Hierarchy of Relaxation times and Residual Entropy: A Nonequilibrium Approach
P. D. Gujrati

UATP/1801


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


We consider nonequilibrium (NEQ) states such as supercooled liquids and
glasses that are described with use of internal variables. We classify the
latter by state-dependent hierarchy of relaxation times to assess their
relevance for irreversible contributions. Given an observation time
τ_obs, we determine the window of relaxation times that divide
the internal variables into active and inactive groups, the former playing a
central role in the NEQ thermodynamics. Using this thermodynamics, we
determine (i) a bound on the NEQ entropy and on the residual entropy, and (ii)
the nature of isothermal relaxation of the entropy and the enthalpy in
accordance with the second law. A theory that violates the second law such as
the entropy loss view is shown to be internally inconsistent if we require it
to be consistent with experiments. The inactive internal variables still play
an indirect role in determining the temperature T(t), the pressure P(t),
of the system, which deviate from their external values.


January 24, 2018]
Hierarchy of Relaxation times and Residual Entropy: A Nonequilibrium Approach
    P. D. Gujrati
    [
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§ INTRODUCTION


Glass such as naturally occurring obsidian, pumice, etc. or man-made Venetian
glass, window glass, etc. is a well-known class of material that has captured
our fascination forever. We can now make a defect-free glass in the laboratory
for a variety of scientific and technological applications. Crudely speaking,
it is an almost solid-like amorphous material that possesses no long range
atomic order and, upon heating, gradually softens as it turns into its
molten state (also known as the supercooled liquid) as it passes through the
glass transition region normally denoted by a suitable chosen single
temperature T_g in this region
<cit.>. For the purpose of
this article, a glass is treated merely as a nonequilibrium (NEQ) state
of matter, which can be made quite homogeneous so to a good approximation it
can be treated as a thermodynamic system that is in internal
equilibrium (IEQ) but not in equilibrium (EQ) as explained later. (At
present, it suffices to say that the entropy in an IEQ state is a state
function of its state variables that now include some NEQ state variables
(commonly known as internal variables)
<cit.> besides those needed
to specify EQ states; see also <cit.>.) This means that
a glass will exhibit relaxation as it strives to come to equilibrium. The
relaxation time is known to be large enough close to T_g that at
much lower temperatures, one can usually treat a glass to be in a almost
frozen state over experimental time scale τ_obs, the time
period over which successive observations are made. We refer the reader to an
excellent monograph by Debenedetti <cit.> on these issues. We will
primarily focus on the thermodynamics of glasses and supercooled liquid in
this work and treat them as NEQ states. Therefore, our discussion will mostly
consider a NEQ system, which we denote by Σ in an extensively large
medium Σ as shown in Fig. <ref>.


As we will not consider a system in isolation in this work, we will always use
EQ or "equilibrium" to mean "equilibrium with respect to the medium
Σ." We will not reserve EQ for the entire system only. We
will also use it for a part of the system, part of the state variables, or
part of the degrees of freedom such as vibrational degrees, of the system, if
they are in equilibrium with Σ. On the other hand, we will
reserve the use of IEQ for the entire system; see also <cit.>.


It is a well-known fact that in glasses, the vibrational modes come to
equilibrium very fast, even though the glass is out of equilibrium. Similarly,
in a sinusoidal variation of T, some degrees of freedom would equilibrate
after a cycle; others would not and would control the temporal behavior of the
system. It seems natural that the sinusoidal variation would give rise to a
distribution of relaxation times. Thus, in general, one of the most important
consequences of the rate of variation of the external stimuli such as the
temperature or pressure is the possibility that the state of the system may be
so far away from equilibrium that the dynamics becomes too complex, involving
multiple relaxation time scales τ_0,τ_1,τ_2,⋯, in
supercooled liquids <cit.>. The
relaxation time is defined as the time required for the corresponding
dynamical variable to come to equilibrium with the medium; see Eq.
(<ref>) for the proper definition of the relaxation time. It
should be emphasized that this interpretation of the relaxation time is
dictated by the experimental setup but does not depend on any particular
mathematical form of the relaxation. An interplay between τ_obs
and relaxation times τ_k's becomes crucial in determining the
thermodynamics of the system and plays a major role in our discussion here. In
fact, one of the following cases for a given τ_k will be usually
encountered in experiments:



  Relax1 τ_k<<τ_obs. In this situation, the kth
relaxing dynamical variable has equilibrated and does not have to be accounted
for in the NEQ thermodynamics.


  Relax2 τ_k≃τ_obs. In this situation, the kth
dynamical variable will continue to relax towards equilibrium during
τ_obs and must be accounted for as the system approaches equilibrium.


  Relax3 τ_k>>τ_obs. In this situation, the kth
dynamical variable will not fully relax and will strongly affect the behavior
of the system. The corresponding dynamical variable is said to be "frozen-in"
over τ_obs.


When there are several relaxation times, it is possible that different
τ_k's will correspond to different cases above. Thus, care must be
exercised in dealing with different relaxation times. The need for  such a
care has been recognized in vitrification for a long time <cit.>.
Relaxation is a universal phenomenon when a system drives itself towards a
more stable state such as an EQ state. In liquids or glasses, relaxations
involving changes of the atomic or molecular positions are generally known as
structural relaxations <cit.>. Recent experimentation advances
have made it possible to directly measure these relaxation processes at the
molecular level simultaneously <cit.>. At sufficiently low
temperatures, the characteristic time for structural relaxations becomes
comparable to the time scale of a macroscopic observation τ_obs
∼ 100 s. For shorter time scales, the supercooled liquid (SCL) exhibits
solid-like properties, while for longer times, it shows liquid-like
properties. Even the dynamics in these cases is not so trivial but has been
investigated for a long time <cit.>
with tremendous success. The glass transition being a "NEQ transition," its
description will require extensive internal variables, collectively
denoted by a vector ξ that are independent of the set
𝐗 of extensive observables (E,V,N,⋯) <cit.>
whenever the system is out of equilibrium
<cit.>
. We denote their collection by 𝐙 in this work. The investigations
of the glass transition invariably assume that the entropy S is a
state function S(𝐙) of the state variables in the
extended state space 𝔖_𝐙⊃𝔖
_𝐗 spanned by 𝐙; here 𝔖_𝐗 is
the state space of the observables. There is a memory of the initial state and
requires the entire history of how the state is prepared to uniquely describe
the preparation. Such a memory in some cases can be described by
ξ. One example is residual stresses <cit.>: if
particle configurations in a glass cannot fully relax to equilibrium, some of
the stresses that build up during flow in the melt persist in the glass; these
stresses cannot be captured by 𝐗. We will say that such a state is
an incompletely described state in terms of 𝐗 but a completely
described state in terms of 𝐙. In contrast, the EQ state
_eq(𝐗) is a completely (i.e., uniquely)
described state by 𝐗 and has no memory of the initial state. This
means that in equilibrium, ξ is no longer independent of
𝐗.

The consideration of dynamics resulting from the simple connectivity of the
sample (also known as the microstate or phase) space has played a pivotal role
in developing the kinetic theory of gases <cit.>, where
the interest is at high temperatures
<cit.>. As dynamics
is very fast here, it is well known that the ensemble averages agree with
temporal averages. However, at low temperatures, where dynamics becomes
sluggish as in a glass <cit.>, the
system can be confined into disjoint components. The confinement
occurs under NEQ conditions, when the observational time scale τ
_obs becomes shorter than the equilibration time τ_eq
such as in glasses, whose behavior and properties have been extensively
studied. These components are commonly known as basins in the energy
landscape picture <cit.>. The
entropy of confinement at absolute zero is known as the residual
entropy and can be observed in glasses or disordered crystals; see below.

















The existence of a nonzero residual entropy does not violate Nernst's
postulate, as the latter is applicable only to EQ states <cit.>. The observation of residual entropy is very common in Nature.
Indeed, Tolman <cit.> devotes an entire section on this
issue for crystals in his seminal work, while Sethna provides an illuminating
discussion for glasses <cit.>. In addition, the existence
of the residual entropy has been demonstrated rigorously for glasses by Pauli
and Tolman <cit.> and for a very general spin model by Chow and Wu
<cit.>; see references in these works for other cases where the residual
entropy is shown to exist rigorously. The numerical simulation carried out by
Bowles and Speedy for glassy dimers <cit.> also supports the existence
of a residual entropy. We refer the reader to consult various publications
<cit.>. Experiment evidence for a nonzero residual
entropy is abundant as discussed by several authors <cit.>
; various textbooks <cit.> also discuss this issue.

We introduce useful notation and concepts in the next section. In the
following section, we introduce the concept of internal equilibrium (IEQ)
states for which the entropy is a state function in the extended state space
𝔖_𝐙.


As we are not interested in ordering phenomena (such as crystallization), we
define a NEQ state with respect to an EQ state that is also disordered,
i.e. with respect to SCL. This is formally done by
considering only disordered configurations and discarding all ordered
configurations in our discussion. We warn the reader that this is different
from the conventional approach in which the equilibrium state is always taken
to be the perfectly crystalline state. This point should not be forgotten. We
then discuss the nature of the nonequilibrium state variables in
𝔖_𝐙 in Proposition 1. The affinity 𝐀
corresponding to ξ is defined so that it vanishes in SCL, the
equilibrium state in our approach.


The concept of a hierarchy of relaxation times is introduced in Sec.
<ref>, which forms a central part of the paper. A given
τ_obs determines a particular time window, which provides a
justification of Proposition 1. We find that internal variable
ξ_E that has equilibrated play no role
thermodynamically since their affinity vanishes during τ_obs. In
Sec. <ref>, we discuss the first law in terms of the
new notation, identify the irreversible work, and the IEQ thermodynamics to be
used in the next two sections on the entropy bound in vitrification and the
residual entropy (Sec. <ref>) and on the properties of the
isothermal relaxation (<ref>). In Sec. <ref>, we
find that ξ_E still indirectly affect thermodynamics
as it is required to have a thermodynamic temperature, pressure, etc. for the
system. The final section contains a brief discussion of the results.



§ NOTATION


Below is a brief introduction to the notation and the significance of various
modern terminology <cit.> for readers who are unfamiliar
with them. As usual, Σ and Σ form an isolated system
Σ_0. Extensive quantities associated with Σ and
Σ_0 carry a tilde □ and a suffix 0,
respectively. As Σ is very large compared to Σ and
is in equilibrium, all its conjugate fields T_0,P_0, etc. carry a suffix
0 as they are the same as for Σ_0, and there is no irreversibility
in Σ. Any irreversibility is ascribed to the system
Σ <cit.>, and is caused by processes such as
dissipation due to viscosity, internal inhomogeneities, etc. that are internal
to the system. Quantities without any suffix refer to the system. Throughout
this work, we will assume that Σ and Σ are spatially
disjoint and statistically quasi-independent
<cit.> so that their volumes,
masses and entropies are additive at each instant. In particular,
dV=-dV, since V_0=V+V remains constant for
Σ_0. We define a quantity to be system-intrinsic (SI) quantity if it
depends only on the property of the system alone and nothing else. For
example, if P is the pressure of Σ and P_0 that of Σ, then PdV is the SI work done by the system, but P_0dV is not
as the latter also depends on Σ through P_0. However,
P_0dV=-P_0dV is the work done by the medium, and this work
can be identified as a medium-intrinsic (MI) quantity. Any extensive SI
quantity q(t) of Σ 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
q(t) with the medium and changes within the system carry the suffix e and i,
respectively:

    dq(t)≐ q(t+dt)-q(t)≡ d_eq(t)+d_iq(t).

For Σ and Σ_0, we must replace q(t) by
q(t) and q_0(t), respectively, so that dq(t)=q(t+dt)-q(t) and dq_0(t)≐
q_0(t+dt)-q_0(t). We will assume additivity so that

    q_0(t)=q(t)+q(t).

For this to hold, we need to assume that Σ and Σ
interact so weakly that their interactions can be neglected. As there
is no irreversibility within Σ , we must have d_i
q(t)=0 for any medium quantity q(t) and

    d_eq(t)≐-dq(t)=-d_eq(t).

It follows from additivity that

    dq_0(t)≡ dq(t)+dq(t)=d_iq(t).

This means that any irreversibility in Σ_0 is ascribed to Σ,
and not to Σ. In a reversible change, d_i
q(t)≡0. For example, the entropy change

    dS≡ d_eS+d_iS

for Σ; here,

    d_eS=-d_eS

is the entropy exchange with the medium and d_iS is
irreversible entropy generation due to internal processes within
Σ; the latter is also the entropy change dS_0 of Σ_0; see
Eq. (<ref>). Similarly, if dW and dQ
represent the 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_iWand
d_iQ 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

    dW_0(t)=d_iW_0(t);dQ_0(t)=d_iQ_0(t).



We have introduced the pressure-volume work. We identify d_e
W=P_0dV=-d_eW,dW=PdV and d_iW=(P-P_0)dV.
In the absence of any chemical reaction, dN_k=d_eN_k
,d_iN_k=0 for the kth species of the particles; otherwise,
d_iN_k is its change due to chemical reaction within Σ. As
the energy of Σ can only change due to exchange with Σ,

    dE=d_eE,d_iE=0.



We now explain the concept of the relaxation time used in this work, which is
a simple generalization of its common usage but which proves useful here.
Consider some dynamical variable Φ(t) as a function of time. Its
dependence on 𝐙(t) is suppressed. Let Φ(∞) denote its
limiting value as t→∞; thus it also represents its EQ value.
In reality, we do not have to wait infinite amount of time as we cannot
distinguish between a nonzero difference |Φ(t)-Φ
(∞)|, which is smaller than some small cutoff value so that
for all purposes it is no different than zero, or a zero difference. Let us
introduce a normalized ratio

    φ(t)=| [Φ(t)-Φ(∞)]/[Φ(0)-Φ(∞
    )]|

to account for this cutoff value, which we denote by e^-λ>0; the
cutoff is primarily determined by the experimental setup. We say that the
dynamical variable Φ(t) has equilibrated when φ(t) equals
the cutoff e^-λ. The relaxation time τ_rel is defined
by

    φ(τ_rel)=e^-λ.

It is clear that for a given choice λ, the relaxation time
τ_rel can be used to describe how rapidly a quantity
effectively reaches its equilibrium value. Usually, one assumes for
φ(t) an exponential form

    φ(t)=exp(-t/τ)

or a stretched exponential form

    φ(t)=exp(-(  t/τ)  ^β),0<β≤1,

also known as the Kohlrausch-Williams-Watts form, which reduces to the simple
exponential for β=1. The relaxation time is

    τ_rel=λ^1/βτ,

and reduces to τ_rel=λτ for β=1, the exponential
form. In this work, we do not make any particular choice for the decay
behavior of φ(t); thus, we do not make any distinction between the two
forms of relaxation given above or any other form. We use a similar cutoff to
identify the equilibration time τ_eq. In reality, the stretched
exponential is very common in glassy dynamics, but its origin is far from
clear at present, even though attempts have been made to express it as a
superposition of simple exponentials with different τ's
<cit.>. It is, therefore, treated as empirical in nature. The
origin for the exponential relaxation, on the other hand, is well known as the
Debye dynamics. For us, what is important is the existence of τ
_rel through Eq. (<ref>) and not the actual form of
φ(t).

We find it very useful in this work to divide all internal variables in
ξ into nonoverlapping groups ξ_n
indexed by n=1,2,⋯. All internal variables in ξ_n are chosen to have the same relaxation time τ_n so that they
equilibrate and are no longer independent of 𝐗 over time interval
Δ t≳τ_n, and that all groups have distinct relaxation times
(τ_i≠τ_j for i≠ j). We supplement ξ by
introducing a new group ξ_0=𝐗 with relaxation time
τ_0=τ_eq in order to compactify our notation so that
𝐙={ξ_k}  _k≥0. We also
introduce the concept of hierarchy of relaxation times τ_0>τ_1
>τ_2>⋯ associated with ξ_0,ξ
_1,ξ_2,⋯, and state spaces 𝔖_0
⊂𝔖_1⊂𝔖_2⊂⋯, where
𝔖_n,n=0,1,2,⋯, is spanned by all ξ
_k,k≤ n, with relaxation times τ_k>τ_n+1. Physically, the
hierarchy of relaxation times means that the longest relaxation time in
𝔖_n is τ_0 corresponding to ξ
_0=𝐗 and the shortest relaxation time is τ_n corresponding
to ξ_n. Thus, if τ_n>τ_obs, any
ξ_k,k>n, with relaxation time shorter than τ_n has
already equilibrated (i.e., is no longer independent of 𝐗)
and does not have to be used to specify the NEQ state. Thus, 𝔖
_n is the state space needed to specify the NEQ state for τ_n
>τ_obs. However, as T_0 is changed, both {τ
_n} and τ_obs can change as shown in Fig.
<ref>. This then affects the choice of the required state space
𝔖_n. Thus, the hierarchy becomes a central concept in our analysis.

One of the most important set of internal variables is that associated with
the vibrational modes in the system. We denote it by ξ_v and seems to have the property that it is always inactive. This
is shown by the lowest lying relaxation time curve corresponding to
τ_v in Fig. <ref>. This is because, we expect
these modes to always come to equilibrium with the medium for any reasonable
τ_obs.



§ GENERALIZED NONEQUILIBRIUM THERMODYNAMICS IN THE EXTENDED SPACE


We are mostly interested in disordered states of a system in this work. Any
ordered state, if it exists, is taken out of the consideration from start.
Thus, the state space 𝔖_𝐗 only contains disordered
states. For vitrification, states in 𝔖_𝐗 refer to the
(physical or hypothetical) EQ states of the supercooled liquid. Defining such
as restricted form of the equilibrium state space is very common in
theoretical physics. For example, when we talk about an equilibrium crystal of
a material, it is also defined in a restricted sense in which its molecules
are not supposed to dissociate into constituent atoms. From now on, we will
denote EQ quantities either by a subscript "eq" or "SCL" and NEQ quantities
without any subscript. If we are interested in a ordered state, we will use a
subscript "CR" to denote its quantity.



 §.§ Equilibrium State


In EQ thermodynamics, a body is specified by a set 𝐗 formed by its
independent extensive observables (E,V,N, etc.); the set also serves the
purpose of specifying the thermodynamic state (also known as the
macrostate) 𝖬 of the system. All EQ states belong to the
state space 𝔖_𝐗 as said above. The thermodynamic
entropy of the body in equilibrium is a state function of 𝐗
and is written as S_eq(𝐗). It is one of the state
functions of the system and is supposed to be differentiable except possibly
at phase transitions, which we will not consider in this review. It satisfies
the Gibbs fundamental relation

    dS_eq(𝐗)=(dE+P_0dV-μ_0dN+...)/T_0
    ,

where we have shown only the terms related to E,V and N. The missing terms
refer to the remaining variables in 𝐗≡{  X_p}, and T_0,P_0,μ_0, etc. have their standard meaning in equilibrium

    ∂ S_eq/∂ E≐1/T_0,∂
    S_eq/∂ V≐P_0/T_0,∂
    S_eq/∂ N≐-μ_0/T_0,⋯
    .

We have used a subscript 0 since in equilibrium, the fields of Σ and
Σ are the same.



 §.§ Nonequilibrium States and Internal Equilibrium
States


The above conclusion is most certainly not valid for a body out of
equilibrium. If the body is not in equilibrium with its medium, its
(macro)state 𝖬(t) will continuously change (relax), which is
reflected in the changes in all of its physical quantities q(t) with time.
Such variations mean that the states no longer belong to 𝔖
_𝐗. These states belong to the enlarged state space
𝔖_𝐙 spanned by 𝐙=(𝐗,ξ). The set ξ of internal variables
<cit.> cannot
be controlled from the outside <cit.>; a readable history of internal
variables is available in a recent paper by Maugin <cit.>. They are
used to characterize internal structures or inhomogeneity
<cit.>
in the system, and are independent of the observables in 𝐗 away
from equilibrium but become dependent on 𝐗 in equilibrium. From
Theorem 4 in <cit.>, it follows that with a proper choice of the
number of internal variables, the entropy can be written as S(𝐙(t))
with no explicit t-dependence. The situation is now almost identical to that
of a body in equilibrium: The entropy is a function of 𝐙(t) with
no explicit time-dependence. This allows us to identify 𝐙(t) as
the set of NEQ state variables. States for which the entropy S
becomes a state function of the state variable 𝐙 are
called internal equilibrium (IEQ) states
<cit.>
and we write

    S_ieq(t)=S(𝐙(t))

for their entropy. This allows us to extend Eq. (<ref>) to

    dS_ieq(t)=
    
    
    ∑_p
    (  ∂ S_ieq(t)/∂ Z_p(t))  dZ_p
    (t)

in which the partial derivatives are related to the fields of the system:

    ∂ S_ieq(t)/∂ E(t)   ≐1/T(t)
    ,∂ S_ieq(t)/∂ V(t)≐P(t)/T(t),
    ∂ S_ieq(t)/∂ N(t)   ≐-μ(t)/T(t),⋯,∂ S_ieq(t)/∂ξ
    (t)≐𝐀(t)/T(t);

these fields will change in time unless the system has reached equilibrium. It
is customary to call 𝐀 the affinity <cit.>. For a
fixed 𝐙, S_ieq does not change in time. Hence, it must
have the maximum possible value for fixed 𝐙
<cit.>. The EQ value of 𝐀
vanishes <cit.>:

    𝐀_eq=0.

In this case, S_ieq is no longer a function of ξ,
which means that ξ  is no longer independent of 𝐗.

We consider the extension of the derivation given earlier <cit.> for
the entropy of Σ_0 by including the internal variable contribution
to obtain as the statement of the second law:

    dS_0(t)/dt    =(  1/T(t)-1/T_0)
    dE(t)/dt+
       (  P(t)/T(t)-P_0/T_0)  dV(t)/dt
    +𝐀(t)/T(t)·dξ(t)/dt
    
        >0;

for a NEQ state. As the entropy of an isolated system Σ_0 can only
increase, dS_0(t)/dt cannot be negative, which explains the last
inequality above for a NEQ process. The strict inequality will be replaced by
an equality for Σ_0 in equilibrium. Each term in the first equation
must be positive in accordance with the second law for a NEQ state.

It follows from Eq. (<ref>) that the above discussion also applies
to an interacting system in a medium for which d_iS/dt is
nonnegative. Thus, we can apply it to a vitrification process in which the
energy decreases with time during isothermal (fixed T_0 of
Σ_0) relaxation. We must, therefore, have

    T(t)>T_0

during any relaxation (at a fixed temperature and pressure of the medium) so
that T(t) approaches T_0 from above [T(t) → T_0^+]
and becomes equal to T_0 as the relaxation ceases and the equilibrium is
achieved; the plus symbol is to indicate that the T(t) reaches T_0 from above.

The relaxation times for different internal variables in ξ
depend on their nature and do not have to be the same. Indeed, the spectrum of
relaxation times in various contexts such as in crystalline solids
<cit.> and glasses <cit.> is intimately related to the
existence of internal variables. Therefore, the spectrum of relaxation times
will be pivotal in our discussion and will be picked up again in Sec.
<ref>.

By attempting to describe NEQ properties of a system by invoking internal
variables, one is able to explain a broad spectrum of NEQ phenomena, but it
should be stated here that the choice and the number of state variables
included in 𝐗 or 𝐙 is not so trivial and must be
determined by the nature of the experiments <cit.>. As we will see in
Sec. <ref>, the observation time τ_obs plays a
central role in determining the relevant state variables during an experiment:


 The state variables that determine the
generalized NEQ thermodynamics are those whose relaxation times are longer
than τ_obs.



The proposition will be justified in Sec. <ref>; see the
paragraph containing Eq. (<ref>).


We will assume here that 𝐙 has been specified. For any IEQ
states _ieq(𝐙), we have τ_ieq
≲τ_obs<τ_eq, where we have introduced the
internal equilibration time τ_ieq required for the system
to come to an IEQ state in 𝔖_𝐙. As expected,
τ_ieq=τ_ieq(𝐙) depends on 𝐙 but
we will not explicitly exhibit its state dependence unless clarity is needed.
These states appear for τ_obs≥τ_ieq. There are
many other states in 𝔖_𝐙 having nonstate entropies
that appear for τ_obs<τ_ieq. As τ_obs
→τ_ieq, we obtain an IEQ state 
_ieq(𝐙). Therefore, there appears a delicate balance
between τ_obs and what internal variables we can describe by our
thermodynamic approach using the concept of IEQ states. This leads us to
consider the hierarchy of relaxation times, which is taken in Sec.
<ref>.

It may appear to a reader that the concept of entropy being a state function
is very restrictive. This is not the case as this concept, although not
recognized by several workers, is implicit in the literature where the
relationship of the thermodynamic entropy with state variables is
investigated. To appreciate this, we observe that the entropy of a body in
internal equilibrium <cit.> is given by the Boltzmann
formula

    S(𝐙(t))=ln W(𝐙(t)),

in terms of the number of microstates W(𝐙(t)) corresponding to
𝐙(t). In classical NEQ thermodynamics <cit.>, the entropy
is always taken to be a state function. In the Edwards approach <cit.>
for granular materials, all microstates are equally probable as is required
for the above Boltzmann formula. Bouchbinder and Langer <cit.> assume
that the NEQ entropy is given by Eq. (<ref>). Lebowitz
<cit.> also takes the above formulation for his definition of the NEQ
entropy. As a matter of fact, we are not aware of any work dealing with
entropy computation that does not assume the NEQ entropy to be  a state
function. This does not, of course, mean that all states of a system are IEQ
states. For states that are not in internal equilibrium, the entropy is not a
state function so that it will have an explicit time dependence. But, as shown
elsewhere <cit.>, this can be avoided by enlarging the space of
internal variables. The choice of how many internal variables are needed will
depend on experimental time scales.



§ HIERARCHY AMONG RELAXATION TIMES AND ENLARGED STATE
SPACES


We now classify state variables in a hierarchical manner as below. In IEQ
states, ξ has had enough time τ_obs
=τ_ieq<τ_eq for _ieq to emerge.
But for τ_obs<τ_ieq, the states in 𝔖
_𝐙 have not had enough time for _ieq to
emerge so that their entropy is a nonstate function, which will continue to
increase if the system is left isolated until it reaches S_ieq
(𝐗,ξ) and becomes a state function. The affinity
𝐀 corresponding to ξ is nonzero in 
_ieq. If there were other internal variables ξ^',ξ^'',ξ^''',⋯ in the system, with relaxation times τ^'
,τ^'',τ^''',⋯, respectively, that are
distinct from ξ, then these must have equilibrated during
τ_ieq so that their affinities 𝐀^'
,𝐀^'',𝐀^''',⋯ have
vanished, implying that they are no longer independent of 𝐗
(𝐀^'=∂ S/∂ξ^'=0). This
means that the entropy does not depend on them. It is clear that
τ_ieq forms an upper bound for the relaxation times
τ^',τ^'',τ^''',⋯. Thus,
they play no role in 𝔖_𝐙. When the process is carried
out somewhat faster (τ_obs<τ_eq) than that required
for obtaining _eq(𝐗), then ξ
has not had enough time to "equilibrate" as we have discussed earlier
<cit.> and 𝐀≠0.

Even if S does not depend on ξ^',ξ^'',ξ^''',⋯, we will see in
Sec. <ref> that they affect the thermodynamics of the system
indirectly, a fact that does not seem to have been appreciated. For the
moment, we will not consider the internal variables ξ^',ξ^'',ξ^'''
,⋯. We will consider them later and will denote them collectively by
ξ_E=(ξ^',ξ^'',ξ^''',⋯).

The discussion below is somewhat abstract and intricate, and requires patience
on the part of the reader. The set-theoretic notation is perfectly suited for
the abstract nature of the discussion. Some readers may find the set-theoretic
notation cumbersome, but this is the price we must pay to make the discussion
comprehensive but compact.

To simplify our discussion, we assume that all internal variables in
ξ are divided into nonoverlapping groups
ξ_n indexed by n=1,2,⋯. We further assume that all
internal variables in ξ_n have the same relaxation time
τ_n so that they equilibrate and are no longer independent of
𝐗 for Δ t≳τ_n. The relaxation times depend
strongly on 𝐗. Let us also define ξ_0=𝐗
in order to compactify our notation below. Because of this, we can include
ξ_0=𝐗 whenever we speak of internal variables from
now on, unless clarity is needed. The groups ξ_n
,n=0,1,2,⋯ are indexed by n so that τ_n's appear in a
decreasing order (with τ_0=τ_eq):

    τ_0>τ_1>τ_2>⋯.

The relaxation times form a discrete set and not a continuum for simplicity.
It is important that the set {ξ_k} has a
finite though large number of elements for a physically sensible thermodynamic
description of the system; having an enormous number of elements will make the
description unnecessarily too complex and completely useless for thermodynamics.

We now introduce the sequence of state spaces {𝔖
_n}, where 𝔖_n,n=0,1,2,⋯ is spanned by the
union

    ξ^(n)≐∪_k=1^nξ_k,n≥1,

of all ξ_k,k≤ n, with relaxation times τ_k
>τ_n+1, with ξ^(0) (not to be confused with
ξ_0=𝐗) denoting an empty set, so that

    𝐙_n≐(𝐗,ξ_1,ξ
    _2,⋯,ξ_n)≡(ξ_0,ξ^(n)),n≥0.

Thus, 𝔖_0=𝔖_𝐗, formed by 𝐙
_0=ξ_0=𝐗, is relevant when τ_0
>τ_obs>τ_1. Similarly, 𝔖_1, formed by
𝐙_1=(ξ_0,ξ^(1))=(𝐗
,ξ_1), is relevant when τ_1>τ_obs>τ
_2, and so on.

It is clear from the construction that the state spaces 𝔖
_n,n=0,1,2,⋯ are ordered with increasing dimensions:

    𝔖_0⊂𝔖_1⊂𝔖_2⊂⋯.

The longest relaxation time in 𝔖_n is τ_0 corresponding
to ξ_0=𝐗 and the shortest relaxation time is
τ_n corresponding to ξ_n. Any ξ
_k,k>n with relaxation time shorter than τ_n need not be considered
as it has already equilibrated and does not affect any state in 𝔖
_n. We can summarize this conclusion as the following


The additional internal variable
ξ_k in 𝔖_k relative to 𝔖_k-1
equilibrates and plays no role (i.e., is absent) in all smaller state spaces
𝔖_l,l≤ k-1 but participate in all state spaces
𝔖_l larger than 𝔖_k-1, i.e., l≥ k.



See the discussion above.

























Let us consider some observation time τ_obs used to observe a
state  of an interacting system. We can always find a pair of
neighboring state spaces 𝔖_n+1⊃𝔖
_n,n≥0 satisfying

    τ_n+1<τ_obs<τ_n;

the two sides define a window Δ t_n≐τ_n-τ_n+1
in which τ_obs must lie. As τ_obs>τ_n+1
, all ξ_k's,k>n, do not have to be considered to
describe the state  as they have already equilibrated (cf. the
discussion of ξ̀ above); thus, 𝔖_k,k>n,
play no role in describing . As τ_obs<τ_n, we
need to consider all ξ_k,k≤ n to describe . We
must, therefore, use 𝔖_n to describe  for a
τ_obs in this window; we denote  by 
(𝐙_n) for clarity in this section. Among all the states in
𝔖_n, there are IEQ states _ieq
(𝐙_n) for which S=S_ieq(𝐙_n). This happens
when τ_obs≃τ_ieq(𝐙_n)≤τ_n,
τ_ieq(𝐙_n) denoting the time required for
(𝐙_n) to evolve into _ieq
(𝐙_n); we will also use τ_ieq^(n) or simply
τ_ieq to denote τ_ieq(𝐙_n) in
𝔖_n if no confusion will arise. For n=0, τ_ieq
(𝐙_0) simply refers to τ_eq.

There exists IEQ states _ieq(𝐙_n) in
𝔖_n for which ξ_n is no longer independent
of 𝐗; for these states, τ_obs≃τ_ieq
(𝐙_n-1,ξ_n(𝐗))≡τ_ieq
(𝐙_n-1). However, ξ_n→ξ_n(𝐗) as t→τ_n even if (𝐙
_n-1) in 𝔖_n-1 has not turned into _ieq
(𝐙_n-1). As achieving internal equilibrium will take some
additional time, we have τ_ieq(𝐙_n-1)>τ_n. We
thus conclude that (with τ_ieq^(0) representing τ
_eq)

    τ_ieq^(n)<τ_ieq^(n-1),n>0,

which will be assumed in this work.

We now consider the window

    τ_1<τ_obs<τ_0.

As τ_0>τ_obs>τ_1, ξ_1 has already
equilibrated so it need not be considered, but ξ
_0=𝐗 has not yet equilibrated. Thus, the entropy must be a
function only of the observables 𝐗, which we must write as
S_ieq(𝐗(t)) as it continues to vary. As τ_obs
→τ_0, 𝐗(t)→𝐗_eq,
S_ieq continues to increase until it finally reaches S_eq; there is no explicit time dependence as all ξ_k's,k>0, have equilibrated; see also Landau and Lifshitz <cit.> and
Wilks <cit.>, where NEQ states with respect to the medium are treated as
IEQ states in 𝔖_𝐗. This is the most common way NEQ
states in the literature are treated when internal variables are not invoked.
This is only possible when τ_obs satisfies Eq.
(<ref>).

We now consider the remaining case

    τ_obs≥τ_0.

This situation corresponds to the quasistatic case so that even
ξ_0=𝐗 has equilibrated to 𝐗_eq
and we are dealing with an EQ state

    S=S_eq=S(𝐗_eq).



We know that {τ_n} depend on the state of the system.
In vitrification that is of our primary interest here, they depend on the
temperature T_0. It is commonly believed that τ_n's increase with
decreasing T_0 as shown in Fig. <ref>, where we show them
as a function of T_0. From this figure, we observe that for a given
τ_obs, drawn as a solid or broken line in red, ξ_E correspond to the internal variables that lie in the
inactive zone lying below τ_obs. (Recall that
ξ_0=𝐗 is now included in internal variables.) They
have all equilibrated. The active zone corresponds to internal
variables that lie above τ_obs. They have not equilibrated. For
higher temperatures (T_0>T_0g), all internal variables are
inactive. At lower temperature, some of them become active and make the system
out of equilibrium. At very low temperatures, all internal variables become
active for their NEQ role. We will discuss this figure further in Sec.
<ref>.



§ GENERAL CONSIDERATION


We have in Sec. <ref> that for a given τ_obs, we
can find the window Δ t_n satisfying Eq.
(<ref>), which then determines the state space
𝔖_n to describe any state  for the given
τ_obs. The internal variables ξ_k's,k>n, do
not have to be considered as their affinities 𝐀_k's have vanished
for the given τ_obs. However, the situation is somewhat
complicated for the following reason. As τ_k's are determined by
time-dependent 𝐙_n, the window will continue to change with time
for a given τ_obs so the value of n will have to adjusted as
τ_k's change. The most simple solution for this complication is to
allow considering all the internal variables regardless of whether they have
equilibrated or not. The fact that 𝐀=0 for equilibrated internal
variables means that their contribution to d_iW will vanish so they
will not affect the Gibbs fundamental relation. Despite this, as we will see
later in Sec. <ref>, these internal variables leave their mark in
relaxation. Therefore, from now on, we will consider the entire set
𝐙 in the thermodynamic approach.



 §.§ First Law


The infinitesimal heat exchange between the medium Σ and
the system Σ will be denoted by d_eQ(t); similarly, the
infinitesimal work done on Σ by Σ will be denoted by
d_eW(t). The subscript "e" is a reminder of the exchange. Then the
first law of thermodynamics is written as

    dE(t)≡ d_eQ(t)-d_eW(t)

in terms of exchange heat and work d_eQ(t)=T_0d_e
S(t) and d_eW(t)=P_0dV(t), respectively; see Sec.
<ref>. If there are other kinds of exchange work such as due to a
magnetic field, an exchange of particles, etc. they can be subsumed in
d_eW(t). However, for simplicity, we will assume only the
pressure-volume work in this work. Both quantities are controlled from outside
the system. If the pressure P(t) of the system is different from the
external pressure P_0 of the medium, then their difference gives rise to
the internal work d_i^VW(t)≐(  P(t)-P_0)
dV(t), which is dissipated within the system; we have added a superscript as
a reminder that this particular internal work is due to volume variation. If
there are internal variables, they do not contribute to d_eW(t) as
the corresponding EQ affinity 𝐀_0=0. Despite this, the internal
variable ξ does internal work given by d_i
^ξW(t)≐𝐀(t)·dξ(t)
and must be added to the internal work due to pressure difference. We thus
identify the internal work d_iW(t) as

    d_iW(t)≐(  P(t)-P_0)  dV(t)+𝐀
    (t)·dξ(t),

and the net work is

    dW(t)=d_eW(t)+d_iW(t)=P(t)dV(t)+𝐀(t)·dξ(t),

a quantity that depends only on Σ and is oblivious to the properties
ofΣ. Such a quantity is called a system-intrinsic (SI)
quantity. Introducing a new quantity <cit.>

    d_iQ(t)≡ d_iW(t),

and the net heat

    dQ(t)≐ d_eQ(t)+d_iQ(t),

we can write the first law as

    dE(t)=dQ(t)-dW(t).

As dE(t) and dW(t) are both SI-quantities, dQ(t) must also be a
SI-quantity. Thus, the above formulation of the first law is in terms of
quantities that refer to the system. There are no quantities that refer to
Σ. We will call dQ(t) and dW(t) as the
generalized heat dQ(t) added to and the generalized
work dW(t) done by the system <cit.>. We will
reserve exchange heat and work for d_eQ(t)=T_0
d_eS(t) and d_eW(t)=P_0dV(t), respectively, throughout
this work; see Sec. <ref>. Remembering this, we will also call
generalized heat and work as simply heat and work, respectively,
for brevity.



 §.§ Second Law


The second law states that the irreversible (denoted by a suffix i) entropy
d_iS generated in any infinitesimal physical process going on
within a system satisfies the inequality

    d_iS≥0;

the equality occurs for a reversible process. For the isolated system
Σ_0, we must have, see Eq. (<ref>)

    dS_0=d_iS_0=d_iS≥0.



As the thermodynamic entropy is not measurable except when the process is
reversible, the second law remains useless as a computational tool. In
particular, it says nothing about the rate at which the irreversible entropy
increases. Therefore, it is useful to obtain a computational formulation of
the entropy, the statistical entropy. This will be done in the next
section. The onus is on us to demonstrate that the statistical entropy also
satisfies this law if it is to represent the thermodynamic entropy. This by
itself does not prove that the two are the same. It has not been possible to
show that the statistical entropy is identical to the thermodynamic entropy in
general. Here, we show their equivalence only when the NEQ thermodynamic
entropy is a state function of NEQ state variables to be introduced below.



 §.§ Internal Equilibrium Thermodynamic


 For a body in internal equilibrium, its entropy S is a function of E,V
and ξ. Introducing the corresponding fields

    (  ∂ S/∂ E)        =β(t)≐1/T(t),(
    ∂ S/∂ V)  =β(t)P(t),
    (  ∂ S/∂ξ)        =β(t)𝐀
    (t),

we can write down the differential

    dS(t)=β(t)[dE(t)+P(t)dV(t)+𝐀(t)·dξ(t)],

which can be inverted to express dE(t) as follows:

    dE(t)=T(t)dS(t)-P(t)dV(t)-𝐀(t)·dξ(t).

Comparing with Eq. (<ref>), we conclude an identity

    dQ(t)=T(t)dS(t),

regardless of the number of internal variables are used to describe Σ.

We now write dQ=T_0d_eS(t)+T_0d_iS(t)+[T(t)-T_0
]dS(t)=d_eQ(t)+T_0d_iS(t)+[T(t)-T_0]dS(t). From this
and using Eq. (<ref>), we conclude that

    d_iQ(t)      =T_0d_iS(t)+[T(t)-T_0]dS(t)
        =Td_iS(t)+[T(t)-T_0]d_eS(t)=d_i
    W(t),

which can be used to express d_iS(t) as follows

    T_0d_iS(t)      =[T_0-T(t)]dS(t)+[P(t)-P_0]dV(t)
         +𝐀(t)· dξ
    (t);
    
    Td_iS(t)      =[T_0-T(t)]d_eS(t)+[P(t)-P_0
    ]dV(t)
         +𝐀(t)· dξ
    (t).

Since dS(t),d_eS(t),dV(t) and dξ(t) are
independent variations, each of the three contributions on the right side in
each equation must be non-negative



    T_0-T(t)]dS(t)    ≥0,
    
     T_0-T(t)]d_eS(t)    ≥
    0,
     P(t)-P_0]dV(t)    ≥0,
    
    𝐀(t)· dξ(t)    ≥
    0,

to comply with the second law requirement d_iS(t)≥0; we are
assuming T_0 and T(t) are positive. The factors T_0-T(t),
P(t)-P_0 and 𝐀(t) in front of the extensive variations are the
corresponding thermodynamic forces that act to bring the system to
equilibrium. In the process, each force has its own irreversible entropy
generation <cit.>. The last inequality implies that each
independent component ξ_k∈ξ must satisfy A_k
(t)dξ_k(t)≥0. There will be no irreversible entropy generation and the
equalities occur when thermodynamic forces vanish, which is the situation for
a reversible process.

It should be noted that Eq. (<ref>) simply
states that heat exchanges (flows) from hot to cold. To see this, we use the
equality d_eS(t)=d_eQ(t)/T_0 to rewrite the equation as
[T_0-T(t)]d_eQ(t)≥0. If T_0>T(t), heat is exchanged to the
system; if T_0<T(t), heat is exchanged from the system.

It follows from the last two inequalities in Eq.
(<ref>) that


    d_iW(t)≥0.

This means that d_iW(t) truly represents irreversibility or
dissipation within the system. We note that while each term in d_i
W(t) is non-negative, this is not so for d_iQ(t) written in the
form



    d_iQ(t)       =T_0d_iS(t)+[T(t)-T_0
    ]dS(t),
         =T(t)d_iS(t)+[T(t)-T_0]d_eS(t)

in which the first term is non-negative, but the second term is non-positive.
This not only means that the physics of d_iQ(t) and
d_iS(t) is very different but also that


    d_iQ(t)≤ T_0d_iS(t) ,dQ(t)≤ T_0dS(t);

the equalities occur only for isothermal (T=T_0) or adiabatic (dS=0) processes.

Let us consider the Helmholtz free energy H(S,V,ξ
,P_0)=E(S,V,ξ)+P_0V(t)
<cit.> in terms of the external pressure P_0 of the
medium. We can treat HH(S,V,ξ,P_0) as an
SI-quantity by treating P_0 as a parameter. It is easy to see that

    dH=TdS(t)-[P(t)-P_0]dV(t)-𝐀(t)· dξ
    (t)+V(t)dP_0.

The above differential clearly shows that the enthalpy H is a function of
S,V,ξ, and P_0. Recall that for an EQ state,
H(S,P_0) is not a function of V so it is a Legendre transform of
E(S,V) with respect to V(t). In other words, ∂ H/∂ V=0.
What we see from above that, for a NEQ states, H is not a Legendre transform
of E with respect to V. This is clearly seen by evaluating

    ∂ H/∂ V=P(t)-P_0≠0,

as the pressure difference need not vanish in an irreversible process. Despite
this, dH has no irreversible component as we easily find that

    dH=TdS(t)-d_iW(t)+V(t)dP_0=d_eQ+V(t)dP_0,

regardless of the number and nature of the internal variables; we have used
here Eqs. (<ref>) and (<ref>). Thus,
dH only contains exchange quantities as both terms on the right side are
controllable from outside the system. As such, it does not have any
spontaneous or irreversible relaxation. For an isobaric process, dP_0=0 so
dH reduces to

    dH=d_eQ.

The above equality, which is well known for a reversible process, remains
valid no matter how irreversible a process is. Thus, it must remain valid for
supercooled liquids and glasses. Observe that just as d_iE=0, see
Eq. (<ref>), so is d_iH=0, with d_e
H=dH=d_eQ.

Let us now consider the Gibbs free energy G(S,V,ξ
,T_0,P_0)≐ E(S,V,ξ)-T_0S(t)+P_0V(t)
<cit.> in terms of the external temperature T_0 and
pressure P_0 of the medium. As is the case with the enthalpy, the Gibbs
free energy is also not a Legendre transform of E(S,V,ξ) with
respect to S(t) and V(t). We find that

    dG      =[T(t)-T_0]dS(t)-d_iW(t)-S(t)dT_0+V(t)dP_0
        =-T_0d_iS(t)-S(t)dT_0+V(t)dP_0,

in which the first term can be identified as d_iG≐
-T_0d_iS(t) and the remainder as d_eG≐
-S(t)dT_0+V(t)dP_0. At fixed T_0 and P_0, we have

    dG=d_iG=-T_0d_iS(t)≤0,

showing that the Gibbs free energy decreases during spontaneous relaxation
such as on a glass.
























§ ENTROPY BOUND DURING VITRIFICATION


We now apply the IEQ thermodynamics of the last section to the vitrification
process, is carried out at some cooling rate as follows. The discussion in
this section is an elaboration and extension of our earlier discussion
<cit.>
and follows the approach first used by Bestul and Chang <cit.> and
later by Sethna and coworkers <cit.>. The temperature of the
medium is isobarically changed by some small but fixed Δ T_0 from the
current value to the new value, and we wait for (not necessarily fixed) time
τ_obs at the new temperature to make an instantaneous
measurement on the system before changing the temperature again. At some
temperature T_0g, the relaxation time τ_0=τ_eq,
which continuously increases as the temperature is lowered [see Fig.
(<ref>)], becomes equal to τ_obs as shown in Fig.
<ref>. The location of T_0g depends on the rate of
cooling, i.e. on τ_obs, which is clear from the figure. The
crossing T_0g is lower for the broken τ_obs than for
the solid τ_obs. There are several other crossings at
T_01,T_02,⋯, see Fig. (<ref>), at which
τ_obs crosses other relaxation curves for τ_1,τ
_2,⋯, respectively. The crossing again depends on whether we take the
solid or the broken curve for τ_obs. Let T_0R
>T_0G denote the temperature of the last such crossing (not shown
in the figure) before T_0G. Just below T_0g, the
structures are not yet frozen; they "freeze" at a lower temperature
T_0G (not too far from T_0g) to form an amorphous solid
with a viscosity η_G≃10^13 poise corresponding to some
time scale t_G, see Fig. <ref>. This solid is
identified as a glass determined by the choice of η_G or
t_G. At T_0G, the relaxation time τ_R is
at least τ_G. Over the glass transition region between
T_0G and T_0g in Fig. <ref>, the NEQ
liquid gradually turns from an EQ supercooled liquid at or above
T_0g into a glass at or below T_0G, a picture already
known since Tammann <cit.>; see also <cit.>. Over this
region, some dynamical properties such as the viscosity vary continuously but
very rapidly. However, thermodynamic quantities such as the volume or the
enthalpy change continuously but slowly. As is evident from Fig.
(<ref>), more and more internal variables become active as the
temperature is reduced and will determine the thermodynamics in this region.
Below T_0G, all of these are almost "frozen" except those in the
inactive zone such as ξ_v corresponding to the
relaxation time τ_v, representing local localized oscillations
within cells in the cell model <cit.>; see the discussion in Secs.
<ref> and <ref>.

As the observation time τ_obs is increased, the equilibrated
supercooled liquid continues to lower temperatures before the appearance of
T_0g. In the hypothetical limit τ_obs
→∞, it is believed that the equilibrated supercooled liquid
will continue to lower temperatures without any interruption, and is shown
schematically by the solid blue curve in Fig. <ref>. We
overlook the possibility of the supercooled liquid ending in a spinodal that
has been seen theoretically <cit.>. It is commonly believed
that this entropy will vanish at absolute zero (S_SCL(0)≡0),
as shown in the figure. As we are going to be interested in S_SCL
(T_0) over (0,T_0g), we must also acknowledge the possibility
of an ideal glass transition in the system. If one believes in an ideal glass
transition, then there would be a singularity in S_SCL(T_0) at
some positive temperature T_K<T_0G, below which the system
will turn into an ideal glass whose entropy will also vanish at absolute zero
<cit.>. The possibility of an
ideal glass transition, which has been discussed in a recent review elsewhere
<cit.>, will not be discussed further in this work. All that will
be relevant in our discussion here is the fact that the entropy vanishes in
both situations (S_SCL(0)≡0). However, it should be emphasized
that the actual value of S_SCL(0) has no relevance for the theorems
we derive below.

It is a common practice to think of the glass transition to occur at a point
that lies between T_0g and T_0G. We have drawn entropy
curves (Glass and SCL) in Fig. <ref> for a process of
vitrification in a cooling experiment. The entropy curves S_g
(T_0,t)for Glass emerges out of S_SCL(T_0) at T_0g for a given τ_obs in such a way that it lies above that of
SCL for T_0g>T_0≥0. At any nonzero temperature T_0,
S(T_0,t) approach S_SCL(T_0) from above during
isothermal (fixed temperature of the medium) relaxation; see the two
downward vertical arrows. These relaxations are discussed in the next section.

The concept of internal equilibrium is also a common practice now-a-days for
glasses <cit.>. Employing the concept of internal
equilibrium provides us with an instantaneous Gibbs fundamental relation, see
Eq. (<ref>), which determines instantaneous
temperature, pressure, etc. of the system.

We now prove the entropy bounds

    S_R≡ S(0)>S_expt(0)>S_SCL(0).

in the form of Theorems <ref> and
<ref>. We will only consider isobaric cooling (we will
not explicitly exhibit the pressure in this section), which is the most
important situation for glasses. The process is carried out along some path
from an initial state A at temperature T_0A in the supercooled
liquid state which is still higher than T_0g to the state A_0
at absolute zero. The state A_0 depends on the path A→A_0,
which is implicit in the following. The change dS between two neighboring
points along such a path is
<cit.> dS=d_e
S+d_iS; for a NEQ system, the two parts of dS are path dependent.
The component

    d_eS(t)=-d_eQ(t)/T_0≡ C_PdT_0/T_0

represents the reversible entropy exchange with the medium in terms of the
heat d_eQ(t) given out by the glass at time t to the medium whose
temperature at that instant is T_0. The component d_iS>0
represents the irreversible entropy generation in the irreversible process;
see Eq. (<ref>). In general, it contains, in addition to
the contribution from the irreversible heat transfer with the medium,
contributions from all sorts of viscous dissipation going on within the
system and normally require the use of internal variables
<cit.>. The equality in Eq.
(<ref>) holds for a reversible process, which we will no
longer consider unless stated otherwise. The strict inequality d_i
S>0 occurs only for an irreversible process such as in a glass.


The experimentally observed (extrapolated) non-zero
entropy S_expt(0) at absolute zero in a vitrification process is a
strict lower bound of the residual entropy of any system:

    S_R≡ S(0)>S_expt(0).





We have along A→A_0

    S(0)=S(T_0)+
    
    
    ∫_A^A_0
    d_eS+
    
    
    ∫_A^A_0
    d_iS,

where we have assumed that there is no latent heat in the vitrification
process. The first integral is easily determined experimentally since it is
expressible in terms of the exchange heat

    ∫_A^A_0
    d_eS=-
    
    
    ∫_A^A_0
    d_eQ/T_0.

The second integral in Eq. (<ref>) is always
positive, but almost impossible to measure as it involves thermodynamic
forces; see Eq. (<ref>):

    ∫_A^A_0
    d_iS=∫_A^A_0{[ T_0-T(t)]dS(t)+[P(t)-P_0]dV(t);                   +𝐀(t)· dξ(t) ]}/T_0>0.

It involves knowing and since the residual entropy S_R is, by
definition, the entropy S(0) at absolute zero, we obtain the important
result

    S_R≡ S(0)>S_expt(0)≐ S(T_0A)+
    
    
    ∫_T_0A^0
    C_PdT_0/T_0.

This proves Theorem <ref>.


The irreversibility during vitrification does not allow for the determination
of the entropy exactly, because evaluating the integral in Eq.
(<ref>) is not feasible
<cit.>. The forward inequality

    S_R-S_expt(0)=
    
    
    ∫_A^A_0
    d_iS>0

is due to the irreversible entropy generation from all possible sources
<cit.>. The inequality is made
strict as we are treating the NEQ glass with τ_obs<τ
_eq(T_0) and clearly establishes that the residual entropy at
absolute zero must be strictly larger than the "experimentally or
calorimetrically measured" S_expt(0).


The calorimetrically measured (extrapolated)
entropy during processes that occur when τ_obs<τ_eq
(T_0) for any T_0<T_0g is larger than the hypothetical
supercooled liquid entropy at absolutely zero

    S_expt(0)>S_SCL(0).





Let Q̇_e(t)≡ d_eQ(t)/dt be the rate of net heat
loss by the system during τ_obs<τ_eq(T_0) as it
relaxes isothermally at some fixed T_0. For each temperature interval
dT_0<0 below T_0g, we have

    | d_eQ|   ≡ C_P| dT_0
    | =
    
    
    ∫_0^τ_obs
    |Q̇_e| dt
        <| d_eQ| _eq(T_0)≐
    
    
    ∫_0^τ_eq(T_0)
    |Q̇_e| dt,      T_0<T_0g

where | d_eQ| _eq(T_0)>0 denotes the
net heat loss by the system to come to equilibrium, i.e. become supercooled
liquid during cooling at T_0. For T_0≥ T_0g, dQ≡
dQ_eq(T_0)≐ C_P,eqdT_0. Thus, the entropy loss
observed experimentally with τ_obs<τ_eq(T_0) is
less than the entropy loss if the system is allowed to come to SCL at each
temperature T_0. We thus conclude that

    S_expt(0)>S_SCL(0).



This proves Theorem <ref>.


The strict inequality above is the result of the fact that glass is a NEQ
state. Otherwise, we will have S_expt(0)≥ S_SCL(0) for
any arbitrary state.

The difference S_R- S_expt(0) would be larger, more
irreversible the process is. The quantity S_expt(0) can be
determined calorimetrically by performing a cooling experiment. We take
T_0A to be the melting temperature T_0M, and uniquely
determine the entropy of the supercooled liquid at T_0M by adding
the entropy of melting to the crystal entropy S_CR(T_0M)
at T_0M. The latter is obtained in a unique manner by integration
along a reversible path from T_0=0 to T_0=T_0M:

    S_CR(T_0M)=S_CR(0)+
    
    
    ∫_0^T_0M
    C_P,CRdT_0/T_0,

here, S_CR(0) is the entropy of the crystal at absolute zero, which
is traditionally taken to be zero in accordance with the third law, and
C_P,CR(T_0) is the isobaric heat capacity of the crystal. This
then uniquely determines the entropy of the liquid to be used in the right
hand side in Eq. (<ref>). We will assume that
S_CR(0)=0. Thus, an experimental determination of S_expt
(0) is required to give the lower bound to the residual entropy in
Eq. (<ref>). Experimental evidence for a non-zero value
of S_expt(0) is abundant as discussed by several authors
<cit.>
; various textbooks <cit.> also discuss this issue.
Goldstein <cit.> gives a value of S_R≃15.1 J/K mol
for o-terphenyl from the value of its entropy at T_0=2 K.
However, Eq. (<ref>) gives a mathematical justification of
S_expt(0)>0. The strict inequality proves immediately that the
residual entropy cannot vanish for glasses, which justifies the curve
Glass in Fig. <ref>. The relevance of the residual entropy
has been discussed by several authors in the literature.
<cit.>


By considering the state A_0 above to be a state A_0 of the glass in
a medium at some arbitrary temperature T_0^' below T_0g,
we can get a generalization of Eq. (<ref>):

    S(T_0^')>S_expt(T_0^')≐ S(T_0)+
    
    
    ∫_T_0^T_0^'
    C_PdT_0/T_0.

We again wish to remind the reader that all quantities depend on the path
A→A_0, which we have not exhibited. By replacing T_0 by
the melting temperature T_0M and T_0^' by T_0, and
adding the entropy S(T_0M) of the medium on both sides
in the above inequality, and rearranging terms, we obtain (with S_L
(T_0M)=S_SCL(T_0M) for the liquid)

    S_L(T_0M)+S(T_0M)≤ S(T_0
    )+S(T_0M)-
    
    
    ∫_T_0M^T_0
    C_PdT_0/T_0,

where we have also included the equality for a reversible process. This
provides us with an independent derivation of the inequality given by Sethna
and coworkers <cit.>.

It is also clear from the derivation of Eq. (<ref>) that
the inequality can be generalized to any temperature T_0<T_0g
with the result

    S_expt(T_0)>S_SCL(T_0),

with S_expt(T_0)→ S_SCL(T_0) as
T_0→ T_0g from below.

While we have only demonstrated the forward inequality, the excess
S_R-S_expt(0) can be computed in NEQ thermodynamics
<cit.>, which provides a clear
prescription for calculating the irreversible entropy generation. The
calculation will, of course, be system-dependent and will require detailed
information. Gutzow and Scmelzer <cit.> provide such a
procedure with a single internal variable but under the assumption of equal
temperature and pressure for the glass and the medium. However, while they
comment that d_iS≥0 whose evaluation requires system-dependent
properties, their main interest is to only show that it is negligible compared
to d_eS.

We have proved Theorems <ref> and
<ref> by considering only the system without paying
any attention to the medium. For Theorem <ref>, we require
the second law, i.e. Eq. (<ref>). This is also true of
Eq. (<ref>). The proof of Theorem
<ref> requires the constraint τ_obs
<τ_eq(T_0) for any T_0<T_0g, which leads to a NEQ
state. The same is also true of Eq. (<ref>).

We have focused on the system in this section. This does not mean that the
conclusion would be any different had we brought the medium into our
discussion. This is seen from the derivation of the inequality in Eq.
(<ref>) from Eq. (<ref>).



§ ENTROPY AND ENTHALPY DURING ISOTHERMAL
RELAXATION


We wish to consider isothermal relaxation in an isobaric cooling
experiment carried out at a fixed pressure P_0. Let us assume that
Σ is in equilibrium at some temperature T_0^'≤
T_0g of some medium Σ^'. We change to a
different medium Σ at T_0<T_0^',P_0 and
bring Σ in its contact. Initially, the temperature T(  0) of Σ is T(  0)  =T_0^'>T_0 so it is out of
equilibrium with the new medium and its temperature T(t) will strive to get
closer to T_0 as we wait for Σ to come to equilibrium with
Σ; see Eq. (<ref>). The initial
entropy S(T_0,0)=S_SCL(T_0^')>S_SCL(T_0). If
the system is now allowed to equilibrate, it will undergo spontaneous
(isothermal) relaxation at fixed T_0 so that S(T_0,t)→
S_SCL(T_0) in time during which its temperature changes. We
assume that the relaxation times of ξ_n as a function of
T(t) is similar to that shown in Fig. <ref>; all we need to
do is to replace T_0 by T(t). During relaxation, the entropy of the
glass is supposed to decrease. This is what we expect intuitively as the
arrows show in Fig. <ref>. We now wish to consider such a
relaxation and determine the behavior of thermodynamic functions such as the
entropy, enthalpy, etc. using IEQ thermodynamics introduced above. We prove
two additional theorems in this section. The theorems are general even though
we have in mind NEQ states including glasses obtained under the condition
τ_obs<τ_eq(T_0) for any T_0<T_0g. We
consider the system to be in internal equilibrium with temperature T(t),
pressure P(t), etc. We remind the reader that all processes that go on
within the medium occur at constant temperature T_0, pressure P_0,
etc. Thus, there will not be any irreversible process going on within the
medium. All irreversible processes will go on within the system.

We will exploit below the strict inequalities in Eq.
(<ref>) to derive a bound on the rate of
entropy variation. For a system out of equilibrium, the instantaneous entropy
S(t) and volume V(t) seem to play the role <cit.> of "internal
variables," whose "affinities" are given by the corresponding thermodynamic
forces T_0-T(t) and P(t)-P_0, respectively. This fact is not commonly
appreciated in the glass literature to the best of our knowledge. Even during
an isobaric vitrification, there is no fundamental reason to assume that the
pressure P of the system is always equal to the external pressure P_0.
However, it is a common practice to assume the two to be the same, which may
not be a poor approximation in most cases. We will not generally make such an
approximation in this work.

We now state Theorem <ref>.


The entropy of a glass reaches that of the
supercooled liquid from above during relaxation at fixed T_0,P_0 of the
mediums. Thus,

    S>S_SCL,

so that the entropy variation in time has a unique direction as shown by the
downward arrows in Fig. <ref>.



It follows from Eqs. (<ref>) and
(<ref>) that for any NEQ state during
relaxation (fixed T_0,P_0)

    dS(t)/dt<0;

the inequality turns into an equality once equilibrium is reached. In other
words, during relaxation,

    S(T_0,P_0,t)→ S_SCL^+(T_0,P_0);

the plus symbol is again to indicate that the glass entropy reaches
S_SCL(T_0,P_0) from above. This completes the proof of Theorem
<ref>.


We have shown T_0,P_0 in S(T_0,P_0,t)≡ S(T(t),P(t),A(t)) to
emphasize that the result is general during any relaxation. In the derivation,
we have only used the second law. Being a general result, it should be valid
for any real glass. Above T_0g, the system is always in equilibrium
with the medium so its temperature is the same as T_0. Below
T_0g, when the system is not in equilibrium with the medium, then
T(t)>T_0 in accordance with Eq. (<ref>) based on the
experimental observation. Any theory, such as the one proposed in
<cit.> and known as the entropy loss
view of the glass transition, in which S(T_0,P_0,t) drops below
S_SCL(T_0,P_0) so that

    S(T_0,t)≤ S_SCL(T_0).

In this case, during relaxation, dS(t)>0 so that (T_0-T(t))dS(t)<0 in
direct conflict with Eq. (<ref>), a
consequence of the second law. Such a theory then violates the second law as
first pointed out by Goldstein <cit.>; we will revisit this issue in
the final section.

We now prove the following theorem:


For a glass, we must have H(T_0
,P_0,t)>H_SCL(T_0,P_0) at all T_0< T_0g, where
S>S_SCL. 



According to Eqs. (<ref>) and
(<ref>), we conclude that d_e
Q=T_0 d_eS<0 [cf. Eq. (<ref>)] while
relaxation is going on and vanishes as T(t)→ T_0^+. It then
follows from Eq. (<ref>) that

    dH(t)/dt≤0,

a result that is consistent with experimental observations
<cit.>. This completes the proof of the theorem.


It follows from the behavior of the Gibbs free energy G(t)=H(t)-T_0S(t)
during relaxation (dG(t)/dt≤0) that dH≤ T_0 dS, i.e.,


    |dH(t)/dt|≥ T_0|dS(t)/dt|

and

    |Δ H(T)|   ≐ H(T_0)-H_SCL
    (T_0)
       ≥ T_0[S(T_0)-S_SCL(T_0)];T_0<T_0g
    .

The equality holds at T_0=T_0g. We can also obtain Eq.
(<ref>) using dH=T_0 d_eS≤ T_0 dS.

From Eqs. (<ref>) and
(<ref>), we also have


    |dH(t)/dt|≥ T(t)|dS(t)/dt| .

The last bound is tighter than the bound in Eq. (<ref>) and reduces
to the equality obtained earlier <cit.> where ξ was
neglected. This equality there was used to infer Eq. (<ref>
). We have just established that the conclusion remains unaltered even if we
consider internal variables.

In summary, the isothermal relaxation originates from the tendency of the
glass to come to thermal equilibrium during which its temperature T(t)
approaches T_0 from above in time. The relaxation process results in the
lowering of the corresponding Gibbs free energy in time, as expected due to
the second law. But it also results in the lowering of the corresponding
entropy as shown in Fig. <ref>, and the enthalpy during
vitrification; the latter is observed experimentally <cit.>.



§ TEMPERATURE DISPARITY DUE TO FAST AND SLOW VARIABLES:
TOOL-NARAYANASWAMY EQUATION


We have shown that for a given τ_obs, we can partition
ξ into two distinct groups: one containing internal variable
ξ_E whose affinity has vanished and the other one,
which we now denote by ξ_N that has not equilibrated
and has a nonzero affinity 𝐀. It is the active internal variables.
As ξ_E has equilibrated, its temperature, pressure,
etc. must be those of the medium, that is, T_0,P_0, etc. It is the
inactive internal variable. On the other hand, the temperature, pressure, etc.
associated with different components of ξ_N must not
be those of the medium as there will be nonzero thermodynamic forces to bring
each to equilibrium in due course. This raises a very interesting question.
Because we are dealing with an IEQ state of the system, there is a
well-defined and unique thermodynamic definition of its temperature
T(t)≐∂ E(t)/∂ S(t). This temperature also satisfies the
identity dQ(t)=T(t)dS(t). How does T(t) relate to  temperatures of
ξ_E and ξ_N? To make some
progress, we assume ξ_E and ξ_N to be quasi-independent over τ_obs. There is a
strong experimental evidence for this <cit.>.
However, there are observables in 𝐗 that also participate in
relaxation. For example, V will relax if P≠ P_0. Similarly, E will
relax if T≠ T_0. As we have discussed earlier <cit.>, one can
treat E,V, etc. in 𝐗 as internal variables with their affinities
1/T-1/T_0,P/T-P_0/T_0, etc. that vanish once equilibrium is reached.
This is also seen from Eq. (<ref>), where the first two
terms have the same form as the last term involving dξ;
recall that dS_0/dt=d_iS/dt. Therefore, in this section, we will
continue to include ξ_0=𝐗 in ξ as
we had done in Sec. <ref>. This should not cause any confusion.
We only have to be careful to always include ξ
_0=𝐗 to specify the system even when τ_obs>
τ_eq=τ_0.



 §.§ A Black Box Model


We consider a simple NEQ laboratory problem to model the above situation.
Consider a system as a “black box”
 consisting of two 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 or ξ
_E and ξ_N, and the insulation allows us
to treat them as independent, having different temperatures. We assume that
there are no irreversible processes that go on within each part so that there
is no irreversible heat d_iQ_1 and d_iQ_2 generated
within each part. We wish to identify the temperature of the system, the black
box. To do so, we imagine that each part is added a certain
infinitesimal amount of heat from outside, which we denote by
dQ_1=d_eQ_1 and dQ_2=d_eQ_2. We assume the
entropy changes to be dS_1 and dS_2. Then, we have for the net heat
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; see Eq. (<ref>). Using dQ_1
=T_1dS_1,dQ_2=T_2dS_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

    1/T=x/T_1+1-x/T_2.

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. Thus, we see from this heuristic model
calculation that the thermodynamic temperature T of the system is not the
same as the temperature of either parts, a common property of a system not in equilibrium.

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.



 §.§ Tool-Narayanaswamy Equation


We turn to the general case of relaxation of thermodynamic properties. At high
enough temperatures, the time variation of T(t) as it relaxes towards
T_0 can be described as a single simple exponential with a characteristic
time scale τ_eq. This happens when all internal variables have
come to equilibrium during τ_obs> τ_eq so no
internal variables besides ξ_0 are needed, a case discussed
by Landau and Lifshitz <cit.> and by Wilks <cit.>.

At low temperatures, this is not true. There are quasi-independent slow
and fast internal variables ξ_N and ξ_E that are well known in glasses and supercooled liquids
<cit.>. The situation is similar to the black
box considered above. Both parts will strive to come to equilibrium with the
medium but they have widely separated relaxation times. As time goes on during
relaxation, some of the groups in {ξ_n
} introduced in Sec. <ref> becomes part of
ξ_E after equilibration as we have discussed there.
We first assume, for simplicity, that all active internal variables in
ξ_N have the same relaxation time τ_1, i.e.,
they equilibrate together but have not equilibrated. The quasi-independence of
ξ_N and ξ_E immediately
leads to the following partition of the S,E,V and ξ into
two contributions, one from each kind:

    𝐙(t)=𝐙_E(t)+𝐙_N(t).

For example, quasi-independence gives the additivity S(t)=S_E
(t)+S_N(t), where S_E(t) and S_N(t) stand
for S_E(E_E(t),V_E(t),ξ_E
(t)) and S_N(E_N(t),V_N(t),ξ_N(t)), etc. Here, we have introduced V_E(t) as the
volume difference V-V_f(  t) in terms of the free
volume V_f(  t) in the cell model in which
V_f(  t) allows for the molecules to move long
distances (liquid-like slow motion) over τ_obs
<cit.>. Thus, V_E corresponds to the fast
center of mass solid-like motion within the cells, which are in equilibrium
with the medium; see also Zallen <cit.>.

Let us now introduce the "energy fraction" x(t) as

    x(t)≡ dE_N(t)/dE(t), 1-x(t)≡ dE_E
    (t)/dE(t),

at a given t, so that

    ∂ S_N(t)/∂ E(t)      =x(t)∂ S_N
    (t)/∂ E_N(t), 
    ∂ S_E(t)/∂ E(t)      =[1-x(t)]∂ S_E
    (t)/∂ E_E(t).



By definition, we have ∂ S_E(t)/∂ E_E
(t)=1/T_0, while ξ_N will have a temperature
different from this. Assuming internal equilibrium, we can introduce a new
temperature T_N(t) by

    ∂ S_N(t)/∂ E_N(t)=1/T_N(t).

The following identity

    1/T(t)=1-x(t)/T_0+x(t)/T_N(t)

easily follows from considering ∂ S(t)/∂ E(t) and using Eq.
(<ref>) for S(t) and Eq. (<ref>). This equation
should be compared with (<ref>) obtained above using a black box model
and is identical to the Tool-Narayanaswamy equation <cit.> in
form, except that we have given thermodynamic definitions of x(t) in
(<ref>) and T_N(t) in Eq. (<ref>).

It is easy to extend the above calculation to the case of different groups
{ξ_n} belonging to ξ_N. The quasi-independence gives

    𝐙(t)=𝐙_E(t)+
    
    
    ∑_n
    𝐙_n(t),

so that S(t)=S_E(t)+


∑_n
S_n(t) with S_E(E_E(t),V_E(t),ξ_E(t)) and S_n(E_n(t),V_n(t),ξ_n̸(t)) as discussed above. For each S_n, we have its own temperature
T_n using. It is now easy to see that Eq.
(<ref>) is extended to

    1/T(t)=1-x(t)/T_0+∑_nx_n(t)/T_n(t),

with x_n(t)≡ dE_n(t)/dE(t) and 1-x(t)=∑_nx_n(t).

Let us now understand the significance of the above analysis. The partition in
Eqs. (<ref>) and (<ref>) along with the
fractions x(t) and x_n(t) shows that the partition satisfies a lever
rule: the relaxing glass can be conceptually (but not physically)
thought of as a "mixture" consisting of different "parts" corresponding to
different temperatures and fractions. However, one of the temperatures is
T_0 of the medium, while T_n(t)'s denote the temperature of the parts
that are not equilibrated yet. As some of these parts equilibrate, their
temperature become T_0 and they add to the weight 1-x(t) for the
equilibrated internal variables. Thus, we see that while ξ_E(t) may play no role in the IEQ thermodynamics, it still plays an
important role in relating the thermodynamic temperature T(t) with those of
various groups of ξ(t). Thinking of a system conceptually as
a "mixture" of "parts" is quite common in theoretical physics. One common
example is that of a superfluid, which can be thought of as a "mixture" of a
normal viscous "component" and a superfluid "component". In reality, there
exist two simultaneous motions, one of which is "normal" and the other one is
"superfluid". A similar division can also be carried out in a superconductor:
the total current is a sum of a "normal current" and a "superconducting current".

Such an analysis has been carried out in detail earlier <cit.>,
where a connection is made with the notion of the "fictive" temperature
<cit.> but in the absence of any internal variables (besides
ξ_0). Here, we will summarize that discussion and refer the
reader to this work for missing details. It is easy to first consider the
simple case in Eq. (<ref>). One can consider the
part ξ_N of the energy fraction x(t) at
T_N to represent a "fictitious" SCL at temperature T_N.
It is fictitious since the entire system does not consist of this part so it
is not in equilibrium as SCL is supposed to be; it is missing the part
corresponding to the fraction 1-x(t). We can supplement mentally the
fictitous SCL by the same SCL of fraction x(t) at the same temperature
T_N to ensure that the entire system consists of ξ_N at T_N. This now represents an IEQ state at
T_N, the left side of Eq. (<ref>).
Thus, T_N represents the thermodynamic temperature of this IEQ
state, which can then be treated as an "unequilibrated" SCL, in thermal
equilibrium with a medium at T_N (but not at T_0). We have
identified it as an "unequilibrated" SCL since there is no reason for
𝐀_N corresponding to ξ_N to
vanish in this SCL, whereas it is required to vanish in equilibrium. This
SCL at T_N is also not identical to the glass as the latter has
ξ_E(t) at T_0, which is absent in this SCL. We
can thus justify T̸  _N as the fictive temperature.

This picture can be extended to Eq. (<ref>
) by introducing T_N as follows:

    x(t)/T_N=∑_nx_n(t)/T_n(t),

which converts it to Eq. (<ref>). We can then
introduce an equilibrated SCL, in equilibrium with a medium at T_N
so that we can treat T_N as the fictive temperature.

Instead of considering a derivative of S with E, we can consider
derivatives with respect to other state variables such as V. In that case, a
similar analysis can be carried out as done in <cit.> to obtain a
similar looking Tool-Narayanaswamy equation for P(t)/T(t). We leave it to
the reader to carry out this simple extension. The result for P=P_0 is
given in <cit.>.



§ DISCUSSIONS AND CONCLUSIONS




 §.§ Consequence of the Relaxation Hierarchy


We have presented a hierarchical classification of relaxation times in
increasing order in Eq. (<ref>), which allows us to
determine a unique temporal window Δ t_n in Eq.
(<ref>) for a given τ_obs as shown by
the two neighboring relaxation curves around red the horizontal line at the
temperature T_0 of interest in Fig. <ref>. The discussion
is valid for any relaxing system with complex relaxation and is not restricted
to only SCL/glass undergoing vitrification. The temporal window is not fixed
as the state of the system changes so it must be adjusted appropriately; see
Fig. <ref>. Let us consider vitrification considered in Sec.
<ref>. Above T_0=T_0g, the system is always in
equilibrium (recall that we have used SCL as the equilibrium state) so
τ_obs≥τ_0; see Eq. (<ref>).
There are no active internal variables. Therefore, the system's temperature
T=T_0. Slightly below T_0g but above T_01, Eq.
(<ref>) is satisfied so ξ
_0=𝐗 is active, but all ξ_k,k≥1 are inactive
so they need not be considered for a thermodynamic description. There are two
different contributions that affect the temporal window that needs to be considered:



  (i) Cooling effect-As we lower T_0 from its previous value
T_0^'=T_0g (τ_obs=τ_0), the system's
initial temperature is T(0)=T_0^'. As the system's temperature
determines τ_1, it has the previous value τ_1^' at
T_0^' initially so it lies below the curve τ_1 at T_0.
But the value of τ_0 at T_0 is determined by the new temperature
T_0 so it increases compared to τ_0^'=τ_0(T_0^'). Consequently, we have τ_obs<τ_0 to satisfy Eq.
(<ref>).


  (ii) Relaxation effect- During isothermal relaxation at the new
temperature, T(t) decreases towards T_0, which increases τ_1 from
τ_1^'=τ_1(T_0^') to τ_1(T_0). This
shrinks the window Δ t_0 in Eq. (<ref>)
in width to the width shown in Fig. <ref> at T_0.


The discussion can be now applied to the sequence of cooling steps to T_0
between T_01 and T_02, between T_02 and T_03, etc. where we
are confronted with the new successive windows Δ t_1,Δ
t_2,⋯. In each window, we need to consider newer internal variables
ξ_1,ξ_2,⋯ so that in a window Δ
t_n, we need to consider 𝐙_n consisting of 𝐗 and ξ^(n) as discussed in Sec.
<ref>. We thus conclude that the dimension of the state space
continues to grow during cooling until all internal variables (presumably
leaving out ξ_v that refers to local vibrations as
noted earlier) become active. Thus, in the glass transition region between
T_0G and T_0g, the irreversibility continues to grow
until all internal variables become active.



 §.§ Residual Entropy


As discussed above, we cannot just consider a fixed, small number of internal
variables (their number keeps changing in the transition region) if we want to
go to some small enough temperatures T_0<T_0G and be able to
describe the cooling process thermodynamically. The best we can do is to
determine a large enough numbers of the internal variables that become active
in the transition region. This requires a deeper understanding of the
structure of glasses and identify these internal variables, which seems to be
an impossible task at present. In our view, this remains an unsolved problem
at present. Despite this, the inequalities in Eqs.
(<ref>), (<ref>) and
(<ref>) remain valid for any choice of 𝐙.

As these inequalities are very important, we summarize them for the benefit of
the reader. According to Eq. (<ref>), the residual
entropy S_R cannot be less than the experimentally measured or
extrapolated S_expt(0) at absolute zero; the latter itself cannot
be less than the entropy of the supercooled liquid at absolute zero. As we
have assumed S_SCL(0)=0, we claim the strict inequality

    S_R>0.

Indeed, the strict inequality between S_expt(0) and S_SCL
(0) holds at all positive temperatures T_0<T_0g as derived in
Eq. (<ref>).

We have not discussed the statistical formulation of the residual entropy,
which has been discussed by us in <cit.> and
<cit.>. The derivation does not require the use
of the second law or entropy maximization. Therefore, it applies to any
nonequilibrium state and is purely combinatorial in nature. For the sake of
completeness, we summarize the result. Let Γ_λ,λ
=1,2,⋯,𝒞 denote the number of disjoint components in the state
space, and let p_k_λ denote the probability of a microstate
k_λ in Γ_λ. The entropy S=-


∑_λ
∑_k_λp_k_λln p_k_λ, ∑_λ
∑_k_λp_k_λ=1, can be written as a sum of two parts:

    S=
    
    
    ∑_λ
    p_λS_λ+S_𝒞,

where p_λ≐∑_k_λp_k_λ is the probability
of the component Γ_λ, and S_λ=-∑_k_λ
p_k_λlnp_k_λ,p
_k_λ≐ p_k_λ/p_λ, is the entropy of the
component, and

    S_𝒞≐-
    
    
    ∑_λ
    p_λln p_λ

is the component confinement entropy. The residual entropy is the component
confinement entropy S_𝒞 at absolute zero, with p_λ=p_λ0 denoting the probability of the component Γ_λ
at absolute zero. We have not imposed any equally probable assumption
in the above derivation so the result is very general. However, to apply IEQ
thermodynamics, we need to impose equally probable assumption.



 §.§ Fate of the Entropy Loss Conjecture


The isothermal relaxation considered in Sec. <ref> shows that
both S and H decrease with time, which is consistent with our intuitive
picture given at the start of that section for S, and experimental evidence
for H. As we have shown, the behavior is a consequence of the second law.
The entropy loss view (ELV) mentioned after Theorem
<ref> and proposed in
<cit.> results in the conclusion that
contradict our results. In particular, the view suggests that during
relaxation, the entropy increase since S(T_0,t)≤ S_SCL(T_0);
see Eq. (<ref>). As Goldstein <cit.> has shown, this
is a violation of the second law. These authors agree that in their view of
the glass transition, the glasses do violate the second law, while others
<cit.> argue
in favor of the second law. For most scientists, the fact that the entropy
loss view violates the second law should be a strong indication that the view
is unrealistic. But the debate persists as is evident from some recent reviews
<cit.>.

Here, we hope to settle the debate by pointing out a hitherto unrecognized
internal inconsistency of the ELV, assuming its premise that the
glasses do violate the second law. In other words, the second law is not the
absolute truth of Nature. This means that all the inequalities in Eq.
(<ref>) must be reversed for the view to
hold. Since dS>0 in ELV during relaxation, it follows from the reverse
inequality in Eq. (<ref>) that
T(t)>T_0, which is the same as Eq. (<ref>). From the
reverse inequality in Eq. (<ref>), we
conclude that

    T_0-T(t)]dH(t)<0.

If we demand that the ELV follow the experimental evidence (dH(t)/dt<0), we
must conclude that T_0>T(t), which contradicts the previous conclusion and
the ELV becomes internally inconsistent. If, however, we accept the previous
conclusion T(t)>T_0 to ensure that the ELV remain internally consistent,
then dH(t)/dt>0 in contradiction with experimental evidence. Thus, the mere
fact that the ELV satisfies the experimental evidence (dH(t)/dt<0) does not
mean that it is internally consistent in the entropy loss view. In other
words, demanding that the ELV is consistent with experiments disproves the ELV
conjecture. Even though we have considered the entropy loss view at different
times
<cit.>
, we believe that the conclusion drawn above is the most direct demonstration
of the internal inconsistency of the ELV, despite the fact that we have
allowed it to contradict the second law.



 §.§ Significance of Inactive Internal variables


Even though the IEQ thermodynamics only involves the active internal
variables, it is clear from Sec. <ref> that even inactive internal
variables such as τ_r indirectly affect the thermodynamics
through the determination of the temperature, pressure, etc. of the system. In
retrospect, this is not so surprising once we recognize that the temperature
of the system is a thermodynamic quantity. However, the division of the
internal variables into active and inactive parts mean that the temperature of
the system must be different from the temperature T_0 during isothermal relaxation.


I wish to acknowledge useful comments from Gyan P. Johari and Sergei V.
Nemilov over the years.


99                                                                                               


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