Continuously Broken Ergodicity

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Continuously broken ergodicity

John C. MauroScience and Technology Division, Corning Incorporated, SP-TD-01-1, Corning, New York 14831

Prabhat K. GuptaDepartment of Materials Science and Engineering, Ohio State University, Columbus, Ohio 43210

Roger J. LoucksScience and Technology Division, Corning Incorporated, SP-TD-01-1, Corning, New York 14831 and

Department of Physics and Astronomy, Alfred University, One Saxon Drive, Alfred, New York 14802

Received 6 December 2006; accepted 28 March 2007; published online 14 May 2007

A system that is initially ergodic can become nonergodic, i.e., display broken ergodicity, if the

relaxation time scale of the system becomes longer than the observation time over which properties

are measured. The phenomenon of broken ergodicity is of vital importance to the study of many

condensed matter systems. While previous modeling efforts have focused on systems with a sudden,

discontinuous loss of ergodicity, they cannot be applied to study a gradual transition between

ergodic and nonergodic behavior. This transition range, where the observation time scale is

comparable to that of the structural relaxation process, is especially pertinent for the study of glass

transition range behavior, as ergodicity breaking is an inherently continuous process for normal

laboratory glass formation. In this paper, we present a general statistical mechanical framework for

modeling systems with continuously broken ergodicity. Our approach enables the directcomputation of entropy loss upon ergodicity breaking, accounting for actual transition rates between

microstates and observation over a specified time interval. In contrast to previous modeling efforts

for discontinuously broken ergodicity, we make no assumptions about phase space partitioning or

confinement. We present a hierarchical master equation technique for implementing our approach

and apply it to two simple one-dimensional landscapes. Finally, we demonstrate the compliance of

our approach with the second and third laws of thermodynamics. 2007 American Institute of

Physics. DOI: 10.1063/1.2731774

I. INTRODUCTION

One of the most prevalent, and often unstated, assump-

tions in statistical mechanics is that of ergodicity, which as-serts the equivalence of time and ensemble averages of the

thermodynamic properties of a system. Ergodicity implies

that, given enough time, a system will explore all allowable

points of phase space. The term broken ergodicity, intro-

duced by Bantilan and Palmer1

in 1981, denotes the loss of

ergodicity that is common in many systems such as spin and

structural glasses.

The question of ergodicity, and that of thermodynamic

equilibrium, is really a question of time scale. We may think

of an experiment as having two relevant time scales: an in-

ternal scale int on which the dynamics of the system occur,

and an external scale ext on which properties are mea-sured. The internal time scale is essentially a relaxation time

over which a system loses memory of its preceding states,

whereas the external time scale defines a measurement win-

dow over which the system is observed. The measurement

can be made either directly by a human observer or by using

an instrument accessing times not available to an unaided

human. A system is in thermal equilibrium if all of the rel-

evant relaxation processes have taken place, while the re-

maining slower processes are essentially frozen on the ex-

ternal time scale.2

The ratio of internal to external time scales defines the

Deborah number3

of an experiment,

D = intext

, 1

named in honor of the prophetess Deborah, who in the Old

Testament sings, the mountains flowed before the

Lord Judges 5:5 . The implication is that mountains,while basically static on a human time scale, do in fact flow

on a geologic time scale inaccessible to mere mortals. For a

human observer, the phenomenon of continental drift is de-

scribed by a large Deborah number D1, intext , wherethe system visits only a small subset of the available points

in phase space during the external i.e., observation time.Such scant sampling of phase space is insufficient for deter-

mining a long-time average of properties, and hence the sys-tem lacks ergodicity. On the other hand, many phenomena

such as relaxations in gases and liquids occur on a time scale

too fast for direct human observation. These phenomena are

described by a very small Deborah number D1, intext , which is indicative of an ergodic system. Since the

system can explore a greater portion of phase space during

the observation time, the measured property values over extare effectively equal to those in the long-time limit.

The observation of ergodic behavior thus depends on

both the internal relaxation time of a system and the external

THE JOURNAL OF CHEMICAL PHYSICS 126, 184511 2007

0021-9606/2007/126 18 /184511/11/$23.00 2007 American Institute of Physics126, 184511-1

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observation time: a system is ergodic when D1 and non-

ergodic when D1. The issue of ergodicity, while inherently

relative, is nevertheless of great physical importance: the

properties we experience are those we measure on our own

finite time scale, not in the long-time limit of ergodicity. It is

thus important for statistical mechanical models to account

for broken ergodicity in order to predict the observed prop-

erties of nonergodic systems.

But what about the process by which ergodicity is lost?The breakdown of ergodicity can occur either discontinu-

ously or continuously. A discontinuous breakdown of ergod-

icity is caused by a sudden decomposition of the phase space

into a set of mutually inaccessible components. An example

of discontinuous ergodicity breaking is glass formation by

sudden quenching from a melt; here, the system experiences

an abrupt loss of thermal energy, thereby trapping it in a

subset of the overall phase space. On the other hand, a glass

can be formed by gradual cooling from a melt. In this case,

the glass-forming system experiences a continuous break-

down of ergodicity as it gradually becomes confined to a

subset of the phase space. Our definition of continuously

broken ergodicity is distinct from what some researchers callweakly broken ergodicity,

46in that we assume that the re-

laxation time is finite but exceeds the observation time.

In a landmark 1982 paper, Palmer7

thoroughly develops

the concept of broken ergodicity and proposes a statistical

mechanical framework with which to treat nonergodic sys-

tems and the discontinuous breakdown of ergodicity. In the

Palmer approach, the phase space of a nonergodic system

is divided into a set of disjoint regions or components ,

where

=. 2

The components are assumed to meet the conditions of con-

finement and internal ergodicity. Confinement indicates that

transitions are not allowed between components; in other

words, if a system has its phase point in a particular compo-

nent at a time t, there is negligible probability of the

system leaving that component during a specified observa-

tion time tobs, i.e., in the time interval t, t+ tobs . The condi-tion of internal ergodicity states that ergodicity holds within

a given component.

The Palmer approach thus assumes a clear separation of

intra- and intercomponent relaxation time scales: intracom-

ponent relaxation i.e., among the various microstates within

a component occurs on a time scale component much shorterthan the observation time D 1 , while intercomponent

transitions occur on a time scale system much longer than tobs D 1 . The assumption that

component tobs system 3

allows components to be in internal equilibrium while the

system as a whole is not. As a result, ergodic statistical me-

chanics can be applied within each of the individual compo-

nents, and the overall properties of the nonergodic system

can be computed using a suitable average over the individual

components.

This has important implications when computing the en-

tropy of a nonergodic system. Before proceeding, it is useful

to review some common definitions of entropy, following the

terminology of Goldstein and Lebowitz:8

1 The thermodynamic entropy of Clausius, given by

dS=dE

T, 4

where E is internal energy and T is absolute tempera-

ture. This definition of entropy is applicable only for

equilibrium systems and reversible processes.

2 The Boltzmann entropy,

SB = kB ln, 5

applicable to the microcanonical ensemble, where kB is

Boltzmanns constant and is the volume of phase

space i.e., the number of microstates visited by a sys-tem to yield a given macrostate.

3 The Gibbs or statistical entropy,

S

= kBi

pi ln pi , 6

applicable to the canonical ensemble, where pi gives

the probability of occupying microstate i, and the sum-

mation is over all microstates. Here, the probabilities

are computed based on an ensemble average of all pos-

sible realizations of a given system. With the assump-

tion of ergodicity, the ensemble-averaged values of piare equal to the time-averaged values of pi.

While there is no disagreement about the definition of

entropy for equilibrium systems, the definition of entropy for

nonequilibrium systems is not established except for the caseof microcanonical isolated systems where there is a general

agreement that Boltzmanns definition is valid and consistent

with the second law.812

For canonical, nonergodic systems, the proper definition

of entropy is a highly contentious topic. For glassy systems,

fortunately, there is a way out. As discussed by Palmer,7

glass can be described as an ensemble of ergodic compo-

nents of the phase space. With the conditions of confinement

and internal ergodicity, we can apply the Gibbs definition of

entropy within each individual component. The resulting en-

tropy of the system, as derived by Palmer7

and provided as

Eq. 12 in Sec. II of this paper, is simply a weighted sum of

the Gibbs entropies of the individual components. In thismanner, the entropy of Palmer can account for the broken

ergodicity of the glassy state.

One particular point of confusion in the glass science

community relates to the application of Eq. 4 for thermo-dynamic entropy in conjunction with differential scanning

calorimetry DSC experiments to compute the entropy of asystem on both cooling and heating through the glass

transition.13

This results in two main findings: a there is nochange in entropy at the laboratory glass transition, and b

glass has a positive residual entropy at absolute zero. We

argue that these results are incorrect since the glass transition

is not a reversible process14

and hence Eq. 4 cannot be

184511-2 Mauro, Gupta, and Loucks J. Chem. Phys. 126, 184511 2007



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applied. Moreover, the ergodic formula for Gibbs entropy,

which approximately equals the thermodynamic entropy,15,16

cannot be used since it cannot account for the defining fea-

ture of the glass transition, viz., the breakdown of ergodicity.

The glass transition, by definition, occurs when tobs =system i.e., a Deborah number of unity .13 The only reason we ob-

serve the glassy state at all is that the relaxation time scale of

the system becomes much longer than the observation time

scale. Hence, the concept of an observation time is central tothe very definition of glass, and the glass transition can only

occur if there is a finite observation time. We will show in

this paper that when a finite observation time is considered,

the glass transition entails a loss of entropy rather than a

freezing in of entropy.

Moreover, we argue that the current belief that glass has

a finite residual entropy at absolute zero is in direct violation

of Plancks statement of the third law of thermodynamics,

which demands that the entropy of any classical system

must vanish at zero temperature.17

Some researchers claim

that glass is somehow exempt from obeying the third law of

thermodynamics since it is a nonequilibrium material.13

However, their result from DSC measurements of heat ca-

pacity that glass has a finite residual entropy at absolute

zero is based on application of equilibrium, reversible ther-

modynamics to the nonequilibrium glassy system. Hence,

their very calculation of entropy is at odds with their argu-

ment that the third law does not apply to glass. We argue that

glass is not exempt from the third law and show later in this

paper that application of Palmers definition of entropy al-

ways leads to zero entropy at zero temperature since there is

no thermal energy to allow for transitions between mi-

crostates .

While the Palmer methodology greatly elucidates the

impact of broken ergodicity on entropy and other properties,many realistic systems do not lend themselves to neat parti-

tioning into discrete components that simultaneously satisfy

the conditions of confinement and internal ergodicity. Also,

the partitioning itself must depend on the transition rates

between microstates and therefore on the temperature of the

system. For a system where temperature evolves with time,

the partitioning would have to be recomputed at each new

temperature step.

In addition to these computational difficulties, there is

the physical problem that transitions between components

are never strictly prohibited at any finite temperature; this is

especially important for Deborah numbers near unity D 1 , where the relaxation is too fast to assume confinement

and yet too slow to assume internal ergodicity. Consequently,

the Palmer approach can consider only the case of discon-

tinuously broken ergodicity, where there are no relaxation

processes with D1; here, the loss of ergodicity can be

modeled well by a discontinuous partitioning of phase space

into multiple components. However, the Palmer approach

needs modification to model a continuous breaking of ergod-

icity, such as in normal laboratory glass formation using a

finite cooling rate. One can argue that the most interesting

physics occurs during the transition between ergodic and

nonergodic states, where D1. This regime corresponds di-

rectly to the glass transition range and is of great scientific

and technical importance.13

In this paper, we present a generalization of the Palmer

approach that accounts for continuously broken ergodicity

and is thus suitable for a realistic modeling of glass transition

range behavior and other phenomena with D1. Our meth-odology is based on a hierarchical master equation approach

that avoids any explicit definition of components and makes

no assumptions of confinement or internal ergodicity. Ourtechnique can thus be applied to any arbitrary system and for

any temperature path. We present equations for the configu-

rational entropy and free energy of a system with continu-

ously broken ergodicity and show results for two simple one-

dimensional landscapes.

II. ENERGY LANDSCAPES AND DISCONTINUOUSLYBROKEN ERGODICITY

The Palmer approach for discontinuously broken ergod-

icity involves a partitioning of the phase space into a set of

components , where each component satisfies the condi-tions of confinement and internal ergodicity.

7It is useful for

our ensuing discussion to translate this terminology into the

energy landscape formalism,18,19

which offers a powerful ap-

proach for studying the thermodynamics and kinetics of mo-

lecular clusters,2024

biomolecules,23

supercooled

liquids,2530

and structural glasses.23,3134

The potential energy landscape of an N-particle system

is a 3N-dimensional hypersurface in the particle configura-

tion space,

U= U r1,r2, . . . ,rN , 7

where r = r1 ,r2 , . . . ,rN R3N are the position vectors of the

N particles. The underlying assumption is a decoupling of

the configurational and vibrational thermal components ofenergy. While the potential energy landscape itself is inde-pendent of temperature, the way in which a system explores

the landscape depends on the available thermal energy. At

high temperatures there is sufficient thermal energy to enable

free exploration of the landscape; for a condensed system,

this corresponds to the case of an ergodic liquid with high

fluidity. As the system is cooled, the transitions slow down

and become thermally activated. This leads to continuously

broken ergodicity as the system gradually becomes trapped

in a subset of the landscape.

The U hypersurface itself contains a multitude of local

minima, each corresponding to a mechanically stable con-

figuration of the system, termed an inherent structure. Theset of 3N-dimensional hyperspace configurations that drains

to a particular minimum via steepest descent is known as a

basin.29

The study of energy landscapes is facilitated by

mapping the continuous U hypersurface to a discrete set of

minima, i.e., basin volumes are mapped to their correspond-

ing inherent structures. The number of inherent structures

scales at least exponentially with N.27

In the energy landscape formalism an individual mi-

crostate corresponds to a single inherent structure or basin. A

component in the Palmer approach corresponds to a so-

called metabasin3539

in the energy landscape, i.e., a group

of basins that are mutually accessible at a given temperature

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and for a given observation time. The metabasins themselves

are separated from each other by large potential energy bar-

riers such that intermetabasin transitions are highly unlikely.

Hence, metabasins satisfy the conditions of confinement and

internal ergodicity.

The probability of a system being confined in a particu-

lar metabasin upon cooling is equal to a restricted summa-

tion of the occupation probabilities of the individual basins

within ,

P=i

pi. 8

As each basin falls within exactly one metabasin, the sum of

metabasin probabilities is unity,

P =

i

pi = 1 . 9

Since ergodicity is maintained within an individual metaba-

sin, we can define a corresponding intrametabasin Gibbs en-

tropy as

S= kBi

pi

Pln

pi

P, 10

where from Eq. 8 we know that

i

pi

P= 1 . 11

Thus, in the Palmer approach see Sec. 5.4 in Ref. 7 , theexpectation value for configurational entropy is a weighted

average of the Gibbs entropies of the individual metabasins,

S =

SP= kB

ipi ln

pi

P. 12

The corresponding free energy can be computed by

F =i

Uipi T S = U T S , 13

where Ui is the potential energy of inherent structure i. Note

that this is a Helmholtz free energy, and the Gibbs free en-

ergy can be computed by

G =i

Hipi T S = H T S , 14

where

Hi = Ui + PVi 15

is the zero-temperature enthalpy of inherent structure i,

which has a system volume of Vi. The enthalpy landscape

approach is an extension of the potential energy landscape

formalism for isobaric systems,40

and is useful for computing

the volume evolution of systems under constant pressure

conditions.34

Equation 12 gives the entropy of a system with discon-

tinuously broken ergodicity. If broken ergodicity were not

accounted for, the entropy would be given by direct applica-

tion of the statistical formula of Eq. 6 . The difference be-

tween the statistical entropy and the nonergodic glassy en-

tropy ofEq. 12 is called the complexity of the metabasin

ensemble7

and is given by

I= S S = kB

P ln P. 16

The complexity is a measure of the nonergodicity of a sys-

tem. For a completely nonergodic system trapped in a single

microstate, each basin itself is a metabasin, and the complex-ity adopts its maximum value of I= S. For an ergodic system,

all basins are part of the same metabasin, and the complexity

is zero. The process of ergodicity breaking necessarily results

in a loss of entropy and increase in complexity as the phasespace divides into mutually inaccessible metabasins. This

loss of entropy is accompanied by an increase in the free

energy of the system, as indicated by Eqs. 13 and 14 .

While the breaking of ergodicity always results in a loss of

entropy S S and a gain in free energy F F, G

G , the energy and enthalpy are unaffected U = U and

H =H .

While the Palmer approach can be used to compute thesudden entropy loss in systems with discontinuously broken

ergodicity e.g., glass formation via instantaneous quench-ing , it cannot be used to compute the gradual loss of entropy

that occurs in systems with continuously broken ergodicity

e.g., laboratory glass formation using a finite cooling rate .

III. CONTINUOUSLY BROKEN ERGODICITY

The limitations of the Palmer approach7

can be over-

come by relaxing the assumptions of metabasin confinement

and internal ergodicity. We consider a nonequilibrium system

with probability distribution pi t at time t. Suppose that we

make an instantaneous measurement of the microstate of asystem at time t. The act of measurement causes the system

to collapse into a single microstate i with probability pi t .

In the limit of zero observation time, the system is confined

to one and only one microstate and the observed entropy is

necessarily zero. However, the entropy becomes positive for

any finite observation time tobs since transitions between mi-

crostates are not strictly forbidden except at absolute zero,barring quantum tunneling . What, then, is the entropy of the

system over the observation window t , t+ tobs ?We can answer this question by following the dynamics

of a system whose microstate is known at t, the beginning

of the observation window. Let fi,j t be defined as the con-

ditional probability of the system occupying microstate j af-ter starting in a known microstate i and subsequently evolv-

ing for some time t, accounting for the actual transition rates

between microstates. The conditional probabilities satisfy

j

fi,j t = 1 , 17

for any initial state i and for all time t. Hence, fi,j tobs gives

the probability of transitioning to microstate j after an initial

measurement in state i and evolving through an observation

time tobs.

While there is no clear agreement on the definition of

nonequilibrium entropy for a canonical system, a couple of




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different expressions have been suggested. One is based on a

relative or conditional entropy and another is time-

dependent Gibbs entropy.41

We choose the latter, notwith-

standing it is not valid for microcanonical systems, for the

following reasons:

1 The time-dependent Gibbs entropy is zero in the limitsof tobs0 and T0;

2 it yields the equilibrium entropy in the long-time limit;

3 it reproduces the Palmer results for discontinuouslybroken ergodicity; and

4 several other authors have also considered the time-dependent Gibbs entropy.

4245

In our case, the nonequilibrium Gibbs entropy is

Si tobs = kBj

fi,j tobs ln fi,j tobs . 18

This represents the entropy of one possible realization of the

system and corresponds roughly to Palmers component en-

tropy of Eq. 10 ; however, we have not made any assump-

tions about confinement or internal ergodicity. Note thatwhile the above equation is of the same form as the Gibbs

entropy of Eq. 6 , the value of fi,j tobs above gives theprobability of the system transitioning from an initial state i

to a final state j within a given observation time tobs. The

probability pi in the Gibbs formulation represents the ergodic

limit of tobs; hence, pi represents an ensemble-averaged

probability of the system occupying a given state and does

not account for the finite transition time required to visit a

state, which may be long compared to the observation time

scale.

The expectation value of entropy at the end of the ob-

servation window t , t+ tobs is the weighted sum of entropy

values for all possible realizations of the system,

S t,tobs =i

Si ttobs pi t . 19

With this approach, there is no need to define components or

metabasins. By considering all possible configurations of the

system and the actual transition rates between microstates,

our approach can be applied to an arbitrary energy landscape

and for any temperature path. Our approach is thus suitable

for modeling systems in all regimes of ergodicity: fully er-

godic, fully confined, and everything in between. We further

note that the combination of Eq. 17 with Eq. 19 reduces

to the Palmer equation of entropy, Eq. 12 , for the case ofdiscontinuously broken ergodicity.

With Eq. 19 , the entropy of the system is zero both inthe limits of tobs0 and T0. The first case tobs0 is in

agreement with Boltzmanns notion of entropy as the number

of microstates visited by a system to yield a given

macrostate:812

with zero observation time, the system re-

mains in the initial microstate. The second case T0 is in

agreement with Plancks statement of the third law,17

which

states that the entropy of any classical condensed system

must vanish at absolute zero. Both limits are equivalent in

the Palmer model of discontinuously broken ergodicity,7

where each microstate would have its own separate compo-

nent metabasin ; no transitions are allowed among the com-

ponents, so the entropy is necessarily zero.

For any positive temperature, the limit of tobs yields

an equilibrated system with complete restoration of ergodic-

ity. In the Palmer view,7

this is equivalent to all microstates

being members of the same component with transitions

freely allowed among all microstates. Both the Palmer ap-

proach and ours yield the same result as the Gibbs formula-

tion of entropy in the limit of t

, i.e., for a fully ergodic,equilibrated system.

In the next section, we describe a hierarchical master

equation approach for implementing the above formalism.

IV. HIERARCHICAL MASTER EQUATIONAPPROACH

The master equation approach is a useful technique for

modeling the dynamics of nonequilibrium statistical me-

chanical systems.46,47

The approach involves constructing a

set of coupled rate equations, with one equation for each

available microstate in the system. For a system with mi-crostates, labeled i ,j 1 , 2 , . . . , , the set of master equa-

tions is given by

dpi

dt=

ji

Wjipj Wijpi , 20

where pi denotes the probability of occupying state i, Wji is

the transition rate from state j to state i, and Wij is the rate

of reverse transition. The occupation probabilities are subject

to the constraint

i

pi = 1 21

for all times t. Assuming fixed rate parameters Wij, the

master equation dynamics of Eq. 20 always follow the re-laxation of a system toward an equilibrium state e.g., during

isothermal relaxation . For an isolated, ergodic system the

Gibbs entropy computed by Eq. 6 increases with time,48

as

required by the second law for a spontaneous, irreversible

process such as relaxation.

A recent application of the master equation approach by

Mauro and Varshneya33

considers the problem of glass tran-

sition in an energy landscape. Rather than starting in a non-equilibrium state and following the dynamics of a system as

it relaxes toward equilibrium, Mauro and Varshneya consider

a liquid system initially at equilibrium. Departure into the

nonequilibrium glassy regime is computed by solving a set

of master equations, where the transition rates are functions

of an arbitrary cooling path, Wij T t . As the system is

cooled, the relaxation time becomes longer than the observa-

tion i.e., simulation time scale. At low temperatures theoccupation probabilities pi are effectively frozen as the sys-

tem becomes trapped in local regions of the energy land-

scape. The macroscopic properties of the system at any point

in time can be computed with the weighted average,




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A T t =i

Aipi T t , 22

where Ai denotes the specific property value associated with

configuration i. For example, Eq. 22 can be used to com-

pute volume-temperature diagrams of glass-forming systemsfor different cooling paths.

34

However, Eq. 6 cannot be used in combination with the

Mauro-Varshneya approach to compute the entropy of a

glassy system, as it does not account for the broken ergodic-

ity inherent in the glassy state.4956

The definition of glass

itself has within it the assumption of an observation time that

is shorter than the relaxation time scale of the system;13

hence, glass is inherently a nonergodic system trapped in a

subset of the overall phase space available during the finite

observation time. Since Boltzmanns definition of entropy

considers only the microstates visited by a system during the

time of observation,812

application of Eq.

6

will lead to an

artificially high value of entropy. Likewise, Eq. 12 cannotbe used since it assumes discontinuously broken ergodicity,

and the glass transition under a finite cooling rate involves a

continuous loss of ergodicity.

Therefore, we propose the following hierarchical master

equation algorithm to compute the dynamics of the system

accounting for continuously-broken ergodicity.

1 Choose an appropriate initial state for the system. In the

Mauro-Varshneya approach,33

the initial configuration

is chosen to be an equilibrium liquid at the melting

temperature Tm ,

pi 0 =1Q

exp UikBTm , 23where Q is the partition function,

Q =i

exp Ui

kBTm. 24

By starting the system in equilibrium at Tm, we are able

to account for all thermal history effects on the final

nonequilibrium state.

2 Compute the master equation dynamics according tothe Mauro-Varshneya approach,

33

dp i t

dt=

ji

Wji T t pj t ji

Wij T t pi t . 25

This gives the ensemble-averaged probability of occu-

pying each of the various basins in the energy land-

scape at any point in time.

3 For any time t when we wish to compute entropy orfree energy, select a particular basin i. Construct a new

set of master equations for the conditional probabilities

fi,j,

dfi,j t

dt = kjWkj T t + t fi,k t

kj

Wjk T t + t fi,j t , 26

using an initial condition of fi,i 0 =1 for the chosen

basin and fi,ji 0 =0 for all other basins. Compute the

dynamics of the system for exactly the observation

time, tobs. The output from this step is fi,j tobs , the con-

ditional probability of the system reaching any basin j

after starting in basin i and evolving for exactly the

observation time. Note that the time scale of fi,j t is

shifted by t relative to pi t .

FIG. 1. One-dimensional fragile landscape with nine basins.

FIG. 2. Evolution of glassy and equilibrium potential energies with respect

to a time and b temperature for the one-dimensional fragile landscape ofFig. 1. We consider linear cooling from 500 to 300 K with a total cooling

time of 1.0 s.




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4 Compute the entropy of the simulation in step 3 usingEq. 18 .

5 Repeat steps 3 and 4, starting from each of the indi-

vidual basins, i 1 , 2 , . . . , .

6 Once all values of Si are computed, the overall entropyof the system is given by Eq. 19 , where the values of

pi t are taken from step 2.

7 Repeat steps 26 for all times of interest.

The above approach is a direct extension of the tech-

nique of Mauro and Varshneya.33

Steps 1 and 2 above are

identical to the Mauro-Varshneya approach, while steps 36

account for continuously broken ergodicity. Whereas thepi t values computed in step 2 give the ensemble-averagedoccupation probabilities of the various microstates, step 3

considers a particular instance of the system. By using the

initial condition fi,i 0 =1 for one particular basin, step 3 es-

sentially constitutes a measurement of system, causing it to

collapse into a single microstate. The probability of collaps-

ing into a particular microstate i is equal to the probability of

the system sampling that microstate at the time of measure-

ment, pi t . Equivalently, pi t can be considered a quench

probability, i.e., the probability of the system becoming

trapped in inherent structure i upon an instantaneous quench

to absolute zero.

The solution to the set of master equations in step 3gives the conditional probabilities of occupying all of the

various j microstates after starting in microstate i and propa-

gating for exactly the observation time tobs. No assumptions

are made regarding which states are accessible versus inac-

cessible, since the dynamics are computed using the actual

transition rates Wjk. In this manner, our approach represents

a generalization of the Palmer approach for broken ergodic-

ity, wherein microstates must be either accessible or inacces-

sible from each other, with no intermediate classification.7

Also, use of the master equations in step 3 allows us to avoid

partitioning into metabasins; in this way, our technique is

generally applicable to any energy landscape and for any

temperature path.The output of steps 35 is a set of Gibbs entropies Si for

each of the possible configurations of the system, accounting

for the actual observation time and transition rates. The ex-

pectation value of the glassy entropy is the sum of these Sivalues weighted by their corresponding quench probabilities

pi t as shown in Eq. 19 . Free energy can then be evalu-

ated using Eq. 13 or 14 .

V. RESULTS AND DISCUSSION

In this section we apply our technique to two simple test

cases. First, let us consider the one-dimensional fragile land-

FIG. 3. Evolution of glassy, statistical, and equilibrium entropy values with

respect to a time and b temperature for the system in Fig. 2. The obser-vation time is 0.01 s.

FIG. 4. Evolution of glassy, statistical, and equilibrium free energies with

respect to a time and b temperature for the system in Fig. 2.




8/11

scape of Mauro and Varshneya,33 depicted in Fig. 1, whichcontains nine basins of varying energy with no clear division

into metabasins. The transitions between basins are thermally

activated and governed by standard transition state theory

details are provided in Ref. 33 . Direct transitions are al-

lowed between all pairs of basins and are not limited to the

immediately adjacent basins. Figure 2 shows the evolution of

potential energy for this system, starting in equilibrium at

500 K and linearly cooling to 300 K over 1 s. Figure 2 a

plots the glassy and equilibrium values of potential energy

with respect to time, and Fig. 2 b plots the same values with

respect to temperature.

Figure 3 shows a similar plot for the glassy, statistical,and equilibrium values of entropy. The glassy entropy S is

computed using Eq. 19 accounting for continuously broken

ergodicity; the statistical entropy S is computed with Eq. 6with the assumption of ergodicity. At high temperatures the

relaxation time of the system is much shorter than the obser-

vation time, and the value of entropy corresponds to exactly

that of the equilibrium liquid. As the system is cooled, the

relaxation slows and the system departs from equilibrium

i.e., undergoes a glass transition . Figure 3 shows that thedeparture of the nonergodic glassy and ergodic statistical

entropies from equilibrium exhibits markedly different be-

haviors. Using the statistical formulation of Eq. 6 , the glass

transition corresponds to a freezing of the occupation prob-

abilities and hence a freezing of the entropy. However, this

does not account for the loss of ergodicity at the glass tran-

sition and results in the nonergodic state always having a

higher entropy than the corresponding ergodic state. In real-

ity, the glass transition must correspond to a loss of entropy,

since the loss of ergodicity limits the number of states that

can be visited during a finite observation time. Whereas the

ergodic formula predicts a large residual entropy of the glass

at absolute zero, in violation of Planks statement of the third

law,17

the nonergodic formalism of Sec. IV correctly predicts

zero entropy at absolute zero.

The corresponding evolution of free energy is shown inFig. 4. In both the nonergodic glassy and ergodic statistical

cases, the glass transition is a nonspontaneous process lead-

ing to an increase in free energy with respect to the equilib-

rium supercooled liquid. In the statistical case, this is due to

a freezing of the potential energy at a higher level than that

of the supercooled liquid i.e., the potential energy of the

glass corresponds to that of the liquid at the glass transition

temperature . This is also true for the nonergodic glassy free

energy, but there is a much greater increase in free energy

due to the loss of entropy upon ergodicity breaking. With the

ergodic formulation of Eq. 6 , the glass is exactly the same

macrostate as its corresponding liquid at the glass transition

FIG. 5. Relaxation of glassy and statistical a entropies and b free ener-gies in the long-time limit toward equilibrium values.

FIG. 6. Impact of observation time on a entropy and b free energyassuming a fixed cooling rate.




9/11

temperature; accounting for loss of ergodicity, as in Sec. IV,

the glass is represented as a different unstable macrostaterelative to the liquid. A thorough discussion of this topic is

the subject of a forthcoming paper by Gupta and Mauro.57

Figure 5 plots the isothermal relaxation of the noner-godic glassy and ergodic statistical entropies over long time.

Whereas both achieve the same equilibrium value in the limit

of long time, the approach to equilibrium is markedly differ-

ent. With the ergodic formulation of Eq. 6 , relaxation is

characterized by a decrease in entropy as it approaches equi-

librium; accounting for the glasss broken ergodicity, relax-

ation is characterized by an increase in entropy as ergodicity

is gradually restored over the increasing observation time.

Relaxation to the equilibrium state is a spontaneous process

in both cases since, as shown in Fig. 5 b , the free energy

decreases monotonically.

The impact of observation time on entropy and free en-

ergy is shown in Fig. 6. As expected, the glass transition

occurs at a higher temperature for shorter observation times.

In the limit of zero temperature, the systems have the same

entropy S = 0 and free energy F = U , regardless of the

observation time.

We now consider a second simple landscape, shown inFig. 7. In contrast to the previous landscape, here there is a

clear division into two metabasins, labeled A and B in the

figure. Metabasin A has three degenerate inherent structures,

and metabasin B has six; the inherent structures in B are at a

potential energy H higher than those in metabasin A. The

transition energy among the inherent structures is H/ 2 within

both metabasins. The transition energy from metabasin B to

metabasin A is four times as large. This effectively sets up

two relaxation modes: fast relaxation within a metabasin

and slow relaxation between metabasins. For this land-

scape, we plot all quantities in nondimensional units of H,

kB, and 1/, where is vibrational frequency.

FIG. 7. Model potential energy landscape with two metabasins. Metabasin

A has three degenerate inherent structures, and metabasin B has six.

FIG. 8. Relaxation of a entropy and b free energy for the two-metabasinenergy landscape in Fig. 7 after quenching from T=4 to T=2 .

FIG. 9. Relaxation of a entropy and b free energy for the two-metabasinenergy landscape in Fig. 7 after quenching from T=4 to T= 1/ 4.




10/11

Figure 8 plots the relaxation of the entropy and free en-

ergy of a glass after quenching from a nondimensional tem-perature of T=4 to T=2, as a function of normalized obser-

vation time. Relaxation proceeds as expected, with

monotonically increasing entropy and decreasing free en-

ergy. However, the relaxation takes on a different character

in Fig. 9, which plots the evolution of entropy and free en-

ergy after quenching from T=4 to T=1/4. While the free

energy still decreases monotonically as the system ap-

proaches equilibrium in agreement with the second law , theentropy actually passes through a maximum during relax-

ation and then approaches equilibrium asymptotically from

above. The maximum in entropy following this quench is

due to a crossover of the system from metabasin B to me-

tabasin A. While equilibrium statistical mechanics favorsmetabasin B at the higher temperatures T=4 and T= 2 due

to its higher degeneracy, metabasin A is preferred at the low

temperature T= 1 / 4 due to its lower energy states. Hence,

after the quench from T=4 to T=1/4, the system relaxes

from metabasin B to metabasin A and passes through a point

of maximum spreading between the two metabasins.

This crossover point corresponds to a maximum in the

intermetabasin entropy, which we define as

Sinter = kBPA ln PA kBPB ln PB, 27

where PA and PB are the total probabilities of occupying

metabasins A and B. Note that for the purposes of this dis-

cussion, the partitioning of the landscape into the two me-

tabasins A and B is kept constant, regardless of the observa-tion time. Figure 10 plots the relaxation of the

intrametabasin and intermetabasin components of entropy

for both quenches, starting with an observation time that al-

lows for transitions within the metabasins but not the

transition between metabasins i.e., zero intermetabasin en-tropy . Whereas the intermetabasin entropy increases mono-

tonically after the quench to T=2, it passes through a maxi-

mum after the quench to T=1/4 due to the crossover effect.

In both cases the relaxation is spontaneous monotonicallydecreasing free energy, in agreement with the second law of

thermodynamics and irreversible dU/dtTdS/dt, asshown in Fig. 11 .

VI. CONCLUSIONS

We have presented a statistical mechanical framework

for treating systems with continuously broken ergodicity.

Our approach is a generalization of Palmers approach7

for

discontinuously broken ergodicity and relaxes the assump-

tions of component confinement and internal ergodicity. It

accounts for the actual transition rates among microstates

and enables the direct computation of entropy loss at the

glass transition. We have implemented our approach using a

hierarchical master equation technique that builds on the

work of Mauro and Varshneya.33

Unlike the traditional mas-

FIG. 10. Relaxation of intra- and intermetabasin entropies after quenching

from T=4 to a T=2 and b T= 1/ 4.

FIG. 11. Plots of dU/dt and T dS/dt after quenching from T=4 to a T=2 and b T= 1/ 4.




11/11

ter equation approach, which assumes ergodicity, our tech-

nique is consistent with Boltzmanns definition of entropy

and the third law of thermodynamics. Using a simple one-

dimensional glass former, we demonstrated the impact of

observation time on glass transition range behavior. Depend-

ing on the energy landscape and cooling path, a crossover

effect may be observed whereby entropy passes through a

maximum during the relaxation process. In all cases relax-

ation toward equilibrium is a spontaneous and irreversibleprocess, in compliance with the second law.

ACKNOWLEDGMENT

The authors gratefully acknowledge valuable conversa-

tions with Arun K. Varshneya.

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Continuously Broken Ergodicity