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PHYSICAL REVIEW E JULY 1999VOLUME 60, NUMBER 1

Ergodicity-breaking transition and high-frequency response in a simple free-energy landscape

M. Ignatiev and Bulbul ChakrabortyThe Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02254

~Received 29 January 1999!

We present a simple dynamical model described by a Langevin equation in a piecewise parabolic free-energy landscape, modulated by a temperature-dependent overall curvature. The zero-curvature point marks atransition to a phase with broken ergodicity. The frequency-dependent response near this transition is reminis-cent of observations near the glass transition.@S1063-651X~99!51007-6#

PACS number~s!: 64.70.Pf, 64.60.My, 82.20.Mj

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Supercooled liquids can undergo an ergodicity-breaktransition at which the time required to adequately samthe allowed phase space becomes longer than observtimes @1,2# and the liquid freezes into a glassy state@1#.Recent experiments indicate that the approach to this gtransition has some universal features@3# when viewed interms of the frequency-dependent response of the systemboth supercooled liquids@3# and spin glasses@4#, the ap-proach to the glass transition is characterized by two genfeatures, a frequency-independent behavior at high frequcies and the presence of three distinct regimes in the reation spectrum. These features are different from thoseserved near a critical point where the high-frequenresponse is thought to be uninteresting@5#. Existing theoriesof the glass transition@1,6# offer no satisfactory explanatioof the unusual frequency-dependent response. In this RCommunication, we present a simple, dynamical mowhich is able to describe the frequency-dependent respobserved near the glass transition.

Time-dependent fluctuations in a system approachincritical point are described, within the spirit of Landatheory, by a Langevin equation for the order parameter@5#.Adopting a similar framework for describing the fluctuationear a glass transition, we study the Langevin dynamicscollective variable,f, relaxing in a multivalleyed free-energy landscape. This collective variable is envisioned toone of the slow variables in a viscoelastic liquid, such adensity fluctuation mode@6# or a component of the averagstrain field @7#. The multivalleyed free-energy surfacemodulated by a temperature-dependent overall curvatThe introduction of this curvature was inspired by simutions of a frustrated spin system in which the role off isplayed by an elastic strain field, and the curvature arises fa coupling between the frustrated spin variables andstrain field @8#. The vanishing of the overall curvatureidentified in our model with the glass transition.

A schematic picture of the free energy is shown in Fig.All valleys ~including the megavalley! in the free-energy surface are assumed to be parabolic. Each valley is parametby its curvaturer n , width Dn , position of the centerfn

0 andposition of the minimumCn . Consequently, each valley icharacterized not only by the time it takes to escape frombut also by its internal relaxation time. The set of$Cn% isfixed by the requirement that the free energy,F(f), is acontinuous function. To simplify the picture even further, wset allDn5D, which then automatically fixesfn

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curvatures of the valleys are taken to be independent ranvariables picked from a distributionP(r ,n). This defines afree-energy function

F~f!51

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2 (n52`

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mn$r n~f2fn0!21Cn%. ~1!

Here$mn% is the set of functions specifying the range of easubwell, i.e.,mn51 if fn

02D/2<f<fn01D/2 and zero oth-

erwise. The curvature of the megavalley is denoted byR.The dynamics is modeled by relaxation in this free-ene

surface and is defined by the Langevin equation

]f

]t52Rf2( mnr n~f2fn

0!1h~ t !, ~2!

whereh is a Gaussian noise with zero average and varia^h(t)h(t8)&5Gd(t2t8). The temperature scale is set byb5D2/G. In the absence of any subvalley structure,~all r n50), Eq. ~2! results in a Debye relaxation spectrum with

FIG. 1. Free-energy landscape for two different values ofR witha fixed distribution of$r n%. The inset shows how the overall curvature modifies the heights of the barriers between the valleys.

R21 ©1999 The American Physical Society

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R22 PRE 60M. IGNATIEV AND BULBUL CHAKRABORTY

relaxation time of 1/R. If R is taken to be of the form assumed in Landau theory, such that it vanishes linearly atcritical temperature, then Eq.~2! provides a mean-field description of critical slowing down@5#. The effect of the sub-valley structure on the relaxation spectrum, and the naand existence of phase transitions in the two-dimensiospace spanned byR andb are the subjects of this paper.

The dynamics of systems approaching the glass transhas been modeled previously by random walks in the eronment of traps@9#. What distinguishes our model is thpresence of an overall curvature modulating the landscand the description of the dynamicswithin the valleys. Asshown below, both these features have nontrivial effectsthe relaxation spectrum.

Specific features of the distributionP(r ,n) affect the de-tailed nature of the response. The assumption that the cuture of each valley is uncorrelated with its position,P(r ,n);P(r ) is the easiest to implement and is the scenariowe examine in detail. A natural candidate forP(r ) is anexponential distributionP(r )5e2b0r /b0, observed in manyspin-glass models@9#. The frustrated spin model that motvated the introduction of the overall curvature also indicaan exponential distribution of barrier heights@8#.

The occurrence of an ergodicity-breaking transition in omodel can be demonstrated explicitly@11#. The equilibriumprobability distribution, predicted by Eq. ~2!, isexp@2bF(f)# as long as the integral of this function overfremains finite@10#. A simple calculation@11# shows that thisintegral diverges as 1/(b02b)AR. As R→0, an equilibriumdistribution can no longer be defined@10# and correlationfunctions become power law in nature with no characteritime scale. The pointR50, b5b0 is special in that thepower-law correlations no longer decay to zero and the stem falls out of equilibrium. The approach to this specpoint, where ergodicity is broken, can be studied by anaing the equilibrium correlation function,C(t)5^f(0)f(t)&,and the response function associated with it through the fltuation dissipation relation@5#. In the absence of the overacurvature, the 1/AR factor gets replaced by the system sizthe probabilitydensitydiverges asb→b0, and the correla-tion functions are always characterized by power laws@9#.The line with R50 is, therefore, special and in the presework, we are mainly interested in studying theR50, b5b0 point as it is approached along a generic line in(R,b) space.

The dynamical processes contributing to the correlatfunction can be roughly subdivided into the internal relaation within each subvalley and the activated motiontween subvalleys, modulated by the presence of the ovecurvatureR. The correlation functions along theR50 linecan be calculated by using the well known mapping ofLangevin equation to a quantum mechanical model@10# andleads to@11#,

C~ t !5(n

ebr nS e2r nt

r n1e2(t/b)e2br nD . ~3!

Physically, the first term within the parentheses represethe superposition of independent relaxations within eachley, while the second term represents activated motion.

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Generalization of these results to nonzeroR involvesevaluating the eigenvalue spectrum of the quantum Hamtonian as modified by the overall curvature. Taking the cvature into account perturbatively leads to the following geeralization ofC(t):

Ctrap~ t !5(n

u~DFn!e2bFnminS e2Rnt

Rn1e2(t/b)e2bDFnD .

~4!

Here Rn5r n1R, Fnmin5n2R2r n , and theu-function ex-

cludes the valleys for which the effective free-energy barrDFn5(Rn/2)(1/22nR/Rn)2, becomes zero due to the preence of the overall curvature. The contribution of these zebarrier valleys cannot be calculated perturbatively. The tievolution of f involves hopping over barriers, relaxatiowithin the subvalleys, and, in the ‘‘free’’ regions, relaxatioin response to only the overall curvature. Since the freegion is expected to be small in the vicinity ofR50, we havesimplified this complex relaxation process by including tfree relaxation within a mean-field type approximation thneglects the positional relationship between the free valland the valleys with barriers. The total correlation functithen reduces to a sum ofCtrap andCf ree with

Cf ree~ t !5(n

u~2DFn!e2bn2R2Rt

R. ~5!

One of the most interesting aspects of our model ishigh-frequency response. The origin of this can be undstood from an analysis of the results along theR50 line @cf.Eq.~3!#. The hopping term inC(t) is then identical to the oneanalyzed in@9#, and in frequency domain, leads to an imagnary part of the susceptibility,x9(v)5vC̄(v), of the formv (b02b)/b at low frequencies and decaying as 1/v at highfrequencies. There is, however, a new feature that arfrom the internal dynamics and drastically changes the hifrequency behavior. The internal relaxation part ofC̄(v) isgiven by

C̄int~v!5Er*

`

dre2(b02b)r

v21r 2. ~6!

The contribution of this term tox9(v) behaves as@p/22arctan(r * /v)# @12# for v<1/(b2b0); a function that de-cays extremely slowly with frequency asb→b0. The totalfrequency-dependent response, therefore, behavesv (b02b)/b for v→0 and is a slowly decaying function ahigh frequencies.

The overall curvature changes the effective barriheights and the effective distribution ofr. The parameterspace of our model is spanned byR andb. In order to sim-plify the analysis, we study the response along the familylines defined bya(b02b)5b0bR, with a51 for most ofthe calculations. The relaxation spectrum, obtained frCtrap(t), is given by

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RAPID COMMUNICATIONS

PRE 60 R23ERGODICITY-BREAKING TRANSITION AND HIGH- . . .

C̄trap~v!5(n

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dre2(b02b)r

3S 1

v21r 21e2bDFn/b

v21e22bDFn/b2D . ~7!

This expression in not analytically tractable and the compfrequency-dependent response can be obtained only fronumerical calculation. The basic features can, howeverunderstood from a simplified analysis. For smallR, the pri-mary effect of the overall curvature is to introduce an upand a lower cutoff to the distributionP(r ). Thee2bn2R termof Eq. ~7! makes the system finite with an ‘‘effective’’ number of wells.1/AR, which in turn leads to an upper cutoof r max52 logR @11#. The lower cutoff arises from theelimination of the zero-barrier valleys that is the sourcethe lower limit on the integral in Eq.~7!. The primary effectof the upper cutoff is to alter the exponent of the lofrequency power law from (b02b)/b to one which goes tozero more rapidly withR. The high-frequency responseaffected by the overall width of the distribution ofr whichincreases asR→0 and leads to an essentially frequencindependent response forAR!v!1/R. A less significant ef-fect of R is to alter the exponent of the power law in thfrequency range.

The results from a complete numerical analysis of Eq.~7!are shown in Fig. 2. It is clear from these results that ther

FIG. 2. Imaginary part of the susceptibility,x9(v)

5vC̄trap(v) @cf. Eq. ~7!# for different values ofR. The peak inten-sities have been matched. Frequencies have been normalized tv0,a microscopic frequency scale determined by the parameters omodel. The inset shows the full relaxation spectra,including thecontribution from the ‘‘free’’ valleys forR50.1, 0.05 and 0.01. Thelong-dashed lines show the high-frequency power law extendingto v.a/R. The parametera51 in the main figure buta5100 inthe inset.

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a slowly decaying high-frequency response. The exponcharacterizing this high-frequency power law is a functionR and approaches zero asR→0. It is also evident that theresponse is characterized by a low-frequency power law wthe exponent approaching zero asR→0.

The free part of the relaxation spectrum is given by

C̄f ree~v!51

R21v2 (n

e2bn2RE0

nR

dre2b0r , ~8!

and leads to a Debye spectrum with peak atv5R. There isa tradeoff between the hopping dynamics and free relaxatwith the hopping contribution decreasing asR increases, andmore valleys become effectively ‘‘free.’’ The inset of Fig.demonstrates the effect of the free part with a shift in pefrequency as well as the appearance of an intermediategime.

Three distinct regimes of the response emerge: a lfrequency power law, associated with hopping betweenferent valleys; a Debye-like peak coming from barrierlerelaxation; and a high-frequency power-law decay resultfrom the superposition of many single-relaxation-time pcesses. The high-frequency power law is intimately relatethe fact that the system explores more valleys as the cuture decreases. The low-frequency power law is controby the effective distribution of barriers, which also depenon R. The curvature, therefore, controls both the high- alow-frequency behavior of the dynamical response. This pture of the dynamical response is very similar to the expmentally observed response in spin glasses@4#, with the re-sponse flattening out at both high- and low-frequency eand the peak moving to zero, but slower than exponentia

In supercooled liquids, only the high-frequency responflattens out, and the peak shifts towards zero accordingVogel-Fulcher law@1,3#. In our model, this difference couldbe ascribed to a difference in the nature of correlations indistributionP(r ,n). In a structural glass, the crystalline stais the absolute global free-energy minimum. The valleysour model correspond to metastable states and the megley represents the states accessible in the supercooled pwith the crystalline minimum lying outside this region. Thsuggests that the depth of a valley and its position are colated, with the deeper valleys situated further away fromminimum of the megavalley. A correlation of this form iP(r ,n) alters theR dependence of the upper cutoff in thdistribution of barrier heights and leads to a maximumcape timetesc;exp(1/Ra). Herea.0 is the exponent of apower law describing the correlation between the depththe position of valleys. For times longer thattesc, there isonly barrier-free motion in our model. At frequencies lowthan vc5(tesc)

21, we, therefore, predict thatx9(v);v,with no flattening out of the low-frequency response. Tupper cutoff does not have a large influence on the respoarising from the internal dynamics of the valleys. The higfrequency cutoff is still'1/R, as seen from Eq.~6!.

The original motivation for constructing the dynamicmodel came from observations in a nonrandomly frustraspin system whose phenomenology is remarkably similastructural glasses@8#. Simulations of this system indicatedfree-energy surface with an overall curvature, and the vishing of this curvature was accompanied by the appeara

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R24 PRE 60M. IGNATIEV AND BULBUL CHAKRABORTY

of broken ergodicity and ‘‘aging’’@8#. The dynamics waseffectively one dimensional, with the shear strain playingrole of f. This frustrated spin system, can, therefore,viewed as a microscopic realization of the dynamical mopresented here, and could provide the connection betwour simple toy model and the dynamics of real glasses.

In conclusion, we have demonstrated that the basictures of the frequency-dependent response near a glasssition can be understood on the basis of a multivalleyed frenergy surface with an overall curvature, which goes to zat the glass transition. This is reminiscent of models whthe glass transition is associated with an instability@13#. In

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our model, the spectrum crosses over from being pure Deat large curvatures to one with three distinct regimes. Tasymptotic, high-frequency power law is characterized byexponent approaching zero as the curvature approachesOur analysis also suggests that the relaxation spectra ofglasses and structural glasses can be described by theunderlying model with different correlations in the distribtion of valleys.

The authors would like to acknowledge the hospitalityITP, Santa Barbara, where a major portion of this work wperformed. This work has been partially supported by NGrant No. NSF-DMR-9520923.

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@1# M. D. Ediger, C. A. Angell, and Sidney R. Nagel, J. PhyChem.100, 13 200~1996!.

@2# D. Thirumalai, R. D. Mountain, and T. R. Kirkpatrick, PhyRev. B 39, 3563 ~1989!; R. D. Mountain and D. ThirumalaiPhysica A192, 543 ~1993!, and references therein.

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@4# D. Bitko et al., Europhys. Lett.33, 489 ~1996!.@5# Nigel Goldenfeld, Lectures on Phase Transitions and th

Renormalization Group~Addison-Wesley, New York, 1992!.@6# U. Bengtzelius, W. Gotze, and A. Sjolander, J. Phys. C17,

5915 ~1984!; W. Kob, e-print cond-mat/9702073, and refeences therein.

@7# S. Dattagupta and L. Turski, Phys. Rev. E47, 1222~1993!.@8# Lei Gu and Bulbul Chakraborty, Mater. Res. Soc. Symp. Pr

455, 229 ~1997!, and~unpublished!.@9# C. Monthus and J. P. Bouchaud, J. Phys. A29, 3847 ~1996!;

J. P. Bouchaud, J. Phys. I2, 1705~1992!.@10# J. Zinn-Justin,Quantum Field Theory and Critical Phenom

enon~Oxford, England, 1989!.@11# M. Ignatiev, Ph.D. thesis, Brandeis University, 1998~unpub-

lished!.@12# The lower cutoffr * is introduced to ensure that one can defi

an internal relaxation process. This arbitrary cutoff is necsary only forR50.

@13# A. I. Mel’cuk et al., Phys. Rev. Lett.75, 2522~1995!.