟩ =
    
    
    ∑_k
    p_kdW_k, dW=⟨ dW⟩ =
    
    
    ∑_k
    p_kdW_k,
    Δ W       =
    
    
    ∫_γ
    dW=
    
    
    ∫_γ
    ⟨ dW⟩ =
    
    
    ∫_γ
    
    
    ∑_k
    p_kdW_k ,
    ΔW     =
    
    
    ∫_γ
    dW=
    
    
    ∫_γ
    ⟨ dW⟩ =
    
    
    ∫_γ
    
    
    ∑_k
    p_kdW_k .





 §.§ Exchange and Irreversible Components


Let us focus on the exclusive approach and express exchange quantities
microscopically. Using the partition of dE_k in dW_k, we have

    d_eW=-
    
    
    ∑_k
    p_kd_eE_k, d_iW=-
    
    
    ∑_k
    p_kd_iE_k≥0.

Similarly, using dp_k≡ d_ep_k+d_ip_k from Eq.
(<ref>), we have

    d_eQ=
    
    
    ∑_k
    E_kd_ep_k, d_iQ=
    
    
    ∑_k
    E_kd_ip_k≥0,

and

    d_eS=-
    
    
    ∑_k
    ln p_kd_ep_k,  d_iS=-
    
    
    ∑_k
    ln p_kd_ip_k≥0.

We finally have

    Δ_eW       =-
    
    
    ∑_k
    
    
    ∫_γ
    p_kd_eE_k,Δ_iW=-
    
    
    ∑_k
    
    
    ∫_γ
    p_kd_iE_k≥0,
    Δ_eQ       =
    
    
    ∑_k
    
    
    ∫_γ
    E_kd_ep_k, Δ_iQ=
    
    
    ∑_k
    
    
    ∫_γ
    E_kd_ip_k≥0,
    Δ W       =-
    
    
    ∑_k
    
    
    ∫_γ
    p_kdE_k,Δ Q=
    
    
    ∑_k
    
    
    ∫_γ
    E_kdp_k,

along with Δ_iW=Δ_iQ. The equation for Δ
W above and its differential form dW provide the correct identification at
the microscopic level of the SI-quantities, and must be used to account for irreversibility.

Furthermore, using Eq. (<ref>) in Eq. (<ref>
), we have

    d_iE     ≐
    
    
    ∑_k
    p_kd_iE_k+
    
    
    ∑_k
    E_kd_ip_k=0,
    
    dE       =d_eE≐
    
    
    ∑_k
    p_kd_eE_k+
    
    
    ∑_k
    E_kd_ep_k,

where we have used the identityd_iW=d_iQ from Eq.
(<ref>) in the top equation to show consistency of the above
approach with the important identity in Eq. (<ref>); the first term
here represents (-d_iW) and the second term stands for
d_iQ.


 Most important conclusion of our approach is that even
if d_iE_k≠0, d_iE=0 as is well known; see Eq.
(<ref>).


The Eq. (<ref>) reproduces Eq. (<ref>), which again shows
consistency of our approach with thermodynamics. 



§ EXCLUSIVE AND INCLUSIVE
HAMILTONIANS


The Hamiltonian ℋ(V,ξ) above is given in terms of SI-variables
and has no information about the medium. Because of this, it has become
customary to call it the exclusive Hamiltonian. However, as V is coupled to
the external pressure P_0, it is also useful to introduce another
Hamiltonian in which V is replaced by P_0. This is a well-known trick in
classical equilibrium thermodynamics (P=P_0), where such a transformation
is known as the Legendre transform. Instead of considering the energy E(V),
we consider its Legendre transform E(V)+P_0V=E(V)+PV. It is well known
that in equilibrium, this transform is nothing but the enthalpy H(P)
<cit.>. A simple example is to consider the situation in Fig.
<ref>(a) in which the system Σ is formed by the gas
in the left chamber delimited by the cylinder and the right chamber with the
movable piston and the cylinder is connected to a large sealed container with
a gas at pressure P_0≠ P; the right chamber and the sealed container
along with the the cylinder and the piston forms Σ.

Taking this cue of the EQ Legendre transform, the inclusive Hamiltonian
ℋ^' is defined as <cit.>

    ℋ^'(V,ξ,P_0)=ℋ(V,ξ)+P_0V,

even when we are dealing with a NEQ situation such as when P≠ P_0
and/or A≠0; here

    P_0≐-∂E/∂V

is the conjugate field of the medium; compare this with P≐-∂
E/∂ V. As we have shown <cit.>, the NEQ
Legendre transform has a very different property. We find that

    V=∂ℋ^'/∂ P_0,A=-∂ℋ^'/∂ξ,P^'≐-∂ℋ^'/∂ V=P-P_0≠0

so that V,ξ and P_0 are parameters in ℋ^' unless
P^' vanishes, which will happen only under mechanical equilibrium.
Therefore, we can think of ℋ^'(V,ξ,P_0) as another
Hamiltonian with the three parameters V,ξ and P_0; each parameter will
have its own contribution to work dWfor the inclusive Hamiltonian
ℋ^'(V,ξ,P_0). However, the main difference is that
P_0 is not an extensive parameter. As it will be treated as a work
variable, the conjugate field will be an extensive variable. The change
dE^'in the inclusive Hamiltonian ℋ^' is

    dE^'=-P^'dV-Adξ+VdP_0,

which will reduce to the EQ form dE^'=VdP_0 for the EQ enthalpy. As
has already been discussed in the literature <cit.> the
inclusive Hamiltonian ℋ^' should be thought of as referring
to a different system Σ^'. This means that ℋ^' or the corresponding energy E^' is an SI quantity for
Σ^'. Thus, there are SI analogs of generalized work dW^', generalized heat dQ^', etc. along with MI analogs like
dW^', etc. Indeed, there is no fundamental
difference between the exclusive and inclusive approaches. All results for the
inclusive Hamiltonian can be simply converted for the inclusive Hamiltonian by
simply adding a prime on all the quantities and adding the contribution from
the parameter P_0. This will become clear as we go along.

We see from Eqs. (<ref>) and (<ref>) that

    d(E^'-E)=d(P_0V).

This is basically what we see from the definition of ℋ^' in
Eq. (<ref>): the difference ℋ^'-ℋ is nothing but P_0V. This remains true also for the
accumulated change

Comparison with Jarzynski's Approach: Finally,  we remark that
Jarzynski's discussion <cit.> of the inclusive Hamiltonian differs
from our discussion in that Jarzynski overlooks the first term -P^'dV
in dE^' above. Accordingly, he assumes the thermodynamic force
P^'=0, which makes his conclusion very different from us; see
Conclusion <ref>. As this is an
important difference, we state it as a


Jarzynski's approach to the
inclusive Hamiltonian sets P^'=P-P_0=0. This results in the
complete absence of dissipation as he does not consider any internal variable
ξ in his analysis.




§ SOME CLARIFYING EXAMPLES


Before proceeding further, we clarify the distinction between various
thermodynamic works dW,d_eW and dW or their
microstate analogs by two simple examples. It is also clear from the previous
section that the pressure difference P^' plays an important role in
capturing dissipation; see Conclusion
<ref>. Only under mechanical
equilibrium do we have P^'=0. The following examples will make this
point abundantly clear that a nonzero force imbalance like P^' is
just as common even in classical mechanics whenever there is absence of
mechanical equilibrium.



 §.§ Force Imbalance in a Mechanical System: A Microstate
Approach




  §.§.§ Exclusive Approach


Consider as our system a general but a purely classical mechanical
one-dimensional massless spring of arbitrary exclusive Hamiltonian
ℋ(x) with one end fixed at an immobile wall on the left and the
other end with a mass m free to move; see Fig. <ref>(b).
The center of mass of m is located at x from the left wall. For the moment
we consider a vacuum instead of a fluid filling system so we do not need to
worry about any frictional drag; we will consider this complication in Sec.
<ref>. The free end is pulled by an external force (not
necessarily a constant) F_0 applied at time t=0; thus x acts as a work
parameter. We do not show the center-of-mass momentum p as it plays no role
in determining work. We treat the system purely mechanically. Therefore, the
exercise here should be considered as discussing a microstate of the system.

Initially the spring is undisturbed and has zero SI restoring spring
force

    F_s=-∂ℋ/∂ x.

The total force

    F_t=F_0+F_s

is the force imbalance F_t≶0. There is no mechanical
equilibrium unless F_t=0 and the spring continues to stretch or
contract, thereby giving rise to an oscillatory motion that will go on
for ever. During each oscillation, F_t is almost always nonzero.
 The SI work done by F_s is the spring work
dW≐ F_sdx performed by the spring (internal approach), while
the work performed by F_0 is dW=F_0dx transferred
to the spring; its negative d_eW=-F_0dx is the exchange work
(external approach). The kinetic energy plays no role in determining work and
is not considered. Being a purely mechanical example, there is no dissipation.
Despite this, we can introduce using the modern notation

    d_iW≐ dW-d_eW≡ dW+dW≡
    F_tdx,

which can be of either sign (no second law here) and represents the
work done by F_t. Thus, dW,d_eW=-dW and
d_iW represent different works, a result that has
nothing to do with dissipation but only with the imbalance; among these, only
the generalized work dW is an SI work. The change in the Hamiltonian
ℋ=E of the spring due to a variation in the work variable x is

    .  dℋ| _w=.  dE| _w
    =-Fdx=-dW≠ dW;

the suffix w refers to the change caused by the performing work by varying x
here. We thus conclude that

    dW=-.  dE| _w,dW=.  dE|
    _w+F_tdx,

which shows the importance of the force imbalance F_t and also
shows that dW≠-dW almost always.



  §.§.§ Inclusive Approach


Let us consider the inclusive Hamiltonian <cit.>
ℋ^'=E^'≐ E-F_0x used in deriving Eq.
(<ref>); it also explains the prime there. We have

    dE^'=dE-d(F_0x)=-F_tdx-xdF_0.

As the force F_t=-∂ E^'/∂ x conjugate to x
does not identically vanish, E^'(x,F_0) is a function of two
work parameters x and F_0. However, Jarzynski, see Conclusion
<ref>, neglects F_t so
E^'(F_0) becomes a function of only F_0
<cit.>. We will see below that the existence of
F_t is very common. As x=-∂ E^'/∂ F_0, x
is the generalized force conjugate to F_0. The SI work dW^'
consists of two contributions :

    dW^'=dW_x^'+dW_F_0^'=F_tdx+xdF_0

and satisfies

    dW^'=-.  dE| _w,

as in the exclusive approach. Furthermore, dW_F_0^'
=xdF_0 represents the external work

    d_eW^'=dW_F_0^'=xdF_0;

hence,

    d_iW^'=dW_x^'=F_tdx

represents the internal work due to F_t, and which appears in the
left side of Eq. (<ref>).

The following identities are always satisfied, whether we consider a
mechanical system as in this subsection or a thermodynamic system as in the
next subsection:



    .  dE^'| _w-.  dE| _w   ≐ dW-dW^'≡-d(F_0x),
    
    
    dW^'-dW   ≐ d_eW-d_e
    W^'≡-d(F_0x).
    
    d_iW^'   ≡ d_iW≡ F_tdx.

For a mechanical system like a microstate 𝔪_k, we should append
a subscript k to each of the quantities in Eq.
(<ref>). For a thermodynamic system, each of the
quantities also refer to thermodynamic average quantities.

Let us investigate the case F_0≡0: F_t=F_s, and
dW=dW^'=F_sdx≠0 and dW=dW^'=0 as a consequence of ℋ=ℋ^'. Such a
situation arises when the spring, which is initially kept locked in a
compressed (or elongated) state is unlocked to let go without applying any
external force. Here, dW=dW^'≠0, while dW
=dW^'=0 as the spring expands (or contracts) under the
influence of its spring force F_s. The reader should notice a
similarity with the free expansion noted above.

Comparison with Jarzynski's Approach: The difference between our
approach and that by Jarzynski should be mentioned. As said earlier, Jarzynski
does not allow the contribution dW_x^'=F_tdx in
dE^' as he treats E^'(F_0) as a function of a single work
parameter F_0. Therefore,


    d_iW^'(J)=0 and dW^'
    ^(J)=-xdF_0=.  dE^'| _w
    ^(J);

we have used a superscript (J) as a reminder for the results by Jarzynski.
Thus, as concluded in Sec. <ref>,
Jarzynski's approach does not allow any force imbalance F_t (see
Conclusion <ref>) so his conclusion
dW^'^(J)=.  dE^'|
_w^(J) fits with the questionable identification
dW^'_k?=.  dE_k^'|
_w whose validity requires d_iW^'=0, i.e.,
F_t=0. As we will see in the next example, F_t is not
only ubiquitous (it is present even in equilibrium) but also necessary for irreversibility.



 §.§ Force Imbalance in a Thermodynamic
System




  §.§.§ Exclusive Approach


To incorporate dissipation, we consider a thermodynamic analog of the above
example by replacing the vacuum with a fluid in the example discussed above;
see Fig. <ref>(b). This example is no different from the one
shown in Fig. <ref>(a). We will actually discuss this
gas-piston system since we have already studied it earlier. This is easily
converted to the spring-mass system by a suitable change in the vocabulary as
we will elaborate below.

The piston is locked and the gas has a pressure P. We first focus on various
work averages to understand the form of dissipation. At time t=0, an
external pressure P_0<P is applied on the piston and the lock on the
piston is released. We should formally make the substitution x→
V,F→ P (or P_k when considering 𝔪_k) and
F_0→-P_0. At the same time, we will also invoke ξ. The gas
expands (dV≥0) and P↘ P_0. The SI work done by the gas is
dW=PdV+Adξ, while dW=-P_0dV=-d_eW. The difference
d_iW=(P-P_0)dV+Adξ≥0 appears as the irreversible work that
is dissipated in the form of heat (d_iQ≡ d_iW) either
due to friction between the moving piston and the cylinder or other
dissipative forces like the viscosity; including them does not change the
first two equations but supplements the meaning of F_t in the last
equation in Eq. (<ref>); it must include all
possible force imbalance as clearly seen in Sec. <ref>. We will
see how internal processes such as friction give rise to internal variables.
Assuming this to be the case, our assumption of the presence of an internal
variable should not come as a surprise.

Let us analyze this model more carefully at a microstate level. Let
ℋ(V,ξ)= E(Vξ) be the exclusive Hamiltonian of the
gas. Let E_k(V,ξ) denote the energy in the inclusive approach of some
𝔪_k; let E(V,ξ)≐⟨ E⟩ and
P(V,ξ)≐⟨ P⟩ be their statistical average over
microstates. We use Eq. (<ref>), and use Eq.
(<ref>) to identify

    dW_k≐-dE_k,d_eW_k≐-d_eE_k,d_i
    W_k≐-d_iE_k,

giving the three work-Hamiltonian relations for a microstate. The statistical
averaging gives

    dW     ≡-.  dE| _w=PdV+Adξ,
    
    d_eW     ≡-.  d_eE| _w
    =P_0dV
    
    d_iW     ≡-.  d_iE| _w
    =(P-P_0)dV+Adξ

just as discussed above; the suffix w refers to these averages determined by
the work variables V and ξ.

There is no reason for P_k=P_0,A_k=0,∀ k, even in a reversible
process for which P=P_0 and A_0=0. As there are pressure fluctuations
even in equilibrium,

    ∑_k
    p_k(P_k-P_0)^2≥0.

For the same reason, we expect affinity fluctuations also in equilibrium.
Thus, P_k-P_0≠0 and A_k≠0 in general; hence d_i
W_k≠0.



  §.§.§ Inclusive Approach


The inclusive Hamiltonian for 𝔪_k appears similar to the
microstate analog of the NEQ enthalpy <cit.>
E_k^'(V,ξ,P_0,)=E_k(V,ξ)+P_0V. It is a function of
V,ξ and P_0 as was discussed in Sec.
<ref>. Therefore,

    dW_k^'     =-(  ∂ E_k^'/∂ V)
    dV-(  ∂ E_k^'/∂ξ)  dξ-(  ∂
    E_k^'/∂ P_0)  dP_0
         =(P_k-P_0)dV+A_kdξ-VdP_0
         =P_kdV+A_kdξ-d(P_0V),

of this, d_eW_k^'=-dW_k^'=-VdP_0 is
spent to overcome the external force -V≐-(  ∂ E_k^'/∂ P_0) conjugate to the work variable P_0 and the
balance is the irreversible work

    d_iW_k^'=(P_k-P_0)dV+A_kdξ=d_iW_k.

Since dW_k=-P_0dV, Eq. (<ref>)
remains satisfied for each microstate, and also for the averages.

As we have seen above that d_iW_k≠0 in general. Thus, we come
to the very important conclusion


 The presence of d_iW_k≡
d_iW_k^' is ubiquitous and must be accounted for even in a
reversible process, let alone an irreversible process; see Eq.
(<ref>). This clearly shows that dW_k (dW_k
^') does not vanish even if dW_k=-d_iW_k
(dW^'=-d_iW_k^') vanishes.



 The microwork dW_k or dW_k^'
continues to contribute to dE_k (such as during 𝒫
in the Jarzynsky process) even if dW_k or dW^' has ceased to exist (during 𝒫).
Therefore, dW_k or dW_k^' contributes over the entire process
𝒫_0.



The evaluation of Δ W_k or Δ
W_k^' becomes extremely easy as we need to focus on the entire
process and do not have to consider the driving and reequilibration stages of
the process separately.As a consequence, we need to determine Δ
E_k between the final and initial states of 𝒫_0. As E_k
is an SI-quantity, its value depend on the state, see Definition
<ref>, so the value of Δ E_k does not depend on the actual
process 𝒫_0. In other words, Δ W_k=-Δ E_k is the
same for all possible processes 𝒫_0 between the same two states.
This observation clarifies a very important Conclusion
<ref> obtained later.


A third example is the spring-mass problem in Fig. <ref>(b),
where we consider the relative motion of the particle. This example can be
considered a prototype of a manipulated Brownian particle undergoing a
relative motion with respect to the rest of the system and is treated within
our internal approach in Sec. <ref>. As we will see, the
irreversible work done by the frictional force is properly accounted by the
generalized work and, in particular, the frictional work is part of
d_iW or of d_iW_k.



§ EMERGENCE OF INTERNAL VARIABLES IN THE
HAMILTONIAN


We wish to show that including other dissipation or internal variables does
not alter the first two equations of Eq. (<ref>).
However, the third equation needs to incorporate additional contributions due
to new forms of dissipation such as new internal variables. The discussion
also shows how the Hamiltonian becomes dependent on internal variables, and
how the system is maintained stationary despite motion of its parts.



 §.§ Piston-Gas System


We consider the second example, which is depicted in Fig.
<ref>(a), for this exercise. To describe dissipation, we
need to treat the motion of the piston by including its momentum
𝐏_p in our discussion. The gas, the cylinder and the
piston constitute the system Σ. We have a gas of mass M_g in
the cylindrical volume V_g, the piston of mass M_p, and
the rigid cylinder (with its end opposite to the piston closed) of mass
M_c. However, we will consider the composite subsystem
Σ_gc=Σ_g∪Σ_c so that with
Σ_p it makes up Σ. The Hamiltonian ℋ of the
system is the sum of ℋ_gc of the gas and cylinder,
ℋ_p of the piston, the interaction Hamiltonian
ℋ_int between the two subsystems Σ_gcand Σ_p, and the stochastic interaction Hamiltonian
ℋ_stoc between Σ and Σ. As is
customary, we will neglect ℋ_stoc here. We assume that
the centers-of-mass of Σ_gc and Σ_p are moving
with respect to the medium with linear momentum 𝐏_gc and
𝐏_p, respectively. We do not allow any rotation for
simplicity. We assume that

    𝐏_gc+𝐏_p
    =0,

so that Σ is at rest with respect to the medium. Thus,
    ℋ(.  𝐱| V,𝐏_gc
    ,𝐏_p)=
    
    
    ∑_λ
    ℋ_λ(.  𝐱_λ| V_λ,𝐏_λ)+ℋ_int,

where λ=gc,p, 𝐱_λ=(𝐫_λ
,𝐩_λ) denotes a point in the phase space
Γ_λ of Σ_λ; V_λis the volume
of Σ_λ, and V=V_gc+V_p is the
volume of Σ. We do not exhibit the number of particles N_g
,N_c,N_p as we keep them fixed. We let 𝐱
denotes the collection (𝐱_g,𝐱_c
,𝐱_p). Thus, the system Hamiltonian ℋ(.
𝐱| V,𝐏_gc,𝐏_p) and the
average energy E depend on the parameters V,𝐏_gc
,𝐏_p. Accordingly, the system entropy, which we assume is
a state function, is written as S(E,V,𝐏_gc,𝐏
_p). Hence, the corresponding Gibbs fundamental relation becomes

    dS=β dE+PdV-𝐕_gc·d𝐏
    _gc-𝐕_p·d𝐏_p],

where we have used the conventional conjugate fields

    c]c
    β≐∂ S/∂ E,β P≐∂ S/∂ V,,
    β𝐕_gc≐-∂ S/∂𝐏_gc
    ,β𝐕_p≐-∂ S/∂𝐏
    _p

as shown elsewhere <cit.>. Using Eq.
(<ref>), we can rewrite this equation as

    dS=β dE+PdV-𝐕·d𝐏_p
    ]

in terms of the relative velocity, also known as the
drift velocity 𝐕≐ 𝐕_p-𝐕_gc
of the piston with respect to Σ_gc. We can cast the drift
velocity term as 𝐕·d𝐏_p≡𝐅
_p·d𝐑, where 𝐅_p≐
d𝐏_p/dt is the force and d𝐑=𝐕dt
is the relative displacement of the piston. The first law applied to
the stationary Σ gives

    dE=T_0d_eS-P_0dV

in terms of the temperature T_0 and the pressure P_0 of the medium.

The internal motions of Σ_gc and Σ_p is not
controlled by any external agent so the relative motion described by the
relative displacement 𝐑 represents an internal variable
<cit.> so that the corresponding affinity 𝐅_p0=0
for Σ. Because of this, Eq. (<ref>) does
not contain the relative displacement 𝐑. We now support this claim
using our approach in the following. This also shows how ℋ(.
𝐱| V,𝐏_gc,𝐏_p)
develops a dependence on the internal variable 𝐑. We manipulate
dS in Eq. (<ref>) by using the above first law
for dE so that

    TdS=T_0d_eS+(P-P_0)dV-𝐅_p·d𝐑,

which reduces to

    T_0d_iS=(T_0-T)dS+(P-P_0)dV-𝐅_p·d𝐑.

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≥0,(P-P_0)dV≥0,𝐅_p·d𝐑≤0,

which shows that each of the components of d_iS is nonnegative. In
equilibrium, each irreversible component vanishes, which happens when

    T→ T_0,P→ P_0, and 𝐕→0
    or 𝐅_p→0.

The inequality 𝐅_p·d𝐑≤0 shows
that 𝐅_p and d𝐑 are antiparallel, which is what
is expected of a frictional force. This causes the piston to finally
come to rest. As 𝐅_p and 𝐕 vanish together, we
can express this force as

    𝐅_p=-μ𝐕f(𝐕^2),

where μ>0 and f is an even function of 𝐕. The medium
Σ is specified by T=T_0,P=P_0 and 𝐕_0=0
or 𝐅_p=0. We will take 𝐅_p and
d𝐑 to be colinear and replace 𝐅_p·d𝐑 by -F_fdx (F_fdx≥0), where the
magnitude F_p is written as F_f as a reminder that this
force is responsible for the frictional force and dx is the magnitude of the
relative displacement d𝐑. The sign convention is that F_f and increasing x point in the same direction. We can invert Eq.
(<ref>) to obtain

    dE=TdS-PdV-F_fdx

in which dQ=TdS from our general result in Eq. (<ref>).
Comparing the above equation with the first law in Eq. (<ref>), we
conclude that

    dW=PdV+F_fdx.

The important point to note is that the friction term F_fdx
properly belongs to dW. As d_eW=P_0dV, we have

    d_iW=(P-P_0)dV+F_fdx.

Both contributions in d_iW are separately nonnegative. The
corresponding inclusive Hamiltonian is given by ℋ^'=ℋ+P_0V. We can easily verify that the first two equations in
Eq. (<ref>) in the main text remain valid with
x→ V,F_0→-P_0 without any modification. The
right-side of the last equation, however, is modified to (
P-P_0)  dV+F_fdx and now contains the internal variable.

We can determine the exchange heat d_eQ=dQ-d_iW

    d_eQ=TdS-(P-P_0)dV-F_fdx



We can now consider a microstate 𝔪_k. For this we need to
consider dE_S≡.  dE| _w, from which we
determine

    dE_k=-dW_k=P_kdV+F_fkdx,

where F_fk is frictional force associated with 𝔪_k.
As d_eE_k=-d_eW_k=-P_0dV, we also conclude that

    d_iE_k=-d_iW=-(P_k-P_0)dV-F_fkdx.

 

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 Σ_gc and Σ_p.
These internal motions within Σ_g can be considered by
following the approach outlined elsewhere <cit.>. We will not
consider such a complication here.



 §.§ 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. <ref>(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_s and F_f, 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. (<ref>) for this case
becomes

    d_iW=(F_s+F_0)dx+F_fdx=F_tdx,

where F_t=F_s+F_0+F_f. The other two works are
dW=(F_s+F_f)dx and dW=F_0dx=-d_e
W. In EQ, F_f=0 and F_s+F_0=0 (F_0≠0) to
ensure d_iW=0. In this case, dW=-dW=F_0dx=dE_S,
but this will not be true for a NEQ state.



 §.§ Particle-Fluid System


In the absence of a spring in the previous subsection, we must set
F_s=0 so

    dW=F_fdx,dW=F_0dx=-d_eW,d_i
    W=(F_0+F_f)dx.

In EQ, F_0+F_f=0 so that F_f=-F_0. This means that
in EQ, the particle's nonzero terminal velocity is determined by F_0 as
expected. In this case, dW=-dW=F_0dx=dE_S, but this will not
be true for a NEQ state.



§ THEORETICAL CONSIDERATION


In this section, we allow not just V and ξ as the two work variables in
the exclusive Hamiltonian ℋ but an arbitrary number of work
variables 𝐖, out of which we single out one special work variable
V that is used to define the exclusive Hamiltonian ℋ^'=ℋ+P_0V. (One can construct NEQ Legendre transform with more
than one variable, but we will not consider such a complication here.) The
lesson from Conclusion <ref> is that d_i
W_k=d_iW_k^' does not normally vanish. It is this
Conclusion that forms a central core of the theoretical development in this
section along with the concept of SI-quantities. The importance of both these
ideas have not been appreciated to the best of our knowledge but will play a
central role in our discussion below.



 §.§ Thermodynamic Work-Energy Principle


We first consider the exclusive Hamiltonian ℋ and consider its
change .  dℋ| _𝐰 due to work variables

    .  dℋ| _w=(∂ℋ/∂𝐖)· d𝐖=-𝐅· d𝐖,

due to all work variables in 𝐖; here

    𝐅≐-∂ℋ/∂𝐖

is the generalized force, from which it follows that the SI-work is

    dW≐𝐅· d𝐖=-.  dℋ| _w.

It is important to recall that dℋ_𝐖 is the change in
ℋ at fixed 𝐖 and should not be confused with the change
.  dℋ| _w due to changes in the work variables in
𝐖. Indeed, .  dℋ| _w ≡
dℋ_S, which follows from Eq. (<ref>), but we use
the current notation .  dℋ| _w instead to
emphasize the role of work parameters in 𝐖.

We first recognize that .  dℋ| _w represents the
change in a microstate energy E_k so we will always think of .
dℋ| _w as dE_k for some 𝔪_k.
Following Sec. <ref>, we break .  dℋ|
_w into two parts, as we did for dE_k for 𝔪_k above:

    .  dℋ| _w=.  d_e
    ℋ| _w+.  d_iℋ|
    _w,

and apply it to some 𝔪_k so that .  d_e
ℋ| _w=d_eE_k≐-d_e
W_k≡ dW_k is due to the exchange work dW_k and .  d_iℋ| _w
=d_iE_k≐-d_iW_k is due to the internal work
d_iW_k performed by the generalized force 𝐅 imbalance
on 𝔪_k. We now make the following claim in the form of a


 𝐓𝐡𝐞𝐫𝐦𝐨𝐝𝐲𝐧𝐚𝐦𝐢𝐜
𝐖𝐨𝐫𝐤-𝐄𝐧𝐞𝐫𝐠𝐲 𝐏𝐫𝐢𝐧𝐜𝐢𝐩𝐥𝐞 The
change .  dℋ| _w=dE_k in the Hamiltonian
ℋ due to work only must be identified with the SI-work dW_k and not with dW_k for 𝔪_k; see Eq.
(<ref>). It has two contributions;
see Eq. (<ref>). The first one corresponds to the
external work dW_k=-d_eW_k performed
by the medium on 𝔪_k and the second one to the
negative internal work -d_iW_k due to the imbalance in the
generalized forces. After statistical averaging, 

    .  d_iE| _w≐⟨.
    d_iℋ| _w⟩≡
    -d_iW≐-
    
    
    ∑_k
    p_kd_iW_k≤0

 results in dissipation in the system. 



Based on the three examples, and the general relation in Eq.
(<ref>) or the earlier result in Eq. (<ref>), the proof
for

    .  dℋ| _w=dE_k=-dW_k

is almost trivial. We only have to recognize that both dE_k and dW_k
are SI-quantities as they appear in the identity in Eq. (<ref>
). The proof here is not restricted to only two work variables. We consider an
arbitrary number of work variables in 𝐖. Consider the first law
representation using the MI-quantities in Eq. (<ref>). The
isentropic change dE_S is identified with

    dE_S=-dW=-𝐅· d𝐖.

It represents the average of -dW_k=dE_k, with

    dE_k≐(∂ E_k/∂𝐖)· d𝐖
    ≡ d_eE_k+d_iE_k.

We thus identify d_eW_k=-d_eE_k and d_i
W_k=-d_iE_k. This proves the first part. The isometric change in
dE represents the generalized heat dQ, as discussed earlier. To proceed
further and better understand the above result and prove the last part, we
turn to the thermodynamics of the system, which we now require to be in
internal equilibrium so that dQ=TdS <cit.>. We
rewrite dE=TdS-dW as

    dE=T_0d_eS-d_eW+T_0d_iS+(T-T_0)dS-d_i
    W,

which leads to

    T_0d_iS=(T_0-T)dS+d_iW,

which is a different way to write Eq.
(<ref>). Each term on the right side,
being independent of each other, must be nonnegative separately to ensure the
second law in Eq. (<ref>); compare with Eq.
(<ref>). This proves the last part

    d_iW≥0.






 §.§ Equivalence of Exclusive and Inclusive Approaches


The discussion above can be extended to the inclusive energy. Indeed, all
relations derived in the exclusive approach have analogs in the inclusive
approach: all we need to do is to insert a prime on each quantity. It should
be stressed that

    -(∂ E_k/∂𝐖) or-(∂ E_k^'/∂𝐖^'),𝐖^'=(𝐖,P_0)

represents the "generalized force" and the work has the conventional
form: "force"×"distance," contrary to what is commonly stated.
According to the theorem, only SI-works (-dW_k,-Δ W_k,-dW,-Δ
W, and -Δ W_k^') must be used on the left sides in Eq.
(<ref>) or its analogs and as shown in Eq.
(<ref>) and its cumulative analog. It is easy to
convince that the Eq. (<ref>) holds in all cases;
see the discussion in Sec. <ref>.



§ A NEW NEQ WORK THEOREM


In a quantum system, the index k denotes a set of quantum numbers, which we
take to be a finite set, i.e., having a finite number of
quantum numbers so that the set {𝔪_k} is
countable infinite. Such is the case for a particle in a box treated quantum
mechanically that we investigate in Sec. <ref>. The energy
E_k(𝐖) of a microstate 𝔪_k depends on the work
parameter 𝐖 so it changes as 𝐖 changes during a process
𝒫_0 according to Eq. (<ref>) but the set k does
not change. This means that 𝔪_k keeps its identity, while its
energy changes during the work protocol 𝒫_0. The change of
energy during this protocol is related to the microwork

    Δ W_k=-(E_kfin-E_kin),

where we have set

    E_kfin≐ E_k(τ),E_kin≐ E_k(  0)
    ;

here, τ is the duration of the process 𝒫_0. As we
discussed in reference to Eq. (<ref>), the change Δ
E_k=E_kfin-E_kin occurs at fixed p_k and equals
-Δ W_k. Any change in p_k requires the microheat Δ Q_k,
i.e., either Δ_eQ_k or Δ_iQ_k. We
finally conclude that


 If we are interested in knowing the
cumulative change Δ W_k, we only need to determine Δ E_k by
following the same 𝔪_k mechanically during a work protocol
𝒫_0. The probability plays no role as Δ Q_k is of no concern.



It should be stated here that for all
different processes 𝒫_0(𝖠_fin|𝖠_in)'s between the same two states 
_fin and _in, not necessarily EQ states,
E_kfinand E_kin are the same, see Definition
<ref>, so

    Δ W_k(𝒫_0)=-Δ E_k(𝖠_fin
    ,𝖠_in),∀𝒫_0(𝖠
    _fin|𝖠_in).

This allows us to determine the dissipated work

    Δ_iW_k(𝒫_0)=-Δ E_k(𝖠_fin
    ,𝖠_in)-Δ_eW_k(𝒫_0)

in which Δ_eW_k(𝒫_0)=-ΔW
_k(𝒫_0) in terms of the external work ΔW
_k(𝒫_0).


What the above remark implies is the following: Different processes between
the same two states _fin and _in
differ not in Δ W_k(𝒫_0) but in ΔW
_k(𝒫_0) or in Δ_iW_k(𝒫_0). This
makes microwork unique in that it does not depend on the nature of
𝒫_0(𝖠_fin|𝖠_in). Despite
this, it contains the contribution of dissipation in it given by the average
⟨Δ_iW_k(𝒫_0)⟩.

However, the property of a quantum 𝔪_k maintaining its identity
during 𝒫_0 is different from the property of a microstate
𝔪_k in a classical system, for which 𝔪_k is a
small dimensionless volume element δ𝐳_k in the phase space
Γ surrounding a point 𝐳_k; the collection {δ𝐳_k} covers the entire phase space Γ. This
microstate changes its identity (δ𝐳_k→δ𝐳_l,k≠ l) as it evolves in time following its Hamiltonian
dynamics; recall that dℋ_𝐖=0 during this dynamics.
Despite this, the evolution is unique so there is a 1-to-1
mapping between 𝔪_k(t)≐δ𝐳_k(t) and
𝔪_k(t^')≐δ𝐳_l(t^'). This
causes no problem as the change Δ W_k=-Δ E_k is not affected
by the microstate evolution past τ, the duration of 𝒫_0;
see Conclusions <ref> and
<ref> for more details. In this case, introducing

    E_k(  τ)  =E(δ𝐳_k(τ)),E_k(
    0)  =E(δ𝐳_k(0),

we can write Δ W_k as in Eq. (<ref>). Thus, whether we
are considering a classical system or a quantum system, we can always express
Δ W_k as in Eq. (<ref>).



 §.§ Derivation


Let us evaluate the particular average ⟨·⟩ _0
in Eq. (<ref>) but of e^β_0Δ W_k using Eq.
(<ref>). We first consider the exclusive approach. We have

    ⟨ e^β_0Δ W⟩ _0   ≐
    
    
    ∑_k
    e^-β_0E_k,i/Z_in(β_0)e^β
    _0Δ W_k
         =
    
    
    ∑_k
    e^-β_0E_k,i/Z_in(β_0)e^–β
    _0(E_kfin-E_kin),

which leads to

    ⟨ e^β_0Δ W⟩ _0=
    
    
    ∑_k
    e^-β_0E_k,fin/Z_in(β_0)=Z_fin(β_0)/Z_in(β_0),

where Z_in(β_0) and Z_fin(β_0) are initial
and final canonical partition functions. Introducing the free energy
difference Δ F≐ F_fin-F_in, we finally have

    ⟨ e^β_0Δ W⟩ _0=e^-β_0Δ F.

This is our new work theorem involving microworks. The same calculation can be
carried out for the inclusive Hamiltonian, with a similar result except all
the quantities must be replaced by their prime analog as said earlier:

    ⟨ e^β_0Δ W^'⟩ _0=e^-β
    _0Δ F^'.

This fixes the original Jarzynski relation in Eq. (<ref>) by
accounting irreversibility in both approaches, and is valid in all cases,
reversible or otherwise.



§ THE FREE EXPANSION 


Consider the case of a free expansion (P_0=0) of a gas in an
isolated system of volume V_fin, divided by an impenetrable
partition into the left (L) and the right (R) chambers as shown in Fig.
<ref>(a). Initially, all the N particles are in the left
chamber of volume V_in in an equilibrium state at temperature
T_0; there is vacuum in the right chamber. At time t=0, the partition is
suddenly removed, shown by the broken partition in Fig. <ref>(b)
and the gas is allowed to undergo free expansion to the final volume
V_fin during 𝒫. After the free expansion, the gas is in
a NEQ state and is brought in contact with Σ_h
during 𝒫 to come to equilibrium at the initial
temperature T_0. If the gas is ideal, there is no need to bring in
Σ_h for reequilibration; we can let the gas come
to equilibrium by itself as it is well known that the temperature of the
equilibrated gas after free expansion is also T_0.
















It should be stated, which is also evident from Fig. <ref>(b),
that while the removal of the partition can be instantaneous, the actual
process of gas expanding in the right chamber is continuous and gradually
fills it. Therefore, at each instant, it is possible to imagine a front of the
expanding gas shown by the solid vertical line enclosing the largest among
smallest possible volumes containing all the particles so that there are no
particles to the right of it in the right chamber in all possible realization
of the expanding gas. By this we mean the following: we consider all possible
realizations of the expanding gas at a particular time t>0 and locate the
front corresponding to the smallest volume containing all the gas particles to
its left. Then we choose among all these fronts that particular front that
results in the smallest volume on its right or the largest volume on its left.
In this sense, this front is an average concept and is shown in Fig.
<ref>(b). We have identified the volume to its right as "vacuum"
in the figure. 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
dW=0. Despite this, as the expansion is a NEQ process,
dW=d_iW>0.



 §.§ Quantum Free Expansion


We now apply Eq. (<ref>) to the free expansion of a
one-dimensional ideal gas of classical particles, but treated quantum
mechanically as a particle in a box with rigid walls, which has studied
earlier <cit.>; see also Bender et al <cit.>. We
assume that the gas is thermalized initially at some temperature
T_0=1/β_0 and then isolated from the medium so that the free
expansion occurs in an isolated system. After the free expansion from the box
size L_in to L_fin>L_in, the box is again
thermalized at the same temperature T_0. The role of V is played by the
length L of the box. This will also set the stage for the classical
treatment later.

As discussed in Sec. <ref> for F_0=0, Δ
W_k≠0 even though ΔW_k=0. Since we are dealing with
an ideal gas, we do not need to bring Σ_h, see
below, so we let the gas to come to equilibrium as an isolated system. As
there is no inter-particle interaction, we can focus on a single particle for
our discussion; its energy levels are in appropriate units

    E_k=k^2/L^2,

where L is the length of the box. We assume that the gas is thermalized
initially at some temperature T_0=1/β_0. It is isolated from the
medium so that the free expansion occurs in an isolated system, during which,
we have Δ_eQ=Δ_eW=0 (but Δ_i
Q=Δ_iW≠0) so that Δ E_free(L_fin
,L_in)=0; see Eq. (<ref>). After the free expansion
from the box size L_in to L_fin>L_in, the box
is allowed to come to equilibrium in isolation so that we have Δ
E_reeq(L_fin)=0. Accordingly, Δ E_eq
(L_fin,L_in)=0 after reequilibration.

The initial partition function is given by

    Z_in(β_0,L)=
    
    
    ∑_k
    e^β_0E_k,in.

Approximating the sum by an integration over k as is common, we can evaluate
Z_in(β_0,L) from which we find that the free energy
F_eq and the average energy E_eqare given by

    β_0F_eq=-(1/2)ln(L^2π/4β_0),E_eq
    =1/2β_0;

while F_eq depends on β_0 and L, E_eq depends
only on β_0 but not on L so that E_eq has the same value
in the final EQ state. This means that the final equilibrium state has the
same temperature T_0. This explains why we did not need to bring
Σ_h in play for reequilibration as assumed above. 

As we have discussed in reference to Eq. (<ref>) and
concluded in Conclusions <ref>,
<ref> and <ref>,
and summarized in Remark <ref>, Δ E_k=-
Δ W_k regardless of whether 𝒫_0 is irreversible or
not. Below we will show that the present calculation here is dealing with an
irreversible 𝒫_0. The energy change between two EQ states is

    Δ E_k=k^2(1/L_fin^2-1/L_in^2).

Let us determine the microwork done to take the initial microstate to the
final microstate by using the internal pressure

    P_k=-∂ E_k/∂ L=2E_k/L≠0

in

    Δ W_k=
    
    
    ∫_L_in^L_fin
    P_kdL.

It is easy to see that this microwork is precisely equal to  -Δ E_k
in accordance with Theorem <ref>, as expected.
It is also evident from Eq. (<ref>) that for each L
between L_in and L_fin,

    P=
    
    
    ∑_k
    p_kP_k=2E/L≠0.

We can use this average pressure to calculate the thermodynamic work

    Δ W=
    
    
    ∫_L_in^L_fin
    PdL=2
    
    
    ∑_k
    
    
    ∫_L_in^L_fin
    p_kE_kdL/L≠0.

as expected. As Δ E=0, it means that Δ Q=Δ W≠0, which
really means Δ_iQ=Δ_iW≠0 in this case. This
establishes that the expansion we are studying is irreversible. This is also
evident from the observation that P≠ P_0=0.

Despite this, Δ W_k is always equal to the same (  -Δ
E_k) regardless of the nature of irreversibility of 𝒫
_0, which is consistent with Conclusion
<ref> and Remark
<ref>. The same Δ W_k will also apply to a
reversible 𝒫_0 as we are considering the energy change between
two EQ states. The only difference is that now Δ Q=Δ W≠0 will
mean Δ_eQ=Δ_eW≠0. It is trivially seen that
Eq. (<ref>) is satisfied for all 𝒫_0, not just
the free expansion.

As P_0=0, there is no difference between the exclusive Hamiltonian and the
inclusive Hamiltonian. Thus, the discussion above is also valid for the
inclusive Hamiltonian and Eq. (<ref>) with E_k^'=E_k and F^'=F.




















 §.§ Classical Free Expansion


We will consider the free expansion of an isolated classical gas in a vacuum
(P_0=0), see Fig. <ref>. We set V_in and
V_fin for simplicity The initial phase space is denoted by the
interior of the solid red ellipse Γ_in on the
left side in Fig. <ref>. The final phase space is
shown by the interior of the broken red ellipse on the left and the solid red
ellipse Γ_fin on the right in Fig.
<ref>. The gas is in a "restricted (i.e.,
being confined in the left chamber)" equilibrium state with equilibrium
microstate probability

    f_0(δ𝐳_0)=e^-β_0E(𝐳_0)/Z_in
    (β_0,V_in)

at t=0; here, the initial partition function in the initial volume
V_in is

    Z_in(β_0,V_in)≐
    
    
    
    ∑_δ𝐳_0∈Γ_in
    
    e^-β_0E(𝐳_0).

We consider the set of microstates in the final phase space
Γ_fin and pick two microstates δ𝐳_0 and δ𝐳 associated with 𝐳_0
∈Γ_in and 𝐳∈Γ≐Γ_fin\Γ_in; here, Γ
≐Γ_fin\Γ_in denotes the difference set of Γ_fin and Γ_in. We use the notation 𝐳_0≐(𝐳_0,𝐳) to denote the two points.
Let us identify (𝐳_γ,𝐳_γ^') as the
unique 1-to-1 phase points obtained by the deterministic
Hamiltonian evolution of (𝐳_0,𝐳) along the deterministic
or mechanical trajectories γ=γ(𝐳_0) and γ
^'=γ^'(𝐳) corresponding to a given work protocol
𝒫_0; see Fig. <ref>. The probabilities
of the two paths are irrelevant for the microworks

    [ Δ W_γ(𝐳_0)=-(E(𝐳_γ)-E(𝐳
    _0)),;  Δ W_γ^'(𝐳)=-(E(𝐳_γ^')-E(𝐳)); ]

see Conclusions <ref> and
<ref>.

While the initial EQ probability distribution f_0(δ𝐳_0) is
nonzero for δ𝐳_0∈Γ_in, it is
common to think of f_0(δ𝐳)=0 for 𝐳∈Γ. This is an ideal situation and requires
taking the energy E(𝐳)=∞, but in reality, f_0
(δ𝐳) falls rapidly as we move into the right chamber away from
the left one in the initial macrostate. Moreover, during free expansion,
f(δ𝐳) at t>0 is not going to remain zero. Therefore, we
formally assume that the initial probability distribution f_0
(δ𝐳) is infinitesimally small by assigning to it a very large
positive energy

    E(𝐳)=e(𝐳)/ε>0,𝐳∈Γ at t=0

by introducing an infinitesimal positive quantity ε. At the end of
the calculation, we will take the limit ε→0^+, which
simply means ε→0 from the positive site. Under this
limit, the contribution from e^-β_0E(𝐳) will vanish:

    e^-β_0E(𝐳)ε→0^+→0.

This allows us to recast the initial partition function as a sum over all
microstates 𝐳∈Γ_fin:

    lim_ε→0^+Z_in^'(β_0
    ,V_fin,ε)≐lim_ε→0^+
    
    
    
    ∑_δ𝐳∈Γ_fin
    e^-β_0E(𝐳)=Z_in(β_0,V_in);

Thus, we can focus on Γ_fin as the phase space to
consider during any work protocol 𝒫_0 instead of
Γ_in. This allows us to basically use a
1-to-1 mapping between initial microstates 𝐳
_0≐(𝐳_0,𝐳) and final microstates 𝐳_γ≐(𝐳_γ,𝐳_γ^'
) discussed above.

We simply denote 𝐳_0 or 𝐳 by 𝐳
∈Γ_fin or 𝐳_γ
∈Γ_fin for the Hamiltonian evolution of
𝐳 along the microwork protocol from now on. We consider
the Jarzynski average of the exponential work in Eq. (<ref>)
for the exclusive Hamiltonian and write it as

    lim_ε→0^+⟨ e^β_0Δ
    W⟩ _0=lim_ε→0^+
    
    
    
    ∑_δ𝐳∈Γ_fin
    e^-β_0E(𝐳)e^-β_0[E(𝐳
    _γ)-E(𝐳)]/Z_in(β_0,V_fin
    ,ε),

where we have used Δ W_γ(𝐳
)=-(E(𝐳_γ)-E(𝐳)) in accordance with Eq. (<ref>). The
initial partition function in the original volume V_in because of
the vanishing probabilities to be outside this volume. Because of the
1-to-1 mapping to 𝐳_γ, we can replace the sum
to a sum over 𝐳_γ, and at the same time cancel
the initial energy E(𝐳) in the exponent; the
cancellation is exact even for 𝐳=𝐳 for
which E(𝐳)→+∞ in the limit ε→0^+. Because of this, the lim operation has no effect in the
numerator. The partition function in denominator reduces to Z_in
(β_0,V_in) as shown in Eq. (<ref>). We finally find

    ⟨ e^β_0Δ W⟩ _0=
    
    
    
    ∑_δ𝐳_γ∈Γ_fin
    e^-β_0E(𝐳_γ)/Z_in(β_0,V_in
    )=Z_fin(β_0,V_fin)/Z_in(β
    _0,V_in),

which is precisely what we wish to prove in Eq. (<ref>).

The situation with the inclusive Hamiltonian is the same as ℋ
^'=ℋ as before. This allows us to also prove Eq.
(<ref>). Moreover, as said in the previous subsection, the
demonstration of Eqs. (<ref>) and (<ref>) is
valid for any arbitrary process, not just the free expansion, the title of
this section.

It should be emphasized that allowing for a negligible probability is a common
practice even in EQ statistical mechanics where we evaluate the partition
function by considering all microstate, regardless of how negligibly small the
corresponding statistical weight is. This probability could even be zero. The
only difference is that the microstate is defined over the volume of the
system and not outside. We have allowed microstates in deriving Eqs.
(<ref>) and (<ref>) with vanishing small or
zero probabilities. Here, we are considering microstates outside the volume of
the system, but mathematically, there is no difference.

By allowing such microstates in Γ, we have
shown that Eqs. (<ref>) and (<ref>) hold even
for free expansion of a classical or quantum gas, where the Jarzynski equality fails.



§ CONCLUSION 


The present work is motivated by a desire to clarify the connection between
work and energy change at the mechanical level. The goal is summarized in
Proposition <ref>. The use of SI-quantities proves useful in
expressing the First law in which the generalized heat dQ is proportional to
dS, while the generalized work dW=-dE_S is isentropic change in the
energy E. This ensures that dW and dQ change E independently so that
we need not consider any effect of dQ while considering the change dE_S;
see Conclusion <ref>.

As the probability during microwork does not change, a microstate can also be
treated as a mechanical system during microwork; see Conclusion
<ref>. As the change Δ E_k is an
SI-quantity, it must be related to the work that is also an SI-quantity. This
gives Δ E_k =-Δ_kW (internal scheme), which contradicts the
current but erroneous practice of using Δ E_k ?=
Δ_kW (external scheme) in diverse applications in
nonequilibrium statistical thermodynamics mentioned earlier; see the
discussion in Sec. <ref>. The mistake results in setting

    Δ_iW_k=0,

which contradicts the most important result of this work that Δ
_iW_k≠0 in almost all cases as concluded in Conclusion
<ref>. This has been justified by a purely mechanical
system in Sec. <ref>, which shows that Eq.
(<ref>) holds only when there is mechanical equilibrium. This
equilibrium occurs under a very special condition of zero force imbalance
F_t. However, as shown in Sec. <ref>, Eq.
(<ref>) does not hold in general for a mechanical system. The
thermodynamic system investigated in Sec.
<ref> also shows that Eq.
(<ref>) does not hold in general.

As Conclusion <ref> shows, the presence
a nonzero force imbalance is necessary (but not sufficient) for
dissipation in the system. The force imbalance is what gives rise to
thermodynamic forces, whose importance does not seem to have been acknowledge
by many workers that consistently use the mistaken identity Δ E_k
=Δ_kW in Eq. (<ref>). This means that
Eq. (<ref>) holds in these studies, which cannot include any
dissipation. This is consistent with Theorem
<ref> that there is no irreversibility
(Δ_iW=0) if we accept Δ E_k =Δ_kW. This also shows that the JE is not a valid identity.

The existence of Δ_iE_k=-Δ_iW_k is one of
the most surprising result of our approach, which appears almost
counter-intuitive and has remain hitherto unrecognized in the field because of
it. It is so because it is well known that it is impossible to have
Δ_iE≠0, where

    Δ_iE=
    
    
    ∫_γ
    d_iE,

in terms of d_iE is given in Eq. (<ref>). However, we have
shown, see Claim <ref> that even if d_iE_k≠0,
d_iE always vanishes.

As we see from Remarks <ref> and
<ref>, the determination of the SI-work Δ
W_k or Δ W_k^' over the entire process 𝒫_0
is extremely easy as it is independent of the nature of the actual process
𝒫_0; it is the same for all processes between the same two
states since Δ E_k or Δ E_k^' is an SI-quantity and
is determined by the initial and the final states but not on the nature of the
process. Because of this, Eqs. (<ref>) and
(<ref>) hold for all processes between the same two states.
This is very different from Δ_eW_k=-Δ_kW
or Δ_eW_k^'=-Δ_kW^', which
depends strongly on 𝒫 during which it is nonzero and vanishes
during 𝒫. The difference is Δ_i
W_k=Δ_iW_k^'.

The correct identification Δ W_k=-Δ E_k or Δ
W_k^'=-Δ E_k^' ensures that

    Δ_iW_k=Δ_iW_k^'≥0

in conformity with its ubiquitous nature as is evident from the existence of
pressure fluctuations in Eq. (<ref>); see Conclusion
<ref>. With this identification, Eqs.
(<ref>) and (<ref>) always hold in the
exclusive and inclusive approaches, respectively. They hold not only for free
expansion, but also for any arbitrary process 𝒫_0 as we show in
Sec. <ref>. Thus, our theorem differs from the JE, which
fails for free expansion.

With the ubiquitous nature of d_iW_k=-d_iE_k and the
SI-equivalance of dW_k=-dE_k as opposed to the mistaken identity
dW_k=dE_k, we believe that the correction requires complete
reassessment of current applications; see Sec. <ref>. In
particular, we need to recognize the importance of d_i
W_k=-d_iE_k, which has not been hitherto recognized. Without
acknowledging this fact in any theory results in a total absence of
dissipation regardless of the time dependence the work protocol such as in the
Jarzynski equality.

99                                                                                               


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