Basis Glass States: New Insights from the Potential Energy Landscape

TL;DR

This paper introduces the concept of basis glass states derived from the potential energy landscape, explaining their role in the thermally activated dynamics, glass transition, and aging processes in glass-forming systems.

Contribution

It proposes a novel framework of basis glass states that tile the configuration space and describe glass dynamics and transitions.

Findings

Basis glass states are confined to single metabasins.

They tile the entire configuration space.

The probability distribution over these states describes glass transition and aging.

Abstract

Using the potential energy landscape formalism we show that, in the temperature range in which the dynamics of a glass forming system is thermally activated, there exists a unique set of "basis glass states" each of which is confined to a single metabasin of the energy landscape of a glass forming system. These basis glass states tile the entire configuration space of the system, exhibit only secondary relaxation and are solid-like. Any macroscopic state of the system (whether liquid or glass) can be represented as a superposition of basis glass states and can be described by a probability distribution over these states. During cooling of a liquid from a high temperature, the probability distribution freezes at sufficiently low temperatures describing the process of liquid to glass transition. The time evolution of the probability distribution towards the equilibrium distribution during subsequent aging describes the primary relaxation of a glass.

Equations

$$\Phi =\Phi ({\text{r}}_1,{\text{r}}_2,\dots ,{\text{r}}_N)$$

$$\mathrm{Prob}({\text{r}}_1,{\text{r}}_2,\dots ,{\text{r}}_N)\propto \exp \left[-\frac{\Phi ({\text{r}}_1,{\text{r}}_2,\dots ,{\text{r}}_N)}{kT}\right],$$

$$\frac{d{p}_i}{dt}=\sum _j[w_{ij}{p}_j(t)-w_{ji}{p}_i(t)].$$

$${\tau }_{\mathrm{fast}}<t_0<{\tau }_{\mathrm{slow}}$$

$$P_{\alpha }(T,t)=\sum _{i\in \alpha }{p}_i(T,t).$$

$$S_c(\mathrm{Liq},T)=-k\sum _i{p}_i(\mathrm{eq},T)\ln p_i(\mathrm{eq},T).$$

$$S_c(\mathrm{Liq},T)=\sum _{\alpha }{P}_{\alpha }(\mathrm{eq},T)S_{c,\alpha }(\mathrm{eq},T)+I(\mathrm{eq},T).$$

$$S_{c,\alpha }(\mathrm{eq},T)=-k\sum _{i\in \alpha }\left(\frac{p_i(T)}{P_{\alpha }(T)}\right)\ln \left(\frac{p_i(T)}{P_{\alpha }(T)}\right)$$

$$I(\mathrm{eq},T)=-k\sum _{\alpha }{P}_{\alpha }(\mathrm{eq},T)\ln P_{\alpha }(\mathrm{eq},T).$$

$$P_{\alpha }(\mathrm{glass}|T_f)=P_{\alpha }(\mathrm{eq},T_f).$$

$$S(\mathrm{glass},T,T_f)=\sum _{\alpha }{P}_{\alpha }(\mathrm{eq},T_f)S_{\alpha }(\mathrm{eq},T).$$

$$S(\mathrm{Liq},T_f)-S(\mathrm{glass},T=T_f,T_f)=I(T_f).$$

$$F(\mathrm{glass},T=T_f,T_f)-F(\mathrm{liq},T_f)=T_{f}I(T_f).$$

$$G(\mathrm{real},T,t|T_f)=\sum _{\alpha }{P}_{\alpha }(T,t|T_f)G_{\alpha }(\mathrm{basis},T),$$

$$⟨E⟩(\mathrm{real},T,t|T_f)=\sum _{\alpha }{P}_{\alpha }(T,t|T_f)⟨E{⟩}_{\alpha }(\mathrm{basis},\mathrm{eq},T)$$

Full text

**Basis Glass States: New Insights from the Potential Energy Landscape  **

Prabhat K. Gupta

Dept. of Materials Science and Engineering,

The Ohio State University

Columbus, Ohio 43210, USA

Walter Kob

Laboratoire Charles Coulomb and CNRS

University of Montpellier

Montpellier, F-34095, France

**Abstract:

**Using the potential energy landscape formalism we show that, in the temperature range in which the dynamics of a glass forming system is thermally activated, there exists a unique set of ‘basis glass states’ each of which is confined to a single metabasin of the energy landscape of a glass forming system. These basis glass states tile the entire configuration space of the system, exhibit only secondary relaxation and are solid-like. Any macroscopic state of the system (whether liquid or glass) can be represented as a superposition of basis glass states and can be described by a probability distribution over these states. During cooling of a liquid from a high temperature, the probability distribution freezes at sufficiently low temperatures describing the process of liquid to glass transition. The time evolution of the probability distribution towards the equilibrium distribution during subsequent aging describes the primary relaxation of a glass.

**1. Introduction

**The difficulty in defining the glassy state [1] at the phenomenological (or macroscopic) level lies not only in the description of the spatial arrangement of constituent atoms which can qualitatively be described by terms such as non-crystalline or amorphous or absence of long range order [2]. An important further challenge is in describing the aging behavior of glasses over long periods of time during which a glass gradually changes from being solid-like at short times to liquid-like at long times. This time-dependent behavior has generated a lively debate as to how to describe a glass: Is it a solid that creeps over long times, or a frozen liquid that relaxes with time, or something in between like a visco-elastic solid? Questions such as “Do cathedral glass windows flow?” are still blowing in the wind [3, 4].

In this paper, we present a semi-microscopic physics-based view of glass forming systems. We refer to this view as the potential energy landscape (PEL) view. This PEL-view is based on the firmly established and widely accepted statistical thermodynamic framework of glass forming systems that provides an understanding of the liquid and supercooled liquid states, of the liquid to glass transition, and of the glassy state  [2, 5, 6]. It is evident that the PEL-view is not a hypothesis or a model, since it is as much a reality as are the interatomic interactions in a system. The PEL-view contains the essential physics that rationalizes the generic features of structural glass-forming systems. Important universal features of these systems are [2, 7]:

• •

Super-Arrhenius increase of the viscosity of the liquid state with decrease in temperature,

• •

Non-linear, nonexponential, non-Arrhenius and spatially heterogeneous primary (or $\alpha$) relaxation in the supercooled liquid state,

• •

Liquid to glass transition temperature (sometimes referred to as the fictive temperature) decreasing with increasing observation time during cooling of a liquid,

• •

Absence of latent heat in liquid to glass transition,

• •

Rounded discontinuous drops in heat capacity and in compressibility during the liquid to glass transition,

• •

Prigogine-Defay ratio value greater than unity during the liquid to glass transition,

• •

Second order properties in the glass state (such as the heat capacity) similar to those in the corresponding crystalline state,

• •

History dependent properties in the glassy state,

• •

Aging of glassy state properties at temperatures below the glass transition temperature.

• •

Near-Arrhenius secondary (also called Johari-Goldstein or $\beta$) relaxation.

The PEL-view was introduced by Goldstein in his classic 1969 paper [8]. The underlying concepts and theories based on the PEL-view have been much reviewed although largely in the condensed matter physics literature [2, 5, 9, 10]. For this reason, details are not presented in this paper. The purpose of this paper is therefore to present just a simplified introduction to the concepts and aspects of PEL that are of particular interest to structural glasses, to describe how the PEL-view rationalizes important features of glass forming systems, and to formulate a new description of real (sometimes termed kinetic or non-equilibrium) glasses in terms of basis glass states that are uniquely determined from the PEL.

The organization of the paper is as follows. In section 2 basic concepts and thermodynamics of a potential energy landscape are described. Section 3 describes the dynamics of a system in the PEL since it is the dynamics that define different states (e.g., the liquid and glassy states) of a system and transitions between them such as the liquid to glass transition. In section 4, we discuss the equilibrium (i.e., ergodic) liquid state in terms of PEL features. Section 5 discusses the liquid to glass transition and the non-equilibrium (broken-ergodic) glassy state. In section 6, we develop the concept of basis glass states and provide a description of a real glass as a superposition of basis glass states. In section 7, we discuss time dependent processes (e.g., structural relaxation) that are always present in a real glass-forming system. This paper is concluded in section 8.

**2. The Potential Energy Landscape

** The PEL is a highly rugged $3N+1$ dimensional continuous hyper-surface representing the total potential energy ($\Phi$) of a system containing a large number ($N$) of atoms (sometimes referred to as particles) as a function of atomic configurations specified by $3N$ atomic coordinates:

[TABLE]

where ${\textbf{r}}_{i}$ is the coordinate of particle $i$. (Note that for non-thermal systems like hard spheres, $\Phi$ is a constant and therefore it is more appropriate to consider the free energy landscape [11]. In this paper, we only consider thermal systems.) The PEL is fully determined from the knowledge of the atomic interactions, composition and volume of the system. We emphasize that the PEL is independent of temperature ($T$). Temperature enters the picture only via the equilibrium probability to find the system at a certain point in the configuration space, i.e., the Boltzmann weight of a configuration:

[TABLE]

where $k$ is the Boltzmann constant.

Since the PEL contains the full information about the potential energy of the system, the thermodynamic properties of the latter can in principle be obtained in an exact manner.  However, the interesting aspect of the PEL approach is related to the fact that many properties of the glass-forming systems can be obtained by a relatively small set of geometric properties of the PEL, thus reducing tremendously the information necessary to describe the system [5, 12]. Important geometric features of a PEL are [6, 10, 13, 14]:

• •

The inherent structures (IS): The local energy minima, called ISs, represent stable configurations of the system and provide important information on the atomic level structure (e.g., the radial distribution function). Note that the set of ISs also contains the ground state of the system which in most cases corresponds to a crystalline phase, although exceptions to this are known [15]. In the following we will assume that the system is a good glass-former and avoids all crystalline configurations, i.e., we consider only the non-crystalline configurations in the PEL.

• •

The basins of attraction (BA): The set of states that upon energy minimization via steepest descent lead to a given IS constitute the basin of attraction (BA) of that IS. The PEL can be partitioned in a unique way into distinct BA’s.  At low temperatures and short times, the system is confined in these BAs and hence they are important for rationalizing the vibrational properties of the system.

• •

The curvatures of the PEL at the ISs: Knowledge of these curvatures provides direct information on the vibrational density of states of the system.

• •

The number of ISs as a function of energy: This dependence determines if the glass-former is strong (weak energy dependence i.e., small configurational heat capacity) or fragile (strong energy dependence i.e., large configurational heat capacity).

• •

The saddles or transition states [16, 17]: These are stationary states of a PEL having at least one negative principal curvature. Their respective energies are related to the non-vibrational (i.e., configurational or structural) relaxation in the system at low temperatures.

• •

The steepest descent paths connecting an IS to a neighboring IS via the common transition state: Such a path corresponds to the reaction coordinate for structural relaxation at low temperatures.

• •

The distribution of the barrier energy of a PEL [18] (defined as energy of the transition state minus the energy of the starting IS): These barriers play an important role in governing the dynamics of transitions among different ISs. This dynamics is thermally activated around temperatures near and below the glass transition temperature.

These features of a PEL refer to the properties of individual ISs and BAs. However, it is also of great interest to go beyond this local view and in particular to see how the different BA’s are connected to each other on larger scales. Previous studies [10] have given evidence that the PEL is not just a random landscape in which BA’s and IS’s are patched together in a random manner, but that instead there is a hierarchy in which IS’s with similar energy are clustered together giving rise to “meta-basins” (MB), i.e., a collection of IS’s that are relatively easily accessible to each other but surrounded by larger barriers from other IS’s. As we will see below, these MBs are important for understanding the behavior of the glassy state.

The MBs give rise to another partition of the PEL, different from the BA partition, that we will refer to as the “meta-basin partition”. It is a unique partition of a PEL and results when inter-IS transitions of the system are sequentially turned off starting from the highest barrier and going down to some threshold barrier height. When a sufficient number of inter-IS transitions are turned off in this manner, a PEL uniquely partitions into a large number of non-overlapping components (namely MBs) among which no transitions exist involving barrier heights greater than the threshold barrier height. Lower barrier transitions paths remain active only within MBs. It should be noted that in general each MB contains many ISs and there are exponentially extensive number of MBs. This MB-partition and the IS-structure of MBs (which IS belongs to which MB) are uniquely determined for a PEL and play a critical role in defining the nature of glass transition and of the glassy state when the PEL dynamics is thermally activated [10].

We also mention that it is possible to extend these concepts to isothermal-isobaric systems using the enthalpy landscape formalism [19, 20]. However, because the primary goal of this paper is to introduce the new concept of basis glass states (see section 6) in a clear and rigorous manner, we limit here the discussion to systems in the canonical ensemble. This in no way limits the general validity of the results and conclusions of this paper.

An important point to emphasize is that while it may be extremely challenging (and may not even be possible) to calculate in an exact manner any or all these geometric features for a given PEL, in principle all these features are uniquely and completely determined for a PEL. In other words, these features are properties of a PEL and therefore of the system. From the theoretical point of view this is an important point since it allows development of relatively simple (abstract) models that describe at least some of the features of a glass-forming system. The well-known “random energy model” is an example of such an abstract PEL [21].

**3. Dynamics in a PEL

** The configuration point of a system in the canonical ensemble (i.e., in contact with a thermal bath at temperature $T$) moves around rapidly on the PEL. The above mentioned properties of the PEL, distribution of the IS-energy, barrier heights, the MB-partition, etc., have a direct consequence on how the system explores the PEL. Previous studies [5, 10, 22] have shown the existence of three dynamical regimes: Above the onset-temperature $T_{0}$, the typical values of the IS-energies that the system explores are independent of temperature and the relaxation dynamics of the system is essentially exponential with time and has an Arrhenius temperature dependence. For temperatures in the range $T_{0}>T>T_{c}$, where $T_{c}$ is the critical temperature of the mode-coupling theory [23], the average IS-energy decreases [24] but the system still mainly probes the PEL close to the saddle points [16, 25, 26]. The relaxation becomes increasingly non-exponential in time and its temperature dependence super-Arrhenius as the temperature is reduced. In practice this means that a typical configuration of the particles has a dynamical matrix (i.e. the matrix describing the local curvatures of the PEL) which has at least one negative eigenvalue. For mean-field like systems this is an exact result [27] but simulations have shown that realistic glass-forming systems show the same behavior. Below $T_{c}$, the system mainly resides deep inside the BAs and explores the local minima in the PEL making only rare transitions between BAs of neighboring local minima. Hence at these temperatures the dynamical matrix has only positive eigenvalues. Thus, only for temperatures below $T_{c}$ one expects the system dynamics to be thermally activated and, at sufficiently low temperatures, transport coefficients such as the diffusion coefficient and viscosity gradually revert back towards Arrhenius behavior [2]. With decreasing temperature, the system spends more time in lower energy ISs and the barrier heights of the transitions increase leading to rapid slowing down of the dynamics.

This temperature dependence of the dynamics of the system in the PEL finds its correspondence in the relaxation dynamics of the glass-former when it is described by the usual dynamic observables, such as the mean-squared displacement of a particle or the intermediate scattering function [28]. For supercooled liquids that show a glassy dynamics, i.e., for temperatures below $T_{0}$, it is well known that the dynamics can be decomposed into two parts [2]: A fast one which corresponds to the rapid vibrational motion of the particles inside the cage formed by their neighbors and a much slower motion that is related to the escape of the particles from this cage and that corresponds to the structural relaxation of the system. While the fast dynamics occurs on the time scale of atomic vibrations, i.e., picoseconds, the relaxation slows down rapidly with decreasing $T$ and hence will be orders of magnitudes slower. As a consequence, we have in the supercooled regime a strong separation of these two time scales. This fact can be used to formulate an approximate description of the dynamics of the system in the PEL: The fast vibrational motion of the particles corresponds to a dynamics inside the BA of an IS. Transitions between neighboring ISs in the same MB (i.e., intra-MB transitions) can be associated with the Johari-Goldstein or $\beta$-relaxation, i.e. the broad peak seen in dielectric measurements that, at low temperatures, is at frequencies higher than the $\alpha$-peak and which shows an Arrhenius dependence on $T$ [29]. Transitions between ISs in different MB (i.e., inter-MB transitions) reflect the $\alpha$-relaxation of the system [30].

The stochastic dynamics for the inter-IS dynamics can be approximated by a master equation for the probability $p_{i}(t)$ for the system to be present at time $t$ in the $i$-th IS-basin [31]:

[TABLE]

Here, $w_{ij}$ represents the $T$-dependent transition rate from the $j$-th IS to the $i$-th IS. Note that by using this master equation one neglects memory effects, i.e., one assumes that the separation of time scales between vibrations and relaxations is large enough so that the system loses its memory of previous IS transitions.

Measurements of an observable property of a system are carried out over some finite period of time, which in the following we will denote “observation time” $(t_{0})$. Whether or not a measured property represents the equilibrium value of the system depends on the relative magnitudes of two different time scales: the time of observation which is an external time scale imposed by the observer, and an internal time scale $(\tau)$ characterizing the kinetics of relaxation processes taking place in the system. Since vibrational times are extremely short, on the order of picoseconds, vibrational properties are usually in equilibrium for typical observation times of an experiment. When the typical structural relaxation time is significantly smaller than the observation time, the system is in thermodynamic equilibrium and is called ergodic.

A non-ergodic system is termed broken-ergodic when the distribution of relaxation times exhibits slow and fast modes such that

[TABLE]

In such a situation, a system is in equilibrium (during the observation time) with respect to the fast modes but is frozen,  i.e., non-ergodic, with respect to the slow modes. This implies that the configuration point of the system is confined (during the observation time) with some probability in a subspace (such as a MB) of the configuration space and is not able to explore a significant portion of the PEL. Note that this probability over the subspace will depend on the history of the system. Palmer [32] was the first to formulate the statistical mechanics of such broken-ergodic systems.

For glass forming systems, it is useful to consider also the configurational changes in terms of MB-dynamics, i.e., the time dependence of metabasin probabilities $P_{\alpha}(t)$ for the system configuration point to be present in the $\alpha^{\rm th}$-MB at time $t$. These MB-probabilities are defined as follows:

[TABLE]

Here the sum is over all ISs that constitute the $\alpha^{\rm th}$-MB. Note that since the equilibrium value of $p_{i}$ depends on temperature, $P_{\alpha}$ will depend on $T$ as well. The MB-dynamics can be approximately described by a suitably aggregated master equation for MB-probabilities [33].

From the formal point of view, in the above equations, one has considered the PEL of only one system. One might thus wonder how in a real experiment on a macroscopic sample the probabilities $p_{i}$ and $P_{\alpha}$ come into play since at any time only one IS and MB will be occupied. In practice a macroscopic sample can be divided in many macroscopically small (but microscopically large) pieces, each of which will have the same properties as the original system. Since the individual pieces, or sub-systems, are independent from each other, apart from small corrections due to presence of common interfaces, the probability that a given sub-sample is in the $i$-th IS-basin is given by $p_{i}$. So, a property of the entire sample is given by the weighted sum of the property of the sub-samples. Thus, it is indeed the probability distribution $\{p_{i}\}$ that determines the behavior of the macroscopic sample.

**4. The liquid state

** For a specified observation time $t_{0}$, a system at sufficiently high temperatures is able to explore the entire configuration space and the MB-probabilities take their equilibrium values $P_{\alpha}({\rm eq},T)$. This equilibrium (ergodic) state corresponds to the liquid or the supercooled liquid state (from now on, we simply refer to it as the liquid state).

A large configurational entropy is a defining characteristic of the liquid state. The configurational entropy, $S_{c}({\rm Liq},T)$, of the liquid state is expressed by the Gibbs entropy equation [34]:

[TABLE]

Here k is the Boltzmann’s constant and the summation is over all ISs of the entire PEL. Note that the probabilities $p_{i}({\rm eq},T)$ depend on temperature and hence at low $T$ only the IS with low energy will be populated. As a consequence, $S_{c}({\rm Liq},T)$ decreases when temperature is lowered [35]. According to Eqn. (6), the configurational entropy will be positive for all temperatures and will vanish only at absolute zero (assuming that the degeneracy of the ground states is sub-exponentially extensive).

The configurational entropy of a liquid can also be expressed in terms of the configurational entropy of MBs:

[TABLE]

Here, $S_{c,\alpha}$ is the configurational entropy of the $\alpha$-th MB and is defined in a manner similar to eqn (6) but with probabilities normalized for the $\alpha$-th MB [36, 37]:

[TABLE]

The quantity $I({\rm eq},T)$, called complexity, is defined as follows [2, 38]:

[TABLE]

In Eqn. (9) the sum is over the all MBs. Complexity represents a global property of the liquid and is that part of the total configurational entropy which is associated with the exploration of different MBs of the PEL. On the other hand, a property of a MB such as $S_{c,\alpha}$ represents a local property (that of a single MB of the PEL). Similar to the configurational entropy, the magnitude of the complexity decreases as $T$ is reduced. It is therefore tempting to argue that at low temperatures the configurational entropy and the complexity are the same, implying that at these $T$’s the system is confined in the BA of just one IS and thus the MB and the BA are identical [35]. However, recent studies indicate that this is not the case, i.e. that even at very low temperatures the configurational entropy is larger than the complexity, or in other words, the MB have a non-trivial structure even at very low $T$ [39]. It is possible that for some systems, the complexity may vanish at some low but finite temperature (similar to but not necessarily equal to the Kauzmann temperature [40, 41]) below which a system is trapped in the ground state MB.

Viscous flow in liquids is a manifestation of $\alpha$-relaxation (i.e., inter-MB transitions) and the viscosity is proportional to the average $\alpha$-relaxation time. A non-Arrhenius behavior of viscosity is a characteristic feature of glass forming liquids [42] and it signals the increase of the barrier heights with decreasing temperature. There is considerable debate about its low temperature behavior: does the viscosity diverge at some finite temperature (in the manner of the Vogel-Fulcher (VF) eqn. [43]), does it revert to the Arrhenius behavior with divergence only at absolute zero [44], or does it follow a different functional form with no divergence at finite temperature [45]. One reason for this debate is the absence of viscosity data at low temperatures near where the divergence is expected based on the VF equation. Empirical equations (such as the MYEGA eqn. [46], Bässler eqn. [47], or the ECG eqn. [48]) with the same (or even lower [47]) number of fitting parameters as the VF equation and that do not exhibit any divergence have been found to fit the experimental data just as well as the VF equation. For the viscosity to diverge at a finite temperature, the barrier must also become infinite at the divergence temperature. Since this is not the case for the PEL of systems that have a finite interaction range (which implies an upper bound for the barrier height), it is impossible to support a Vogel-Fulcher type of temperature dependence at low temperatures. Note that a maximum (but finite) barrier height at low temperatures will imply a change to an Arrhenius behavior at low temperatures as has been reported by some investigators [44].

In the PEL formalism, the increase in barrier heights with decreasing temperatures is largely due to the fact that at low $T$ the liquid starts to explore ISs with low energies. These IS’s are connected to neighboring IS with barriers that become increasingly large since the local arrangement of the particles becomes more and more optimized and breaking up this local arrangement becomes increasingly difficult. A consequence of this picture is that the temperature dependence of the barrier height should be approximately proportional to the configurational heat capacity of the liquid. This correlation between temperature dependence of the barrier height and the configurational heat capacity of the liquid is indeed supported by experimental data and has been noted by several investigators [49, 50]. For example, pure silica (a strong glass forming liquid) exhibits nearly Arrhenius behavior and has extremely small configurational heat capacity while OTP (a fragile liquid) is highly non-Arrhenius and has a large configurational heat capacity [51].

**5. The glass transition and the glassy state

**Upon cooling a liquid, the structural relaxation dynamics slow down and inter-IS transitions begin to freeze sequentially starting with the highest barrier. At some low temperature, when a sufficient number of inter-IS transitions are blocked, the PEL partitions into MBs that are no longer mutually accessible on the time scale of the experiment. This MB-partitioning of the PEL corresponds thus to the liquid to glass transition. The partitioned state is broken-ergodic and corresponds to the glass state [52, 53]. The temperature at which the partitioning occurs depends on the cooling rate and is called the fictive temperature $T_{f}$ of the glass. Technically, $T_{f}$ should be called the glass transition temperature. Unfortunately, the term glass transition temperature customarily refers to the temperature where the viscosity is $10^{12}$  Pa.s (i.e., average relaxation time about 100 sec) and hence $T_{f}$ is also named “kinetic glass transition temperature”. We emphasize that in the liquid to glass transition the system is trapped in MBs and not in individual basins of attraction. Thus, even in the glass state the system will undergo some localized intra-MB relaxation dynamics at low temperatures, the details of which we will discuss below.

At the glass transition, $P_{\alpha}$, i.e, the probability that the system is in the $\alpha$-MB, freezes at a value equal to that with which the liquid was exploring the MB immediately prior to the glass transition:

[TABLE]

We emphasize that while the values of the MB-probabilities $P_{\alpha}$ depend on the time of observation via the fictive temperature, the geometry of the MB-partition does not.

It is easy to see that the energy of the system does not change during glass transition since the basin probabilities $p_{i}$ are the same in glass and in the liquid at $T_{f}$ . However, this is not true for the entropy. The entropy of a glass with fictive temperature $T_{f}$ and at a bath temperature $T$ is the thermal average of the entropy of the MBs weighted at $T_{f}$ [36]:

[TABLE]

Note that on the right hand side, $S_{\alpha}({\rm eq},T)$ is evaluated at $T$ and not at $T_{f}$, because each MB remains ergodic, so only the weights $P_{\alpha}$ depend on $T_{f}$.

The comparison of liquid and glass entropy expressions shows that there is a net entropy loss during the discontinuous liquid to glass transition. (Note that the discontinuous nature of glass transition is an intrinsic feature of partitioning of the PEL and is not associated with cooling rate or observation time used.) This is simply because the average configuration space volume of a MB is much smaller than that of the configurational space volume of the entire system. Eqs. (7) and (11) show that the magnitude of the entropy loss is given by the complexity evaluated at $T_{f}$:

[TABLE]

This loss of entropy entails a drop in the heat capacity during the liquid to glass transition. Further, since there is no change in the energy at the glass transition, the free energy, $F$, of the system increases discontinuously at $T_{f}$ with respect to the free energy of the liquid state:

[TABLE]

It must be emphasized that a glass transition is not an equilibirum thermodynamic transition but a dynamic transition. Because of its unusual features (no change in energy, a loss in entropy, and an increase in the free energy), the glass transition is sometimes also referred to as a “Random First Order Transition”(RFOT) [54, 55].

With no change in energy (i.e., no latent heat), the notion of a loss in entropy may appear to violate the second law. However, in reality there is no violation because a liquid to glass transition is not a spontaneous process. Instead, it is a process caused by an externally imposed constraint of finite observation time. Unlike this view, the conventional picture assumes entropy continuity during a glass transition [56, 57]. This difference in the two views has a direct consequence on the magnitude of the residual entropy, i.e., the entropy at absolute zero. According to the PEL-view, since no inter-IS transitions are kinetically possible at $T=0$, the configurational entropy vanishes at absolute zero whereas in the traditional view, glasses do have a finite residual entropy [58]. So far, it has not been possible to test experimentally the entropy change during a glass transition because entropy is not an experimentally observable quantity (unlike energy, it is not a dynamic variable) and there is no defined way of calculating the entropy change from the experimentally measurable heat capacity in a non-equilibrium, non-spontaneous process like the liquid to glass transition.

**6. Basis glass states and real glasses

** Since the IS-structure of a MB is fixed and there is no overlap among MBs, one can use the MBs to define a “basis-glass state”. In other words, for a basis-glass based on the ${\alpha}-$MB, we have $P_{\alpha}({\rm Basis})=1$ and the weights for all other MBs are zero. Note that we distinguish between the terms MB and basis-glass state: A MB is a region in configuration space defined by the MB-partition while a basis-glass represents a (hypothetical) state of the system with some probability distribution for the ISs within the MB. By definition, a basis-glass cannot exhibit ${\alpha}$-relaxation but it can undergo $\beta$-relaxation. It is worth emphasizing that a basis-glass may exhibit some temperature dependence since not all ISs within a MB have the same energy. This in turn makes the position and the width of the $\beta$-peak depend on temperature. The configurational entropy of a basis-glass is the same as the configurational entropy of the corresponding MB and hence is much less than the total configurational entropy of the liquid. Because of its low configurational entropy, a basis-glass will behave like a solid.

Let us consider a real glass which we symbolically denote by $G({\rm real},T,t|T_{f})$, formed at a fictive temperature $T_{f}$ and aged at temperature $T$ for some time $t$. For times longer than the observation time used during prior cooling, the MBs are no longer dynamically isolated and hence the system starts to make some inter-MB transitions even at temperatures lower than $T_{f}$. This causes the MB-probabilities $\{P_{\alpha}(T_{f})\}$ to depend on time $t$ as well as the temperature $T$ of the sample. Let us denote the probability that the system is found in the the MB $\alpha$ by $P_{\alpha}(T,t|T_{f})$. The real glass can thus be considered as a linear combination of basis-glasses $\{G_{\alpha}\}$ with $T$ and time dependent weights $P_{\alpha}(T,t|T_{f})$:

[TABLE]

where $G_{\alpha}({\rm basis},T)$ is a glass that is confined in MB $\alpha$.

Thus, in the PEL view, a real glass is described by the set of (time and temperature dependent) MB-probabilities,$\{P_{\alpha}(T,t|T_{f})\}$. As a real glass ages, it evolves irreversibly through a continuum of different glassy states. This implies, for example, that an aged glass and a freshly formed glass of the same system are not the same but two different glasses. It should be noted that the time dependence of a real glass is contained primarily in the time-dependence of the MB-probabilities. On the other hand, the basis-glasses are the primary reason for the solid like behavior of real glasses at short observation times.

Any observable property of a real glass is thus given by the average of its values over all basis-glass states. For example, the average energy, $\langle E\rangle$, of a real glass is the average energy of basis-glasses:

[TABLE]

It should be noted that because a real glass is a superposition of basis-glass states, a real glass is truly heterogeneous given by a mosaic of basis-glass states in the physical 3-dimensional space. The size of the mosaic “tiles” is controlled by the small amorphous-amorphous interfacial energy between different basis-glass states [59].

**7. Structural relaxation in the glass state

** Real glasses exhibit two types of relaxation processes: fast secondary ($\beta$) due to intra-MB transitions and slow primary ($\alpha$) due to inter-MB transitions. For short times, the inter-MB transitions remain frozen and the system exhibits only secondary relaxation without altering the MB-probabilities since the barriers associated with inter-MB transitions are larger than the barriers for inter-IS transitions within the same MB. This is consistent with the observation that the range of barrier heights for the secondary relaxation is on average less than that of the $\alpha$-relaxation [60]. Note that the identity of a real glass does not change during secondary relaxation. Over longer times, inter-MB transitions allow the system to relax slowly, i.e. the glass will age.

The PEL-view readily explains several observed features of the $\alpha$-relaxation in glasses. For example, one should note that because a real glass is a mosaic of basis-glasses, the $\alpha$-relaxation process is spatially heterogeneous. During $\alpha$-relaxation, the MB-probabilities $P_{\alpha}(t)$ evolve with time from their initial value $P_{\alpha}(\rm eq,T_{f})$ in the glassy state towards their final equilibrium value $P_{\alpha}(\rm eq,T)\}$ in the liquid state. This time dependence of the $\alpha$-relaxation is nonexponential because of the distribution of inter-MB barrier heights that become active during relaxation. While the non-exponentiality may resemble a stretched-exponential behavior [61], in general the $\alpha$-relaxation may not be a true stretched exponential because there is an upper bound to $\alpha$-barrier heights which is mathematically not the case in a stretched exponential function.

We emphasize that real glasses exhibit $\alpha$-relaxation even for temperatures much less than $T_{f}$ provided the observation time is sufficiently increased as is the case in aging experiments. The initial stages of $\alpha$-relaxation in a glass (generally referred to as “iso-structural” relaxation) have been investigated in a variety of glasses [62]. The iso-structural flow is nearly Arrhenius and its activation energy is observed to be significantly less than the activation energy for the equilibrium $\alpha$-relaxation in the corresponding liquid at $T_{f}$, the observed values of the ratio are in the range from 0.13 - 0.68 [62]. This can be understood by realizing that the inter-MB barriers which begin to unfreeze during the early stages of iso-structural relaxation in a glass are precisely the ones that were the last (and hence the smallest) to freeze at $T_{f}$ just before the MB-partitioning during the prior cooling of the liquid.

According to the MB-based view advanced here, the conceptual foundations of the physics underlying relaxation processes are simple and follow the principle of “last to freeze is first to relax”. The roadmap essentially consists of sequential unfreezing of thermally activated transitions starting with the smallest barriers. In principle, one could develop the mathematical formalism of relaxation dynamics rigorously by solving the non-isothermal non-linear time evolution of the associated master equation for the MB-probabilities starting from the moment when the liquid was last in equilibrium. This will require a detailed knowledge of the barrier height distribution of the hyper-dimensional PEL and is beyond the purpose and scope of this paper.

**8. Concluding Remarks

** In this paper, we have argued that the PEL view of glassy forming systems provides a rational framework to describe the observed universal behavior of kinetic glass transition and of structural relaxation. This framework is rooted in the fundamental constructs of statistical mechanics and an assumption of thermally activated stochastic dynamics in the PEL. In addition to rationalizing the observed behavior, this formalism leads to new concepts such as basis-glass states (that are unique to a system) and provides a description of real glasses as a superposition of basis-glass states. Further, it leads to new consequences such as the loss of complexity (a portion of the liquid configurational entropy) during the glass transition and of zero residual entropy in the glassy state. Last but not least the concept of MBs and basis glass states allows one to rationalize the fact that even deep in the glass state the particles are not completely frozen but still can undergo a local dynamics that is not of vibrational type and which is related to the Johari-Goldstein $\beta$-relaxation.

Acknowledgments:

Prabhat Gupta is extremely grateful to all his colleagues, collaborators, coauthors, and friends in the glass community for their generous support and encouragement throughout his career. Walter Kob acknowledges the support of ANR grant ANR-15-CE30-0003-02.