# Nature of the order parameter of glass

Koun Shirai1,2 1Vietnam Japan University, Vietnam National University, Hanoi,
Luu Huu Phuoc Road, My Dinh 1 Ward, Nam Tu Liem District, Hanoi, Vietnam
2SANKEN, Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

###### Abstract

In glass physics, order parameters have long been used in the thermodynamic
description of glasses, but the nature is not yet clear. The difficulty is how
to find order in disordered systems. The conventional treatments of glass
cause serious internal inconsistencies in thermodynamics. The issue turns out
to be ascribed to a fundamental problem of defining state variables in
thermodynamics. This study provides a coherent understanding of the nature of
order parameters for glasses and crystals, starting from the fundamental issue
of defining state variables in thermodynamics. The state variable is defined
as the time-averaged value of a dynamical variable under the constraint
conditions, when equilibrium is established. It gives the same value at any
time it is measured as far as the equilibrium state is maintained. This
constancy is here called time invariance. From this definition, it is deduced
that the state variables of a solid are the time-averaged positions
$\\{\bar{\bf R}_{j}\\}$ of all atoms constituting the solid, regardless of
periodicity. The general properties of order parameters are fully compatible
with the time invariance of state variables. For glass, $\\{\bar{\bf
R}_{j}\\}$ are state variables and order parameters too. Because $\\{\bar{\bf
R}_{j}\\}$ are order parameters, any property determined by $\\{\bar{\bf
R}_{j}\\}$ can also be an order parameter. This explains why many quantities
can be used for order parameters. Symmetry introduces redundancy among the
averaged positions of the atoms, reducing the number of independent state
variables. The often-claimed association of order to periodicity lacks
generality as shown by abundant examples of well-ordered material, such as
DNA. Order in thermodynamics means the time invariance.

##  I Introduction

###  I.1 Historical background

In glass physics, since Prigogine and Defay introduced the idea Berthier and
Biroli (2011); Berthier and Ediger (2016); Prigogine and Defay (1954); Davies
and Jones (1953a, b); Nemilov (1995), order parameters have been indispensable
for theoretical studies in the glass transition. Despite their importance, the
nature of order parameters of glass has not been clarified. For crystals, the
meaning of the order parameter is intuitively clear. It is a quantity
indicating a long-range ordered state, such as magnetization for ferromagnetic
materials Blundell (2001); Sethna (2017). Landau originally introduced the
notion to indicate the emergency of symmetry in phase transitions (Landau and
Lifshitz (1976), chap. 14). From this original meaning, it is no sense to
speak of order for disordered systems. Thus physicists have attempted to find
physical meaning in the order parameter of glass. Naturally, several ways have
been proposed to define the order parameter. They extend from geometrical
parameters, such as four-point correlation function Berthier and Biroli
(2011), polyhedron order Xia et al. (2015), and orientational order Yang et
al. (2019), to more abstract ones, such as the overlap function or caging
order parameter Parisi (1983); Franz and Parisi (1997); Charbonneau et al.
(2014), which are derivatives from Edwards-Anderson order parameter in spin
glass not (a). In the following, we summarize a general (but not necessarily
universal) understanding of the order parameter of glass. In this paper, the
order parameter (OP) is denoted by $z$.

#### Kinetic parameter.

Since a chemical reaction is a process to approach an equilibrium state, $z$
is considered a kinetic parameter called advancement indicating the degree to
which a chemical reaction proceeds Prigogine and Defay (1954). Since the glass
transition is the process of approaching the glass state, this analogy makes
sense. In general, $z$ is not considered a state variable in a chemical
reaction because a reaction is a nonequilibrium process. However, by
introducing affinity $A$, the kinetic parameter $z$ is coped in the standard
formulation of thermodynamics, as

$dG=-SdT+Vdp-Adz,$ (1)

where $G$ is Gibbs free energy, $S$ is entropy, $p$ is pressure, and $V$ is
volume. Affinity is the driving force for a reaction and it is a function of
state variables $T$ and $p$ plus $z$, as $A=A(T,p,z)$. With appropriate
scaling, $z$ can vary from 1 to 0, with 0 corresponding to equilibrium. As the
system approaches equilibrium, $A$ vanishes. Hence, no kinetic parameters
remain at equilibrium, as should be. However, equilibrium is never established
in the glass transition because of its large and slow retardation of atom
rearrangement. Structural changes end at a finite value $z_{0}\neq 0$ below
the transition temperature $T_{g}$. This glass state is called the frozen
state. The value $z_{0}$ is subject to the energy-minimum condition

$\left(\frac{\partial G}{\partial z}\right)_{T,p}=0,$ (2)

which, in turn, gives the dependence of $z_{0}$ on $T$ and $p$, as
$z_{0}=z_{0}(T,p)$. Accordingly, $z_{0}$ of the frozen state is not an
independent variable. Although it is originally a kinetic parameter, it
becomes a state variable in the frozen state Prigogine and Kondepudi (1999);
Nemilov (1995). This is not a contradiction. The difference in the character
of the state and dynamical variables of the frozen state is tuned based on
local equilibrium Ansatz Prigogine and Kondepudi (1999). The problem in the
case of glass is that it is difficult to give a specific interpretation to
“the degree of advancement” for the glass transition. This is in contrast to
the case of chemical reactions, where $z$ has a clear meaning of a change in
the components of molecular species. The difficulty increases when one
recognizes that there are multiple OPs (see Sec. III.1).

#### Structural parameter.

At this point, we have to declare the scope of this study. The main interest
of this study is not the process but the material property. Therefore, we
restrict our argument to OP as material property. This only focuses on the
state variable $z_{0}$ of the frozen state. Liquid states are structureless.
Instead, they have a large number of atom configurations. This freedom in
configuration is reduced at the glass transition. Hence, it is natural to
consider that this reduction in configurational freedom indicates the
occurrence of structural fractions. By observing the short-range order of the
radial distribution function (RDF), this short-range order deserves a glass OP
Davies and Jones (1953a); Elliott (1990). However, the differences observed in
RDF between glass and liquid are not sharp. On account of the drastic change
in atom mobility at the glass transition, it is unsuitable to characterize the
glass transition with such a dull quantity. The obvious difference between
glass and liquid is atom mobility: atoms are mobile in liquids but not in
solids. This aspect can be presented by time-correlation function. Thus, the
dynamic density-density correlation functions become essential tools for
studying the glass transition Radons et al. (2005). On the other hand, dynamic
correlation functions do not provide more accurate structural information than
RDF does. They focus on the time dependence, $z(t)$, but do not resolve the
detailed dependence on $z_{0}$ of various frozen states.

The difficulty is that the same kind of glass exhibits different properties
depending on the thermal history, although their atomic arrangements look
similar Berthier et al. (2019a). For this reason, glass manufacturers
particularly focus on preparation conditions. RDF is not sensitive to the
structural details of the frozen state. Beyond RDF, no unique way exists to
define “higher-order” OPs. With this difficulty, researchers sometimes refer
to it as hidden order Binder and Young (1986); Wang et al. (2014); Xia et al.
(2015); Tong and Tanaka (2018); Berthier et al. (2019b); Tanaka (2022), a term
that gives a somewhat mysterious tone to the glass OP. The problem is
essentially ascribed to how the glass structure is identified. Consider two
samples of the same glass. Despite the same chemical composition, generally
they have different atomic arrangements. The question is whether the two
samples have the same OPs. Even it may be unclear what the “same” glass means
Shi .

#### Hysteresis problem.

The last issue raises the entangling problem between the properties of glass
and the process. It is known that the current properties of a glass are
affected by its history, including the cooling rate at which it was obtained.
This behavior is known as hysteresis. Previously, we said that the glass OP,
$z_{0}$, is a state variable, and $z_{0}$ is controlled only by $T$ and $p$.
We have learned that the current properties of glass are not uniquely
determined by current $T$ and $p$ alone. It is really a contradiction. When
hysteresis appears in a material property, the material is generally
considered to be in a nonequilibrium state. Describing hysteresis
thermodynamically in terms of state variables has long been a conundrum since
it was first addressed by Bridgman Bridgman (1950). The thermodynamic
description of hysteresis by adding internal variables to usual state
variables has been actively studied Coleman and Curtin (1967); Rice (1971);
Muschik (1993); Maugin (1993); Maugin and Muschik (1994); Restuccia and
Kluitenberg (1988). (We use internal variables and OP interchangeably here.)
Although this approach may succeed to some extent, the nature of the internal
variable also is not known. Because of this difficulty, some authors argue
that “internal variables can eventually be measured by a gifted
experimentalist but cannot be controlled” (in Ref. Maugin (1993), p. 224).

The serious problem in the conventional view that hysteresis is a
nonequilibrium property is that it leads to destructive consequences for
thermodynamics Kestine and Rice (1970); Kestin (1992). Close observations
reveal that hysteresis is a property that any solid has to some degree Shi .
The mechanical properties of metals depend on their thermal treatment history.
For materials used for memory, the current state is determined by a field
previously applied. The electrical properties of semiconductors are affected
by the preparation conditions due to the presence of intrinsic defects and
dislocations. Viewing glass as nonequilibrium state merely because of
hysteresis leads to the unpleasant conclusion that all solids are in a
nonequilibrium state. No one has solved this difficulty until now.

#### Central contention.

In the last argument, we noticed that the OP problem of glass is part of a
general problem that seriously affects our understanding of the thermodynamics
of solids. By focusing on the arguments from (a) to (c), we can figure out
that the serious difficulty (all solids turn to be nonequilibrium) is based on
the understanding that the equilibrium properties of a solid are a function of
$T$ and $p$ (or $V$) only. This is true in the gaseous state, but there is no
rigorous proof for solids. The problem eventually goes back to the fundamental
problem what an equilibrium is for solids Shi . Without revisiting the
fundamental problem of thermodynamics, the elusive notion of glass OP cannot
be understood.

Another aspect of the OP problem in glass is the notion of disorder in
thermodynamics. In solid state physics, order means periodicity. However,
thermodynamics is a closed theory and does not need the help of other
disciplines. Therefore, the theory should not depend on the details of the
internal structure of the system. The term disorder in thermodynamics is most
often used to describe entropy. Traditionally, entropy has been interpreted as
“the degree of disorder”. However, a more appropriate interpretation is
missing information Ben-Naim (2008) (see Sec. III.2). From this perspective,
the meaning of order for glass is investigated here.

###  I.2 Purpose and strategy

This paper presents a coherent view of glass OP from fundamental laws of
thermodynamics. The glass problem is not treated as an exception, but as part
of a general theory. The conclusion is that the OP of glass as well as other
solids are equivalent to state variables in thermodynamics. The state
variables in a solid are the equilibrium positions of atoms in the solid. To
draw this conclusion, we begin our argument with the definition of equilibrium
(Sec. II.1). Next, we explain how the state variables are defined and show the
results for solids (Sec. II.2). Based on this, we find equivalence between the
OP and the state variables. The arguments in Secs. II.1 and II.2 were
presented in an early work Shi  but are included here because of completeness
of theory. Finally, we discuss the key aspects of OP in the glass issue (Sec.
III).

#### Scope.

Before proceeding to the main theory, we must set a limit because the topics
are too broad to treat in a single paper. The question is, what a
thermodynamic approach means for those phenomena which are traditionally
considered nonequilibrium?

Figure 1: Specific heat $C_{p}$ of glass at the glass transition. The
transition region is defined by the region where $C_{p}$ rapidly changes. The
glass transition temperature $T_{g}$ is roughly given by the middle point of
the region. Solid curves indicate the actual cooling or heating processes.
Dashed curves indicate the equilibrium curves obtained from the nonequilibrium
states in adiabatic relaxation processes.

Since a process represents changes in state over time, the state in the
process must, by definition, be nonequilibrium. At this point, there is no
point in discussing it. Figure 1 shows the specific heat $C_{p}$ of glass
around the glass transition temperature $T_{g}$ in the cooling or heating
processes. This exhibits hysteresis between the cooling and heating processes.
When glass is cooled from liquid state $A$ at a constant rate, a $C_{p}-T$
curve like a solid line $a$ is obtained. The states on this line are
nonequilibrium because they have time dependence. Now, let us stop the
temperature scan at $T$ (gray squares) and quickly isolate the sample from the
heat bath. The glass will then undergo an adiabatic relaxation and eventually
reach a state that is no longer to change. This static state (open circles) is
called equilibrium in the present context. The important question here is
whether this is an equilibrium state. The definition of equilibrium is the
subject of this paper, which is discussed in Sec. II.1. Repeating this
relaxation process at various temperatures on the original nonequilibrium
curve $a$ yields a continuous equilibrium curve $b$ (dashed lines).

The thermal path $b$ thus obtained is called a quasi-static process in the
thermodynamics context. This study focuses on the static states on this quasi-
static path. In usual sense of thermodynamics, all states in a quasi-static
process must be in equilibrium. The big question is how they are
thermodynamically described. The question is answered in Sec. II.2. By
restricting the OP of a glass to those static states only, we can construct
theory in a clear manner. Any thermodynamic function, such as entropy, can be
obtained by integrating $C_{p}$ along the path $b$. Note that this restriction
on the thermal path does not necessarily mean that the final state $B$ of the
quasi-static cooling process is unique. Many different quasi-static paths
$b^{\prime}$ starting from the same state $A$ exist, ending at different
equilibrium states $B^{\prime}$.

##  II Theory

###  II.1 Thermodynamic equilibrium

Classical thermodynamics consists of four laws, from the zeroth to the third
Law. The zeroth law describes a property of thermodynamic equilibrium. It is a
transitive property: if the systems $A$ and $B$, and independently $A$ and
$C$, are in equilibrium, the systems $B$ and $C$ will also be in equilibrium.
Bringing a block of 50∘C copper into contact with a block of 50∘C aluminum
does not cause any change. Glass is no exception. No changes occur when a
piece of glass at 50∘C and a block of aluminum at 50∘C are brought into
contact. Although the transitive property is a property of equilibrium, it is
not a sufficient condition for equilibrium. Further conditions are required
that the other state variables also remain unchanged. The zeroth law provides
the definition of the empirical temperature as a universal measure of
equilibrium but does not provide the definition of equilibrium.

One may define equilibrium as the state that all the state variables do not
change. This definition naturally raises another question: which are state
variables? Obviously, temperature and pressure are the answer for gases.
However, are there any other variables independent of $T$ and $p$? (In this
study, chemical reactions are not treated, and hence mole fraction is
excluded.) This question is by no means trivial for solids. Callen posed this
question, in his textbook, when the definition of equilibrium is discussed
(Callen (1985), Sec. 1.5). He noticed that, for many solids, the current
properties are affected by the past history. Plastically deformed solids
cannot be described by $T$ and $V$ solely Berdichevsky (2006). Callen finally
failed to answer the question whether the current properties can be specified
by adding appropriate state variables. Nobody has succeeded. The difficulty is
that the word state variable is already used to define equilibrium, but the
state variable presupposes equilibrium. The argument falls into an egg and
chicken problem. This dilemma has been recently solved by Gyftopoulos and
Berreta Gyftopoulos and Beretta (2005). They use only the second law for
defining equilibrium. No law is more fundamental than the second law, so that
the derived definition is quite general. From the second law, we know the
property of equilibrium as

###### Definition 1

(Equilibrium) it is impossible to extract work from a system in equilibrium to
the outside without leaving any effects on the environment. (Gyftopoulos and
Beretta (2005), p. 58)

We can now proceed in the reverse order and use this as the definition of
equilibrium. No use is made of the word state variable, and thus no self-
contradiction occurs. Definition 1 is consistent with our understanding for
the equilibrium character of existing solids. At this point, glasses are no
exception. Glasses must be in equilibrium, because work cannot be extracted
from them.

Definition 1 says that work can be obtained only when a system undergoes
spontaneous change. In the glass literature, it is often claimed that glass
changes slowly toward equilibrium. Since this is a spontaneous change, readers
might argue that work can be obtained in that period. However, in the real
world, there is no eternal equilibrium. Such a sluggish change commonly
occurs. The state of the gas in a gas cylinder is equilibrium. However, even a
robust metal wall is subject to gas leakage, and the gas will eventually
escape from the cylinder. A mixture of nitrogen and hydrogen molecules is
chemically inactive at normal temperatures. However, the mixture will react to
produce ammonia gas in the age of Jupiter: the free energy of the latter is
lower than the former. A metal block is in equilibrium but will be oxidized
for years. Although diamond does not change under normal conditions, an
energetic investigation shows that it will eventually be transformed into
graphite. Considering nuclear reactions into account, only iron should be in
stable equilibrium. These observations indicate that equilibrium must be
argued under certain restrictions. Namely, when changes in property can be
ignored within a certain time, we can argue equilibrium not (b). The
characteristic time is the relaxation time of that property.

###  II.2 Constraints and state variables

What factors determine the relaxation time? This problem was addressed by
Gibbs in his classical book more than a century Gibbs (1906). He investigated
the distinction between equilibrium and static states. Equilibrium is a static
state sustained by two competitive active tendencies that work in opposite
directions. There is also another type of static states. For example, a
mixture of nitrogen and hydrogen gases is stable at normal conditions,
maintaining a static state. However, this static state varies at high
temperatures, so they can react to produce ammonia. Gibbs called it passive
resistance, which prevents them from reacting. Since then, different names
have been used in different areas, such as inhibitor and anti-catalysis
Hatsopoulos and Keenan (1965). In today’s context of thermodynamics, the word
“constraint” is the most familiar; hence, it is employed here. The nature of
constraints in materials was not known at that time. Nowadays, we know that
the substance of constraint is an energy barrier of any kind. In the
introductory course of thermodynamics, the word constraint is used to mean
macroscopic walls like a rigid wall of a container. The wall is visible, and
hence the meaning is clear. In contrast, since the energy barrier in materials
is invisible, the role of the energy barrier in thermodynamics is clouded.
However, by considering the physical reality of the macroscopic wall of a
container, which may be made of metal, we will find that the substance of the
macroscopic wall is another kind of energy barrier built inside the metal. The
height $E_{b}$ of any energy barrier is finite. This determines the relaxation
time $\tau$, as

$\tau=\tau_{0}\exp\left(\frac{E_{b}}{k_{\rm B}T}\right),$ (3)

where $\tau_{0}$ is the inverse of the attempt frequency of the corresponding
motion, and $k_{\rm B}$ is Boltzmann’s constant. A rigid wall is a
mathematical tool to raise $E_{b}$ and $\tau$ to infinity to simplify the
analysis.

###### Definition 2

(Constraint) A constraint $\xi_{j}$ is an object to inhibit the change in
$j$th-type motions of the constituting particles of a system.

A constraint $\xi_{j}$ is substantiated by the energy barrier $E_{b,j}$ for
$j$th-type motions. Since in the real world the relaxation time $\tau_{j}$ is
always finite due to the finite value of barrier height $E_{b,j}$, the $j$th
constraint works only in a timescale shorter than $\tau_{j}$.

A dynamical variable $X(t)$ is subject to a certain constraint $\xi_{j}$. The
constraint $\xi_{j}$ on $j$th-type of motions means the imposition of a range
of permissible changes in $X(t)$.

###### Definition 3

(State variable) If the time average $\overline{X(t)}$ has a definite value
when the system is in equilibrium, the state variable of $j$th type, $X_{j}$,
is defined by

$X_{j}=\frac{1}{t_{0}}\int_{\xi_{j}}X(t)dt,$ (4)

where $t_{0}$ is the period of time average.

By the definite value, we mean that the time-averaged value is independent of
the time period $t_{0}$, in which the time average is taken. For example, for
a gas in a box with volume $V$, the constraint is the wall of the box. The
position, $x(t)$, of a gas molecule is constrained within the wall. In
equilibrium, the molecule visits everywhere in the box with equal probability.
The time average of the molecule’s position, $\overline{x(t)}$, is
indeterminate. Thus, $\overline{x}$ is not a state variable. Now consider the
spatial distribution function, $\delta(x^{\prime},x(t))$, of a molecule at
$x^{\prime}$. The integration of this function over the inside of the box

$X(t)=\int_{V}\delta(x^{\prime},x(t))dx^{\prime},$ (5)

is interpreted as the density-weighted volume. When the distribution is
uniform in space, which is obtained in equilibrium, the time-averaged value
$\overline{X(t)}$ gives the volume $V$. The time-averaged value
$\overline{X(t)}$ turns to a state variable $V$. The non-vanishing property
after time average is the most important property of state variables, and here
it is called the time invariance against time averaging: it is different from
the usual one, i.e., the constant in the equation of motion. Or we may call it
the invariance against noise. This invariance is exploited to identify state
variables in the following.

A thermodynamic system consists of many constraints $\\{\xi_{j}\\}$ (Ref.
Gyftopoulos and Beretta (2005), p. 13). Indeed, the substances that
characterize the structure of a system are constraints. The wall of a
container determines the shape of the gas in the container. At a fixed
temperature, only one independent state variable $V$ exists. When a mobile
wall is inserted into this container, the structure of the whole system is
specified by two volumes $V_{1}$ and $V_{2}$, separated by the internal wall.
The state variables are these two volumes $V_{1}$ and $V_{2}$. This
partitioning can be continued to any number of compartments. We find that one
new state variable is created each time a constraint is inserted. This one-to-
one correspondence between constraint and state variable was clarified by
Reiss Reiss (1996). The structure of a system is fully specified by a set of
constraints $\\{\xi_{j}\\}$.

The heuristic finding of this one-to-one correspondence between constraint and
state variable is proven rigorously from the second law of thermodynamics. The
second law of thermodynamics is exceptional in physics laws in that it has
many different ways of expression, namely, 21 different ways $\breve{C}$ápek
and Sheehan (2005). The most recent one, given by Gyftopoulos and Berreta,
reads as follows:

The second law of thermodynamics

There is one and only one equilibrium state for a set of constraints
$\\{\xi_{j}\\}$ and a given $U$. (Ref. Gyftopoulos and Beretta (2005), p. 63)

This guarantees that the state variable $X_{j}$ is uniquely determined for a
constraint $\xi_{j}$ (uniqueness). As stated in Definition 3, the uniqueness
is the definiteness that the state variables must satisfy. For example, the
time averaged positions of atoms in a gas do not have a unique value; hence,
they cannot be state variables. Furthermore, this expression of the second law
also claims that equilibrium is uniquely specified by a set of constraints
$\\{\xi_{j}\\}$ (completeness).

By defining state variables rigorously, we find that the equilibrium positions
of atoms in solids are state variables. An atom $j$ in a solid is so
constrained that it can move only within its unit cells $\xi_{j}$. It has a
unique value $\bar{\bf R}_{j}$ in equilibrium. Therefore, we conclude

###### Corollary 1

(State variables of solids) State variables of a solid are the time-averaged
positions, $\bar{\bf R}_{j}$, of all atoms that comprise the solid.

The full set $\\{\bar{\bf R}_{j}\\}$ of a solid uniquely determines the
structure of a solid. This means that completeness is satisfied by this set.
The internal energy of a solid is expressed as a function of $\\{\bar{\bf
R}_{j}\\}$ in addition to a trivial variable $T$,

$U=U(T,V,\\{\bar{\bf R}_{j}\\}).$ (6)

Besides $T$, there are $3N_{\rm at}$ state variables in a solid, where $N_{\rm
at}$ is the number of atoms in the solid. This is the most contrasting
thermodynamic property of solids compared to that of gases, in which only
volume $V$ is a state variable. Up to here, no assumption has been made about
the periodicity or symmetry of solids. Hence, Corollary 1 also applies to
glasses.

The functional relation of Eq. (6) is taken for granted for today’s
microscopic theory of solids, although $\\{\bar{\bf R}_{j}\\}$ were not
acknowledged as state variables. A displacement of only one atom to an
interstitial position modifies the internal energy $U$. Interstitial positions
are also equilibrium positions in the present context, although they are often
regarded as metastable states in materials science. However, a metastable
state is also an equilibrium state within the timescale where the interstitial
atom stays at that position. Therefore, as in Eq. (6), $3N_{\rm at}$
equilibrium positions are independent variables. When a solid is transformed
into a liquid, the atom positions lose their role as state variables because
the time average of atom position becomes indeterminate. A block of ice in a
room is thermodynamically stable in a few minutes, and we can measure its
thermodynamic properties on this time scale. Equilibrium atom positions
$\bar{\bf R}_{j}$ of the ice are state variables. We can regard the ice as an
equilibrium state. However, it will thaw out in a day. The ice state must be
considered as a nonequilibrium state on this timescale. Equilibrium atom
positions $\bar{\bf R}_{j}$ become indeterminate. After melting, atom
positions disappear from the arguments of Eq. (6).

By the arguments made up to here, it is concluded that glasses are in
equilibrium states, similar to other solids. They do not change their
structures within their own relaxation times. Each state of the glass on the
quasi-static path $b$ in Fig. 1 is fully specified by state variables
$\\{\bar{\bf R}_{j}\\}$ through Eq. (6). This conclusion is quite different
from the traditional view of nonequilibrium character of glass. Over a
century, it has been believed that glasses are in nonequilibrium states.
However, it is noted that previous studies cannot be rigorous proof for this:
it is illogical to prove nonequilibrium when the rigorous definition of
equilibrium is not known. Previous studies constructed their theories by
placing the nonequilibrium character of glass at the starting point, while
this character itself was taken for granted. The main reasoning behind this
characterization is that the current properties of a glass cannot be specified
by $T$ and $V$ solely not (c). However, as already pointed out, there is no
proof that only $T$ and $V$ are independent state variables for solids.
Rather, it is shown that $\\{\bar{\bf R}_{j}\\}$ are needed for complete
thermodynamic description of a solid. Now, it is demonstrated that the
properties of glasses are better described by adapting this equilibrium view:
these include a free-energy behavior Shirai (2020, 2021a), the jump behavior
of specific heat at the glass transition Shirai et al. (2022, 2023), and the
large activation energies of the glass transition Shirai (2021b).

Lastly, we comment on two words, “macroscopic” and “few numbers”, when
discussing state variables. It is commonly said that state variables are
macroscopic quantities, which cannot accept atom positions as state variables.
However, how large is macroscopic and how small is microscopic are ambiguous.
Often, we distinguish them on a human scale. This manner is subjective, which
is not suitable for describing physics. Atom positions are microscopic on a
human scale. However, we can observe it, for example, using the X-ray
diffraction method. The diffraction pattern exposed on a film is macroscopic.
If we do not know the mechanism of the X-ray diffraction method, we might
recognize that the crystal structure is characterized by a macroscopic entity.
Also, it is widely believed that the number of state variables must be small,
which cannot accept the Avogadro’s number of variables. However, again, it is
ambiguous how big a large number is and how small a small number is. Infinity
is a large number. However, there are many infinities. For example, the number
of integers is infinity, while it is quite smaller than the number of real
numbers. By definition, an average $\bar{\bf R}$ presumes the existence of a
large number of microscopic states ${\bf R}(t)$.

###  II.3 Order parameters of glass

The immediate consequence of the preceding argument is that OPs are state
variables. An OP is a physical property that does not vanish by time averaging
in equilibrium. It retains a constant value as long as an equilibrium does not
break, namely, the time invariance. This property is fully compatible with
Definition 3. Therefore, from Corollary 1, we conclude the following.

###### Corollary 2

(Order parameters of solids) Order parameters of a solid are state variables
of that solid.

A familiar example of OP is magnetization in magnetic materials. Above the
Curie temperature $T_{c}$, the orientation of local moments ${\bf m}_{j}(t)$
is random, and hence its time average vanishes. Below $T_{c}$, the time
average does not kill the nonzero component of the local moment $\bar{\bf
m}_{j}$. Hence, the time-averaged moment $\bar{\bf m}_{j}$ can be a state
variable.

The total magnetization ${\bf M}$ is considered the OP for the ferromagnetic
case. However, vanishing total moment, ${\bf M}=0$, does not necessarily mean
lack of order. For ferromagnetic crystals, the time-averaged local moments
$\bar{\bf m}_{j}$ are all the same, $\bar{\bf m}$, yielding the nonzero total
magnetization ${\bf M}=\sum_{j}\bar{\bf m}_{j}=N_{\rm at}\bar{\bf m}$. The
number of OPs, $n_{\rm OP}$, is one. For anti-ferromagnetic crystals, there
are two types of sites ($k=1$ and $2$). There is the same number for each
site, $N_{\rm at,k}$. In each type of sites, the time-averaged local moments
$\bar{\bf m}_{j}$ all are the same, $\bar{\bf m}_{k,j}=\bar{\bf m}_{k}$,
yielding the subtotal ${\bf M}_{k}=N_{\rm at,k}\bar{\bf m}_{k}$. The
orientations of ${\bf M}_{k}$ of the two groups are anti-parallel so that the
total moment ${\bf M}$ vanishes. In this case, we can take ${\bf M}_{1}$ or
${\bf M}_{2}$ as the single OP with an additional constraint ${\bf
M}_{2}=-{\bf M}_{1}$. Let this constraint be an additional OP, then $n_{\rm
OP}=2$. We can continue creating different kinds of magnetic order by
increasing the number of magnetic sites $k$ to an arbitrary number. There is
no macroscopic value in the total magnetization ${\bf M}$, but an OP still
exists, and there may be more OPs. Spin glass is a magnetically ordered state
in the limit of an infinite $n_{\rm OP}$. Each local moment has a random
orientation but a non-vanishing averaged value $\bar{\bf m}_{j}$. This is a
contrasting property to paramagnetism, where all the time-average local
moments vanish, i.e., $\bar{\bf m}_{j}=0$. A local moment $\bar{\bf m}_{j}$ is
a microscopic quantity on a human scale but has the time-invariance property,
and therefore is a state variable and also an OP.

Overwhelming magnetism is carried by electron spins, although spin freedoms
are not considered here. If the readers do not want to be bothered with extra
freedoms, they may replace the above examples of magnetism with
ferroelectricity (or antiferroelectricity). This can be done simply by
reinterpreting the local moment $m_{j}$ as the local dipole moment. The dipole
moment is presented by atom displacement $\bar{\bf u}$, as $\bar{\bf
m}_{j}=q_{j}\bar{\bf u}_{j}$, where $q_{j}$ is an effective charge. In this
manner, all quantities can be represented by atom positions $\bar{\bf R}_{j}$,
so all arguments up to here can be directly applied. Similar to magnetism, we
can create as many OPs as desired. The limiting case $n_{\rm OP}=3N_{\rm at}$
is “dipolar glass”, in analogy with spin glass. Indeed, this name is used for
some kind of asymmetric molecular crystals such as rubidium cyanide Shimada et
al. (1986). Such molecular crystals are known to have non-vanishing entropy at
$T=0$ (in Ref. Fowler and Guggenheim (1952), §532).

An apparent OP presenting the solid structure is an X-ray diffraction pattern
(listed in Table 1 of Ref. Anderson (1984)). They appear when a liquid is
crystallized and they change when a phase transition occurs between different
structures. However, it is rare to acknowledge X-ray diffraction patterns as
OP, probably because it is too obvious, so it is difficult to find merit in
treating them as OPs. The position of atoms appearing in the X-ray diffraction
is the time-averaged position $\bar{\bf R}_{j}$ but not instantaneous position
${\bf R}_{j}(t)$. (An interesting historical debate is written in Ref. Kittel
(2005), p. 641). Therefore, the X-ray diffraction patterns are qualified as OP
of the crystal.

Certainly, X-ray diffraction patterns are obtained because of crystal
periodicity. As a result, it is liable to consider OP as the periodicity in
this case. This is also why it is difficult to find OP in amorphous materials
not (d). Although periodicity is a necessary condition for diffraction, the
condition of the definiteness of $\bar{\bf R}_{j}$ precedes the presence of
periodicity. In the case of solids, regardless of periodicity, the same-time
position-position correlations $\langle{\bf R}_{k}(t),{\bf
R}_{j}(t)\rangle_{t}$ have definite values, $\bar{\bf R}_{k}\otimes\bar{\bf
R}_{j}$, but not for liquids and gases. Therefore, $\\{\bar{\bf R}_{j}\\}$
should be OP as well as state variables, and this is true for amorphous solids
too. One might complain that there is no method to observe correlation
functions $\langle{\bf R}_{k}(t),{\bf R}_{j}(t)\rangle_{t}$ of individual
pairs of atoms and argue that non-observable quantities cannot be OPs. The
X-ray diffraction method extracts spatial periodicity from the correlation
function $\langle{\bf R}_{k}(t),{\bf R}_{j}(t)\rangle_{t}$ by averaging over
all the atoms. Typically, experimentalists use the powder X-ray diffraction
method among many variants. Using this method, the periodicity of the three-
dimensional space is averaged to reduce to one-dimensional information. If the
powder diffraction method were the only available experiment, only the
orientational average of diffraction peaks would correspond to OP. This is
tantamount to say that the definition of OP changes as our technology is
developed. This is undesirable for describing the principles of physics. The
measurability in diffraction methods is merely a problem of currently
available experimental techniques and should be irrelevant to the definition
of state variables. Direct imaging methods, such as transmission electron
microscopy, are now available. Although their utilization is still limited,
giving one counterexample is enough for denying the impossibility of direct
observation of microscopic structures.

For the glass transition, in old theories, the OP $z$ is interpreted to
present a nonequilibrium parameter, as described in Introduction. It is
considered that a glass-forming liquid freezes in the cooling process before
reaching equilibrium. Thus, $z$ is frozen at nonzero value $z=z_{0}$. The
value $z_{0}$ is determined by the condition of minimum free energy $G$,
according to Eq. (2). Thus, $z_{0}$ is considered not to be an independent
variable, as $z_{0}=z_{0}(T,p)$ Landau and Lifshitz (1976). In the new theory,
the OPs are equilibrium positions, $\bar{\bf R}_{j}$, for each frozen state.
Furthermore, these OPs are independent of $T$ and $p$. In the new theory, time
averaging must be performed within the constraints $\\{\xi_{j}\\}$. With these
constraints, the position of $j$th atom, ${\bf R}_{j}(t)=\bar{\bf R}_{j}+{\bf
u}_{j}(t)$ can vary only in the vicinity of $\bar{\bf R}_{j}$. Hence, the
accurate form of Eq. (2) must be interpreted as follows:

$\left(\frac{\partial G}{\partial{\bf u}_{j}}\right)_{T,p,\\{\bar{\bf
R}_{j}\\}}=0.$ (7)

It has been known that there are infinite numbers of local minima in the
energy functional $E=E(\\{{\bf R}_{j}\\})$ in a solid Payne et al. (1992);
Oganov and Glass (2006). This fact can be understood by Eq. (7). There is no
surprise to find infinite equilibrium configurations in a single glass. All
such states can be described by thermodynamic methods.

##  III Properties of the glass order parameter

###  III.1 Plurality and arbitrariness

In this section, some OP features in glass are discussed. The first one is
arbitrariness in choosing the specific form of OPs. The definition of OP in
glasses varies in the literature. Some properties of glasses are often used
for the OP. In Sec. II.3, on the other hand, we have demonstrated that OPs are
the equilibrium positions of the atoms. Here, the gap between the present
conclusion and others’ view is explained. The most widely-used OP in the glass
literature may be the fictive temperature, which was phenomenologically
introduced by Tool about a century ago Tool and Eichlin (1931); Tool (1946).
He defined the fictive temperature $T_{f}$ as the temperature at which the
glass would find itself in equilibrium if brought there quickly from its
current state. Later, his formula was better treated in a thermodynamic
framework by Davies and Jones Davies and Jones (1953a). The idea of
introducing the fictive temperature is to facilitate thermodynamic treatments
by bypassing the relaxation process. The equilibrium condition (2) is
equivalent to $A(T,p,z)=0$. Hence, the fictive temperature is defined by:

$A(T_{f},p,z)=0.$ (8)

Thus, $T_{f}$ is a function of $p$ and $z$. In this manner, the fictive
temperature can be qualified as a state variable and also an OP, provided that
there is a unique $T_{f}$ satisfying Eq. (9) for a given $z$.

Equation (9) is not too useful unless the concrete form $A(T,p,z)$ is known.
The response of glass to a given external influence exhibits non-linearity and
non-exponential time response Hodge (1994). By acknowledging that the value of
fictive temperature depends on the measured property $P$, Narayanaswamy
defined $T_{f}$ by the response function $M$

$M(t,\Delta
T)=\frac{P_{2}(t)-P_{2}(\infty)}{P(0)-P_{2}(\infty)}=\frac{T_{f}-T_{2}}{\Delta
T},$ (9)

for a step temperature change $\Delta T=T_{2}-T_{1}$ Narayanaswamy (1971).
Here, $P_{2}(t)$ stands for the time-dependent $P$ after changing $T$ from
$T_{1}$ to $T_{2}$. Using $M(t)$, the current value $P(t)$ is calculated by
time convolution of the history of thermal treatment $T(t^{\prime})$, where
$t^{\prime}\leq t$. Further approximations are needed to work on the formula,
but it is not the subject of this study. Equation (9) shows that any property
of glass can be used as an OP through the fictive temperature. In fact,
enthalpy, thermal expansivity, and refractive index represent OPs, with
different $T_{f}$ Rit ; Kovacs et al. (1979); Hutchinson and Kovacs (1976).
However, it is noted that, from the present view, there is a problem in the
previous usage of $T_{f}$. The problem is the assumption for the final state
of the relaxation process. Many researchers believe that, in the transition
region, the glass state is metastable. In stead, supercooled liquids are
considered as equilibrium, and the glass state is expected to be eventually
transformed into a liquid state over a long period Davies and Jones (1953a).
This assumption is not approved by the energetic argument made by the author
Shirai (2020), which shows that the free energy of glass is lower than that of
the parent supercooled liquid. The supercooled liquid state cannot be restored
at $T<T_{g}$.

The plurality of OPs mentioned above is evident from the present theory. Since
the properties of solids are determined by their structures, any physical
property $P_{k}$ can be taken as OP,

$P_{k}=P_{k}(\\{\bar{\bf R}_{j}\\}).$ (10)

There are $N_{\rm at}$ independent properties $P_{k}$. Among them, any set
$\\{P_{k}\\}$ can be used for a set of OPs. The fictive temperature,
$T_{f,k}$, associated with the property $P_{k}$ is determined by a set of
constraint, $\\{\xi_{j}\\}$. As described in Sec. II.1, every constraint has
its own relaxation time $\tau_{j}$. Even if the volume is fixed, a relaxation
occurs in the enthalpy. Accordingly, the relaxation times of volume
expansivity and of enthalpy are different. The refractive index is affected by
the inhomogeneity of the sample, whose relaxation process is different from
other quantities.

###  III.2 Randomness and order

The second issue is the meaning of disorder in thermodynamics. This issue was
already discussed in Sec. II.3. However, it is useful to investigate it from a
different view point of the notion missing information, which may be a better
interpretation for entropy than the conventional interpretation, i.e., the
degree of disorder is Ben-Naim (2008). In solid state physics, disorder means
a break of periodicity. This traditional interpretation penetrates so deeply
into our minds that we are liable to understand that the periodicity is akin
to the order state in thermodynamics too. However, by developing the material
research, many solids with non-periodic structures have been discovered,
including incommensurate compounds and quasicrystals. It is difficult to say
whether they are ordered states or not. Outside solid state physics, the
traditional interpretation of disorder for entropy was already criticized Ben-
Naim (2008); Rosenkrantz (1983); Grandy-Jr. (2008); Denbigh (1989); Styer
(2000). Thermodynamics is a universal theory whose laws must remain
independent of the details of material structures.

The meaning of order/disorder in thermodynamics was seriously studied in
information theory, from which the notion of missing information emerged Jay ;
Tribus and McIrvine (1971); Wehrl (1978); Jaynes (1979); Zurek (1999); ter
Haar (1966); Ben-Naim (2008). If we look at the number 3.14159$\dots$, it
looks like a random number but yet conveys useful information Cha . An English
text would appear to be just a series of random letters to someone who does
not know English. If we do not know the meaning of the arrangement of
nucleotides in DNA, they seem merely a random array. These examples urge us to
reflect on the definition of randomness. In information theory, where there is
no idea of crystal symmetry, the missing information has generality. An atom
arrangement is shown in Fig. 2(a). There is no periodicity in the atom
arrangement. However, this figure was drawn by removing the lines in Fig.
2(b), in which a square lattice is drawn in a distorted space. In a square
lattice in the Cartesian coordinates, each atom is identified by the lattice
address $(m,n)$. The information of the address of atoms is not missed by
deforming the space. Topologically, the two figures are equivalent.

Figure 2: (a) Random arrangement of atoms. This figure is the same as (b),
except the lines are hidden.

The geometrical arrangement of homes in our town does not have periodicity.
However, a unique address is assigned to each home. A mail carrier can deliver
postal materials to any home once he retains the town map. However, if
residents move frequently, the mail carrier cannot deliver the items to the
residents correctly. Information on addresses is missing. In order to
correctly deliver the postal items, the mail carrier must ensure that the
address of today is the same as yesterday. As far as the word “apple” is
always associated with the real object apple, this word conveys useful
information. If everyone uses it differently, this word does not make sense.
The common property of information in the above examples is the time
invariance: that is, it always gives the same value at any time it is
measured. This requirement of the time invariance is the same as for state
variables, as stated in Definition 3.

###### Corollary 3

(Information) Information in thermodynamics means values of state variables.

State variables (and OPs) play the role of language, conveying the information
of a material not (e).

Sometimes, the notion of missing information is misconstrued. There is a long
debate about the nature of information when the meaning of entropy is
investigated Jaynes (1979); Denbigh (1981). It is said that there are two
approaches to statistical thermodynamics: objective and subjective information
(in Ref. Callen (1985), p. 385). The issue is, who is the subject of missing
information Denbigh (1981)? The entropy value of DNA is a property of that DNA
and should not be changed if we learn the meaning of the arrangement of
nucleotides. The essential matter is not the missing of our information but
the time invariance of material properties. A specific arrangement of
nucleotides will have the same result whether we observe it today or tomorrow.
The arrangement of nucleotides is thus state variables and, thereby, OPs.

There are often redundancies in a full set of state variables $\\{\bar{\bf
R}_{j}\\}$. Crystal symmetry yields redundancy. In the case of ferromagnetic
materials, individual moments $\bar{\bf m}_{j}$ are all the same. Hence, only
one OP of the total magnetization $\bar{\bf M}$ is a useful state variable. In
crystals, the periodicity significantly reduces the independent state
variables among $\\{\bar{\bf R}_{j}\\}$. Rotational symmetries impose further
restrictions on the remaining state variables. In this manner, the role of
crystal symmetries in thermodynamics reduces the number of independent OPs but
does not change the essential nature of state variables, namely, the time
invariance.

###  III.3 Prigogine-Defay ratio

Interesting evidence for the plurality of OPs in glasses stems from the
inequality relationship for the Prigogine-Defay (PD) ratio
${\it\Pi}\equiv\Delta C_{p}\Delta\kappa/TV(\Delta\alpha)^{2}$. Here, $\Delta
C_{p}$, $\Delta\kappa$ and $\Delta\alpha$ are the jumps in specific heat, in
compressibility, and in thermal expansion coefficient, respectively, at the
transition Prigogine and Defay (1954). In standard phase transition theory for
crystals, the same quantity is defined for the second-order transition and is
proven as ${\it\Pi}=1$ Pippard (1957). For the glass transition, the
overwhelming experimental data indicate inequality, Davies and Jones (1953a);
O’Reilly (1962); Moynihan et al. (1976); Moynihan and Lesikar (1981)

${\it\Pi}>1.$ (11)

Davies and Jones explained this inequality (11) by the presence of more than
one OPs Davies and Jones (1953a). Later, further proofs were given from
various points of view Goldstein (1963, 1973); Gupta and Moynihan (1976);
Goldstein (1975); Lesikar and Moynihan (1980); Nieuwenhuizen (1997); Schmelzer
and Gutzow (2006); Tropin et al. (2012). So far, these proofs are based on the
analytical behaviors of thermodynamic functions, leaving the actual substance
of OPs untouched.

From the present theory, it is evident that there is more than one OP. Here, a
further significance of the PD ratio is mentioned. Recently, an attractive
physical meaning has been found for this inequality (11) by Shirai et al.
Shirai et al. (2023). The internal energy $U$ is expressed as a function of
the equilibrium positions $\\{\bar{\bf R}_{j}\\}$ of all atoms, as shown by
Eq. (6). This can be broken into

$U=E_{\rm st}(T,\\{\bar{\bf R}_{j}\\})+E_{\rm ph}(T)+E_{\rm te}(T,V),$ (12)

where $E_{\rm st}$, $E_{\rm ph}$, and $E_{\rm te}$ are the components of the
structure (configuration), phonon, and thermal expansion, respectively.
Correspondingly, the total specific heat is written as

$C_{p}=C_{\rm st}(T,\\{\bar{\bf R}_{j}\\})+C_{\rm ph}(T)+C_{\rm te}(T,V).$
(13)

They showed that the PD ratio is rewritten as

${\it\Pi}\equiv\frac{\Delta C_{p}}{\Delta C_{\rm te}}=\frac{\text{(Change in
the total energy)}}{\text{(Contribution to isotropic volume change)}}.$ (14)

This shows that ${\it\Pi}$ is in fact equal to one only when the structural
part $\Delta C_{\rm st}$ continuously vanishes at the transition. The equality
holds for the second-order phase transition, because there is no discontinuity
in the crystal structure. The glass literature often argues that the glass
transition is a kinetic transition because the lack of latent heat is
interpreted as no structural change. The fact is that the structural change
occurs in the transition region with a finite width $\Delta T_{g}$. This was
recently shown by first-principles calculations Shirai et al. (2022, 2023).
These authors showed that the structural part $\Delta C_{\rm st}$ dominates in
$\Delta C_{p}$. Therefore, the glass transition can be essentially classified
as the first-order transition. The lack of latent heat is due to broadening
the transition temperature $T_{g}$. Latent heat $Q_{l}$ emerges only due to
the occurrence of transition at a discrete $T$. Pippard points out that what
matters is whether there is an energy difference before and after the
transition, not whether latent heat is observed (in Ref. Pippard (1985), Chap.
8). Defects and impurities destroy the long-range order. This breaks latent
heat at the discrete temperature $T_{c}$, which turns to sensible heat in a
transition range with a width $\Delta T_{c}$, resulting in a substantial
increase in $C_{p}$ in this region.

##  IV Conclusions

We have shown that the notions of the order parameter, information, and state
variable are the same in thermodynamics. They have a common property of the
time invariance against time averaging. Their measurement returns the same
value every time it is performed once equilibrium was established. Based on
the time-invariance property, we conclude that the glass OPs are the
equilibrium positions $\\{\bar{\bf R}_{j}\\}$ of all the atoms that constitute
that glass. Any glass property that is determined by the structure
$\\{\bar{\bf R}_{j}\\}$ can be an OP. Crystal symmetry brings about redundancy
in the OPs and reduces the number of independent OPs, whereas all OPs remain
independent variables of a glass. The often-claimed association of orders with
periodicity is disproven by abundant examples of well-ordered materials, such
as DNA. The orders in thermodynamics instead mean the time invariance.

The above conclusion is derived from the rigorous definition of equilibrium,
which used only the second law of thermodynamics. There is no assumption that
the ordered state is the lowest-energy state, as is often claimed. Indeed,
this assumption is inconsistent with experimental facts. The never-proven
speculation “an ordered state will be obtained if we wait a long time” has
been eliminated from the theory. All equilibria can be retained only within
the timescale in which a given set of constraints is maintained. Accordingly,
OPs, state variables, and information about states can only remain meaningful
within that timescale.

Usefulness of the present theory in current research interests is that it
provides firm grounds on the machine leaning of glasses Bapst et al. (2020);
Minamitani et al. (2023) and phase change materials Sosso and Bernasconi
(2019). The theory guarantees that the attempt to specify the current
properties of these materials by static parameters in their language must in
principle succeed.

###### Acknowledgements.

The author thanks H. Yoshino for the valuable discussion on the Edward-
Anderson model. We also thank Enago (www.enago.jp) for the English language
review. We received financial support from the Research Program of “Five-star
Alliance” in “NJRC Mater. & Dev.” (2020G1SA028) from the Ministry of
Education, Culture, Sports, Science and Technology of Japan (MEXT).

## References

  * Berthier and Biroli (2011) L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587 (2011).
  * Berthier and Ediger (2016) L. Berthier and M. D. Ediger, Phys. Today 69, (1) 40 (2016).
  * Prigogine and Defay (1954) I. Prigogine and R. Defay, _Chemical Thermodynamics_ (Longmans, London, 1954).
  * Davies and Jones (1953a) R. O. Davies and G. O. Jones, Adv. Phys. 2, 370 (1953a).
  * Davies and Jones (1953b) R. O. Davies and G. O. Jones, Proc. Roy. Soc. A 217, 26 (1953b).
  * Nemilov (1995) S. V. Nemilov, _Thermodynamic and Kinetic Aspects of the Vitreous State_ (CRC Press, Boca Raton, 1995).
  * Blundell (2001) S. Blundell, _Magnetism in Condensed Matter_ (Oxford Univ. Press, Oxford, 2001).
  * Sethna (2017) J. P. Sethna, _Statistical Mechanics: Entropy, Order Parameters, and Complexity_ (Clarendon Press, Oxford, 2017).
  * Landau and Lifshitz (1976) L. D. Landau and E. M. Lifshitz, _Statistical Physics_ (Nauka, Moscow, 1976), 3rd ed., japanese translation by A. Kobayashi, I. Ogawa, G. Tominaga, T. Hamada, and I. Yokota (Iwanami, Tokyo, 1980), §. 142.
  * Xia et al. (2015) C. Xia, J. Li, Y. Cao, B. Kou, X. Xiao, K. Fezzaa, T. Xiao, and Y. Wang, Nature commun. 6, 8409 (2015).
  * Yang et al. (2019) J. Yang, Y.-J. Wang, E. Ma, A. Zaccone, L. H. Dai, and M. Q. Jiang, Phys. Rev. Lett. 122, 015501 (2019).
  * Parisi (1983) G. Parisi, Phys. Rev. Lett. 50, 1946 (1983).
  * Franz and Parisi (1997) S. Franz and G. Parisi, Phys. Rev. Lett. 79, 2486 (1997).
  * Charbonneau et al. (2014) P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Nat. Commun. 5, 3725 (2014).
  * not (a) The random first-order transition (RFOT) theory has been developed for glass transition in analogy with the Edward-Anderson (EA) model in spin glass Kirkpatrick et al. (1989); Mézard and Parisi (2012). The original EA theory Edwards and Anderson (1975) uses a mathematical tool of replica ensemble. Physical reality of such a model requires further interpretation, and hence it is out of the present scope.
  * Prigogine and Kondepudi (1999) I. Prigogine and D. Kondepudi, _Thermodynamique, Des Moteurs Thermiques aux Structures Dissipatives_ (Editions Odile Jacob, Paris, 1999), japanese trans. by M. Seo and K. Iwamoto, (Asakura, Tokyo, 2001), Sec. 5.1.
  * Elliott (1990) S. R. Elliott, _Physics of Amorphous Materials, 2nd ed._ (Longman Scientific and Technology, Hong Kong, 1990).
  * Radons et al. (2005) G. Radons, W. Just, and P. Häussler, eds., _Collective Dynamics of Nonlinear and Disordered Systems_ (Springer, Berlin, 2005).
  * Berthier et al. (2019a) L. Berthier, G. Biroli, J. Bouchaud, and G. Tarjus, J. Chem. Phys. 150, 094501 (2019a).
  * Binder and Young (1986) K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
  * Wang et al. (2014) Z. Wang, B. A. Sun, H. Y. Bai, and W. H. Wang, Nature Commun. 5, 5823 (2014).
  * Tong and Tanaka (2018) H. Tong and H. Tanaka, Phys. Rev. X 8, 011041 (2018).
  * Berthier et al. (2019b) L. Berthier, M. Ozawa, and C. Scalliet, J. Chem. Phys. 150, 160902 (2019b).
  * Tanaka (2022) H. Tanaka, J. Non-Cryst. Solids: X 13, 100076 (2022).
  * (25) K. Shirai, cond-mat.stat-mech/1812.08977.
  * Bridgman (1950) P. W. Bridgman, Rev. Mod. Phys. 22, 56 (1950).
  * Coleman and Curtin (1967) B. D. Coleman and M. Curtin, J. Chem. Phys. 47, 597 (1967).
  * Rice (1971) J. R. Rice, J. Mech. Phys. Solids 19, 433 (1971).
  * Muschik (1993) W. Muschik, ed., _Non-Equilibrium Thermodynamics with Applications to Solids_ (Springer, Wien, 1993).
  * Maugin (1993) G. A. Maugin, in _Non-Equilibrium Thermodynamics with Applications to Solids_ , edited by W. Muschik (Springer, Wien, 1993), pp. 205–258.
  * Maugin and Muschik (1994) G. A. Maugin and W. Muschik, J. Non-Equilib. Thermodyn. 19, 217 (1994), 19, 250 (1994).
  * Restuccia and Kluitenberg (1988) L. Restuccia and G. A. Kluitenberg, Physica A 154, 157 (1988).
  * Kestine and Rice (1970) J. Kestine and J. R. Rice, in _A Critical Review of Thermodynamics_ , edited by E. B. Stuart, B. Gal-Or, and A. J. Brainard (Mono Book Corp., Baltimore, 1970), pp. 275–298.
  * Kestin (1992) J. Kestin, Int. J. Solids Structures 29, 1827 (1992).
  * Ben-Naim (2008) A. Ben-Naim, _Entropy Demystified_ (World Scientific, Singapore, 2008), expanded ed.
  * Callen (1985) H. Callen, _Thermodynamics and an Introduction to Thermostatistics_ (Wiley, New York, 1985), 2nd ed.
  * Berdichevsky (2006) V. L. Berdichevsky, Scripta Mater. 54, 711 (2006).
  * Gyftopoulos and Beretta (2005) E. P. Gyftopoulos and G. P. Beretta, _Thermodynamics - Foundations and Applications_ (Dover Pub., New York, 2005).
  * not (b) Now, the traditional view that the glass state is a nonequilibrium state is subjected to serious criticisms. Many pieces of compelling evidence supporting the stability of silica glass exist. Silica glass in geology retains its structure more than one million years Berthier and Ediger (2016), while normal metals degrade only in a thousand years. Recent experiments do not support the predicted long-term changes Brüning and Samwer (1992); Hecksher et al. (2008); Zhao et al. (2013); Ponga et al. (2015). Theoretically, it is shown that the free energy of glass is lower than that of supercooled liquid at $T<T_{g}$ Shirai (2020). In first-principles calculations, there is no sign for the mechanical instability of structures.
  * Gibbs (1906) J. W. Gibbs, _Scientific Papers_ , vol. I: Thermodynamics (Longmans, Green and Co., New York, 1906).
  * Hatsopoulos and Keenan (1965) G. N. Hatsopoulos and J. H. Keenan, _Principles of General Thermodynamics_ (John Wiley & Sons, Inc., New York, 1965).
  * Reiss (1996) H. Reiss, _Methods of Thermodynamics_ (Dover Pub., New York, 1996).
  * $\breve{C}$ápek and Sheehan (2005) V. $\breve{C}$ápek and D. P. Sheehan, _Challenges to the Second Law of Thermodynamics - Theory and Experiment_ (Springer, New York, 2005).
  * not (c) Historically, there is another reasoning for the nonequilibrium characterization of glass, namely, the violation of the third law. The author dares not to discuss it here, because the subject is too big to treat. For full account, refer to a preprint: K. Shirai, arXiv: 2207.11421. Here, the author only notes that the same line of thought about state variables solves the third law problem.
  * Shirai (2020) K. Shirai, J. Phys. Commun. 4, 085015 (2020).
  * Shirai (2021a) K. Shirai, J. Phys. Commun. 5, 015004 (2021a).
  * Shirai et al. (2022) K. Shirai, K. Watanabe, and H. Momida, J. Phys.: Condens. Matter 34, 375902 (2022).
  * Shirai et al. (2023) K. Shirai, K. Watanabe, H. Momida, and S. Hyun, J. Phys.: Condens. Matter 35, 505401 (2023).
  * Shirai (2021b) K. Shirai, J. Phys. Commun. 5, 095013 (2021b).
  * Shimada et al. (1986) T. Shimada, T. Matsuo, H. Suga, and F. Luty, J. Chem. Phys. 85, 3530 (1986).
  * Fowler and Guggenheim (1952) R. Fowler and E. A. Guggenheim, _Statistical Thermodynamics_ (Cambridge, London, 1952), 3rd ed.
  * Anderson (1984) P. W. Anderson, _Basic Notions of Condensed Matter Physics_ (Benjamin, Menlo Park, 1984).
  * Kittel (2005) C. Kittel, _Introduction to Solid State Physics_ (John Wiley, New York, 2005), 8th ed.
  * not (d) In this paper, the terms of amorphous and glass are used as the same meaning. Conventionally, they are considered as distinct classes of materials in terms of the forming process. In this study, processes are not our concern.
  * Payne et al. (1992) M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992).
  * Oganov and Glass (2006) A. R. Oganov and C. W. Glass, J. Chem. Phys. 124, 244704 (2006).
  * Tool and Eichlin (1931) A. Q. Tool and C. G. Eichlin, J. Am. Ceram. Soc. 14, 276 (1931).
  * Tool (1946) A. Q. Tool, J. Am. Ceram. Soc. 29, 240 (1946).
  * Hodge (1994) I. M. Hodge, J. Non-Cryst. Solids 169, 211 (1994).
  * Narayanaswamy (1971) O. S. Narayanaswamy, J. Am. Ceram. Soc. 54, 491 (1971).
  * (61) H. N. Ritland, J. Am. Ceram. Soc. 37, 370 (1954); 39, 403 (1956).
  * Kovacs et al. (1979) A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson, and A. R. Ramos, J. Polym. Sci.: Polym. Phys. Ed. 17, 1097 (1979).
  * Hutchinson and Kovacs (1976) J. M. Hutchinson and A. J. Kovacs, J. Polym. Sci.: Polym. Phys. Ed. 14, 1575 (1976).
  * Rosenkrantz (1983) R. D. Rosenkrantz, ed., _E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics_ (Reidel Pub., Dordrecht, 1983).
  * Grandy-Jr. (2008) W. T. Grandy-Jr., _Entropy and the Time Evolution of Macroscopic Systems_ (Oxford, Oxford, 2008).
  * Denbigh (1989) K. G. Denbigh, Brit. J. Phil. Sci. 40, 323 (1989).
  * Styer (2000) D. F. Styer, Am. J. Phys 68, 1090 (2000).
  * (68) E. T. Jaynes, Phys. Rev. 106, 391, (1957); 106, 620\.
  * Tribus and McIrvine (1971) M. Tribus and E. C. McIrvine, Sci. Amer. 225 (3), 179 (1971).
  * Wehrl (1978) A. Wehrl, Rev. Mod. Phys. 50, 221 (1978).
  * Jaynes (1979) E. T. Jaynes, _The Maximum Entropy Formalism_ (MIT Press, Cambridge, Massachusetts, 1979).
  * Zurek (1999) W. H. Zurek, _Feynman and Computation: Exploring the Limits of Computers_ (Perseus, Reading, 1999).
  * ter Haar (1966) D. ter Haar, _Elements of Thermostatistics_ (Holt, Rinehart and Winston, New York, 1966), 2nd ed.
  * (74) G. J. Chaitin, Sci. Amer., 232 47 (1975); 259 80 (1988); 294 74 (2006).
  * not (e) We note a risk of careless use of information. Making too much analogy to information theory will fall in endless arguments on the concept of information Jauch and Baron (2003); d. Beauragard and Tribus (2003); Denbigh (1981); Jaynes (1979). To avoid this, we restrict the usage of information to the information of material properties only. The above arguments on numbers and languages are cited only because of usefulness of understanding randomness.
  * Denbigh (1981) K. G. Denbigh, Chem. Brit. 17, 168 (1981).
  * Pippard (1957) A. B. Pippard, _Elements of Classical Thermodynamics_ (Cambridge, Cambridge, 1957).
  * O’Reilly (1962) J. M. O’Reilly, J. Poly. Sci. 57, 429 (1962).
  * Moynihan et al. (1976) C. T. Moynihan, P. B. Macedo, C. H. Montrose, P. K. Gupta, M. A. Debolt, J. F. Dill, B. E. Dom, P. Z. Drake, A. J. Easteal, P. B. Elterman, et al., Ann. New York Acad. Sci. 279, 15 (1976).
  * Moynihan and Lesikar (1981) C. T. Moynihan and A. V. Lesikar, Ann. New York Acad. Sci. 371, 151 (1981).
  * Goldstein (1963) M. Goldstein, J. Chem. Phys. 39, 3369 (1963).
  * Goldstein (1973) M. Goldstein, J. Chem. Phys. 77, 667 (1973).
  * Gupta and Moynihan (1976) P. K. Gupta and C. T. Moynihan, J. Chem. Phys. 65, 4136 (1976).
  * Goldstein (1975) M. Goldstein, J. Appl. Phys. 46, 4153 (1975).
  * Lesikar and Moynihan (1980) A. V. Lesikar and C. T. Moynihan, J. Chem. Phys. 73, 1932 (1980).
  * Nieuwenhuizen (1997) T. M. Nieuwenhuizen, Phys. Rev. Lett. 79, 1317 (1997).
  * Schmelzer and Gutzow (2006) J. W. P. Schmelzer and I. Gutzow, J. Chem. Phys. 125, 184511 (2006).
  * Tropin et al. (2012) T. V. Tropin, J. W. P. Schmelzer, I. Gutzow, and C. Schick, J. Chem. Phys. 136, 124502 (2012).
  * Pippard (1985) A. B. Pippard, _Response and Stability: An introduction to the physical theory_ (Cambridge, Cambridge, 1985).
  * Bapst et al. (2020) V. Bapst, T. Keck, A. Grabska-Barwinska, C. Donner, E. D. Cubuk, S. S. Schoenholz, A. Obika, A. W. R. Nelson, T. Back, D. Hassabis, et al., Nat. Physics 16, 448 (2020).
  * Minamitani et al. (2023) E. Minamitani, I. Obayashi, K. Shimizu, and S. Watanabe, J. Chem. Phys. 159, 084101 (2023).
  * Sosso and Bernasconi (2019) G. C. Sosso and M. Bernasconi, MRS Bull. 44, 705 (2019).
  * Kirkpatrick et al. (1989) T. R. Kirkpatrick, D. Thirumalai, and P. G. Wolynes, Phys. Rev. A 40, 1045 (1989).
  * Mézard and Parisi (2012) M. Mézard and G. Parisi, in _Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications_ , edited by P. G. Wolynes and V. Lubchenko (John Wiley, 2012), p. 151\.
  * Edwards and Anderson (1975) S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).
  * Brüning and Samwer (1992) R. Brüning and K. Samwer, Phys. Rev. B 46, 11318 (1992).
  * Hecksher et al. (2008) T. Hecksher, A. I. Nielsen, N. B. Olsen, and J. C. Dyre, Nature Phys. 4, 737 (2008).
  * Zhao et al. (2013) J. Zhao, S. L. Simon, and G. B. McKenna, Nat. Commun. 4, 1783 (2013).
  * Ponga et al. (2015) E. A. Ponga, C. Rodriguez-Tinoco, G. Cerullo, C. Ferrante, J. Roriguez-Viejo, and T. Scopigno, Proc. Natl. Acad. Sci. 112, 2331 (2015).
  * Jauch and Baron (2003) J. M. Jauch and J. G. Baron, in _Maxwell’s Demon 2: Entropy, Classical and Quantum information, Computing_ , edited by H. S. Leff and A. F. Rex (IOP Pub., Bristol, 2003).
  * d. Beauragard and Tribus (2003) O. C. d. Beauragard and M. Tribus, in _Maxwell’s Demon 2: Entropy, Classical and Quantum information, Computing_ , edited by H. S. Leff and A. F. Rex (IOP Pub., Bristol, 2003).