We examine the density-density correlation function in a model recently
proposed to study the effect of entropy barriers in glassy dynamics. We find
that the relaxation proceeds in two steps with a fast beta process followed by
alpha relaxation. The results are physically interpreted in the context of an
adiabatic approximation which allows to separate the two processes, and to
define an effective temperature in the off-equilibrium dynamics of the model.
We investigate the behavior of the response function associated to the density,
and find violations of the fluctuation dissipation theorem.Comment: 4 Pages including 3 Figures, Revte

Angell C A

Bouchaud J P

Cugliandolo L F

Felix Ritort

Franz S

Godreche C

Gotze W

Monthus C

Odagaki T

Silvio Franz

English

arXiv

Franz, S.

Ritort, F.

NARCIS 

Relaxational processes and entropic traps in the Backgammon model

Crossref

Relaxation processes and entropic traps in the Backgammon model

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Relaxational processes and entropic traps in the Backgammon modelFranz, S.; Ritort, F.DOI10.1088/0305-4470/30/11/003Publication date1997Published inJournal of Physics. A, Mathematical and GeneralLink to publicationCitation for published version (APA):Franz, S., & Ritort, F. (1997). Relaxational processes and entropic traps in the Backgammonmodel. Journal of Physics. A, Mathematical and General, 30(L), L359-L365.https://doi.org/10.1088/0305-4470/30/11/003General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:15 Apr 2023J. Phys. A: Math. Gen.30 (1997) L359–L365. Printed in the UK PII: S0305-4470(97)80216-8LETTER TO THE EDITORRelaxation processes and entropic traps in theBackgammon modelSilvio Franz†§ and Felix Ritort‡‖† International Center for Theoretical Physics, Strada Costiera 11, PO Box 563, 34100 Trieste,Italy‡ Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65,1018 XE Amsterdam, The NetherlandsReceived 9 December 1996, in final form 5 March 1997Abstract. We examine the density–density correlation function in a model recently proposedto study the effect of entropy barriers in glassy dynamics. We find that the relaxation proceedsin two steps with a fast beta process followed by alpha relaxation. The results are physicallyinterpreted in the context of an adiabatic approximation which allows one to separate the twoprocesses and define an effective temperature in the off-equilibrium dynamics of the model. Weinvestigate the behaviour of the response function associated with the density and find violationsof the fluctuation dissipation theorem.The relaxation in supercooled liquids near the glass transition has a characteristic two-step form. Experiments on very different materials reveal the existence of a first fastrelaxation process, called beta relaxation, followed by a much slower one, called alpharelaxation [1]. One of the most striking successes of mode coupling theory [2] is its abilityto capture this phenomenon and to give a correct prediction for the relation among theexponents characterizing the two relaxations. However, we believe that comprehensionof the basic mechanisms underlying the relaxation in glasses is missing. Experiments inglasses have been recently interpreted in terms of traps models [3, 4]. In these models, thesystem evolves among traps—or metastable states—which have a lifetime that grows withdecreasing temperature, and finally diverges at the glass transition. The two-step relaxationfollows naturally from the hypothesis that equilibration inside a trap occurs much faster than‘jumps’ among different traps. How a trap may be defined and described for real systemsor microscopic models is an interesting open problem. If the traps have to be interpretedas the result of energy barriers in a rough energy landscape, one finds the difficulty thatthe relaxation should appear as a random process even on a large scale. A ‘jump’ amongtwo different traps should imply a discontinuity in various quantities as the energy or thecorrelation function. This problem was already noted in [4], where it was proposed, asa way out, that real systems could be composed of a large number of quasi-independentsubsystems leading to the observed self-averaging properties for the different quantities. Inthis direction, it can be instructive to investigate a different mechanism for slow relaxationand, in particular, the role of entropy barriers.§ E-mail address: franz@ictp.trieste.it‖ E-mail address: ritort@phys.uva.nl0305-4470/97/110359+07$19.50c© 1997 IOP Publishing Ltd L359L360 Letter to the EditorIn this letter we study the nature of density fluctuations in the so-called ‘Backgammonmodel’, a microscopic model that has proved useful in studying some mechanismsunderlying the glassy relaxation and, in particular, the role of entropy barriers. The model[5] is a Boltzmann gas withN particles in anN site space with Hamiltonian given by thetotal number of empty sites,H = −N∑r=1δnr ,0 (1)where r = 1, . . . , N denotes the sites of the space andnr the occupation number of thesite r. The system evolves following a single-particle Metropolis dynamics: at each sweepa particle to move and an arrival site are chosen at random. The particle is moved withprobability 1 if the energy does not increase and with probability exp(−β) if the move costsone unit of energy. The relaxation of the energy has been studied in detail in [5–9]; webriefly re-assume here the main results. At zero temperature the dynamics are slower andslower as time goes by: the average density of particles in the occupied states increasesand as a consequence the dynamics slows down. This observation has allowed for theidentification of fast and slow degrees of freedom: the relaxation within the occupied sitesat a given time proceeds much faster than the variation of the energy and the diffusionof ‘towers of particles’. This allows a self-consistent treatment of the dynamics which isin very good agreement with the exact solution. At large times the occupation probabilityP(n, t) mimics the equilibrium probability,P(n, t) = eβ(t)δn,0−ζ(t) ζ(t)n−1n!(2)with an effective time-dependent temperatureT (t) = 1/β(t) larger than the real one, andeffective fugacityζ(t) related toβ(t) by the condition of constant density〈nr(t)〉 = 1,i.e. eβ(t) + eζ(t) − 1 = ζ(t) eζ(t). At zero temperature, the decay of the energy follows thelaw E(t) ∼ −1+ O(1/ log(t)), whereas at small but finite temperatures this behaviour iscut off for times of the order of the relaxation timeτ ∼ exp(β)/β2 when the relaxationbecomes exponential. Probing the system on finite time scales, for example mimickingheating–cooling experiments, one finds a characteristic glassy behaviour as hysteresis loopsfor the energy [6]. Additional information on the off-equilibrium dynamics is gained bystudying the energy–energy autocorrelation function [5]CE(t, s) = 〈δnr (t),0δnr (s),0〉 − E(t)E(s)E(s)(1− E(s)) .This quantity shows aging at zero temperature with a scaling behaviourCE(t, s) ∼(s/t)12 (log(s)/ log(t)) [9]. Again, at finite temperature this scaling is observed up to timesof the order of the relaxation time.Here we concentrate our attention on the relaxation of the density–density correlationfunction, a quantity that is measured in experiments on real systems [1]. This is a betterquantity to use when studying fast processes in systems since on short timescales localdensities vary while the energy stays essentially constant.The fluctuations of the local density around its average〈nr(t)〉 = 1 are studied byintroducing the density–density correlation functionC(r)(t, s) = 〈nr(t)nr(s)〉 =∑n,mnmP (r)(n, t |m, s)P (r)(m, s) (3)and its associated responseR(r)(t, s) = δ〈nr(t)〉δhr(s)(4)Letter to the Editor L361having denoted asP (r)(n, t) the probability that stater is occupied byn particles at timet ,P (r)(n, t |m, s) the same probability conditioned to havem particles at times andhr as aninfinitesimal inhomogeneous ‘pressure field’ coupled linearly with the local density in theHamiltonian†. We will present data for the normalized correlationCnorm(t, s) = 〈nr(t)nr(s)〉 − 1〈nr(s)2〉 − 1 .Choosing a site-independent initial distribution, the correlation function remains siteindependent while the response at finite time depends on the distribution ofhr ′ n thedifferent sites in the form of a dependence onµ(r) = Prob(hr ′ > hr) andν(r) = Prob(hr ′ <hr). Despite this, in the following we will drop the indexr from the various quantities.At equilibrium, correlation and response are time translation invariant and related by thefluctuation–dissipation theorem relationT R(t − s) = ∂C(t − s)/∂s. In off-equilibriumconditions it has been often proved useful to characterize the violation of equilibrium bythe ‘fluctuation dissipation ratio’ [10, 11]x(t, s) = T R(t, s)(∂C(t, s)/∂s). (5)The evolution of the previously defined functions can be studied starting from thehierarchy of equations forP(n, t), P(n, t |m, s) and related quantities following a proceduresimilar to the one used in [7]. One can then derive closed integral equations in terms of afew functions that can be integrated numerically [7] or analytically in the long time limit [9].Details of this analysis, and in particular the full set of hierarchies for the density–densitycorrelation (3) and the associated response function (4), will be presented elsewhere. Inthis paper we integrate directly, truncating the hierarchy to some large order. In practicewe have found that for low enough temperature and not too large times the truncation atn = 100 yields excellent results.Let us now discuss the form of the density correlation function in equilibrium. Startingfrom a random initial configuration, after times of the order ofτα ≈ eβ/β2 the systemeventually reaches equilibrium. The correlation function is then time translation invariantand can be studied exactly as a Laplace transform. It turns out that for temperatures smallenough the dominant contribution to the equilibrium correlation function for all times isgiven by the superposition of two Debye processes,Ceqnorm(t) =ζ − 1+ e−t/ζζe−t/τα (6)with τα = 2((ζ −1) eζ +1)/ζ 2 ≈ 2 eβ/β2 ζ . The fast and slow processes at equilibriumare related to the relaxation within occupied states and to diffusion, respectively. We notethat the value of the plateau at equilibrium is(ζ − 1)/ζ , which corresponds to completedecorrelation within occupied states but no diffusion. The equilibrium curve forT = 0.05 isthe dashed curve in figure 1. In the time window shown in figure 1 the correlation functionstays essentially constant and equal to the plateau value(ζ −1)/ζ ' 0.94, decaying to zeroafterwards. Here the alpha relaxation time isτα ' 3× 106.Regarding the off-equilibrium correlation function, we show in figure 1 the data fortheCnorm(t, s) for T = 0.05 as a function oft − s for various values ofs. Over the timewindow we explore, the system is far from being thermalized. The integration atT = 0 onthe same time window leads to almost identical results. Theα andβ processes are well† A homogenous pressure would not have any effect in the system due to the constraint on the global density∑Nr=1 nr = N .L362 Letter to the EditorFigure 1. The density–density correlation functionCnorm(t, s) as a function oft−s for differentvalues ofs = 1, 10, 102, 103, 104 (from left to right) atT = 0.05 (continuous curves). Theexistence of a plateau separates the beta and alpha regimes. The dashed curve corresponds tothe equilibriumCeqnorm(t − s). The curves withs = 103, 104 are excellently fitted by the form(9) with b(s) = 2.20, 3.85, respectively. For the same times,ζ(s) = logP(1, s) = 8.93, 11.51.separated for large enoughs. In this case, the shape of the off-equilibrium relaxation curvecan be understood qualitatively within the framework of the adiabatic approximation. Thedecorrelation time for the density among filled states is much smaller that the time neededto diffuse and/or change sensibly the energy; there must then be a timescale such that wecan approximateP(n, t |m, s) ≈ δm,0δn,0+ (1− δm,0)(1− δn,0) P (n, s)1− P0(s) . (7)This equation reflects the simple fact that empty states never become occupied. Thesecond factor is determined by the closure condition∑n P (n, t |m, s) = 1. ConsequentlyCnorm(t, s) = ζ(s)− 1ζ(s). (8)For short times (t − s  s), ζ(s) = − log(P1(s)) does not vary too much; it is then naturalto describe the beta relaxation by form (6) just substitutingζ by ζ(s). In this way we havechecked that we correctly describe even the beginning of the alpha relaxation. However,the best combined description of slowα relaxation together withβ relaxation is obtainedby functions of the aging form:Cnorm(t, s) ≈ ζ(s)− 1+ e−(t−s)/ζ(s)ζ(s)(1+ b(s))(1+ (b(s)√t log(t)/√s log(s))) (9)a form inspired by that of the energy–energy correlation function found in [9]. At finitetemperature,b(s) is a crossover function to form (6).In figure 2 we see that the two relaxation processes are also manifest in the responsefunction. In order to characterize better the two processes in off-equilibrium conditionswe study the relation between the response and correlation during the dynamics. The firstLetter to the Editor L363Figure 2. The response function for the values ofs = 1, 10, 102, 103, 104 (from top to bottom)at T = 0.05.quantity of interest is the value of the fluctuation–dissipation ratio at equal times. We seethatx(t, t) reaches values close to one long before total equilibrium sets in. This is a furtherindication that the system is in local equilibrium and yields in a natural way the notion ofa trap in this system. Note that the traps in the Backgammon model are purely entropic.This means that the system escapes from the trap even at zero temperature when thermalexcitations are absent. In other words, the dynamics are slowed down by entropic trapswhich can be considered as metastable states with a finite (energy-dependent) lifetime.Most interestingly, we find that for large times (large values of) x(t, s) remains nearlyconstant in each of the two processes. This is shown in figure 3 where we plotx(t, s) asa function of 1− Cnorm(t, s) for different values ofs. We find thatx(t, s) is close to 1in the beta relaxational process and jumps to as-dependent smaller valuexα(s) at a valueCnorm(t, s) = qEA which remains constant in the alpha process.It is rather natural, with the adiabatic approximation in mind, to look for a possibleinterpretation ofx as the ratio between the effective and actual temperature. Unfortunatelywe found negative evidence for such interpretation of the data. For low enough temperatureand times smaller thanτα the actual value ofx(t, s) is nearly temperature independent.The step behaviour ofx(t, s) is quite reminiscent of the dynamics in spin glasses with onestep of replica symmetry breaking [10]. This in turn represents the suitable off-equilibriumgeneralization of mode coupling theory [12] in the case of a Whitney fold glass singularity,i.e. precisely the case where there are well separatedα andβ relaxations. We do not knowat this stage if this corresponds to a deep analogy or is a mere coincidence. We stress thatat equilibrium, the ending of the beta relaxation, as well as the onset of theα relaxation,are described by power laws in mode coupling theory. As is seen in (6), in our case theequilibrium relaxation is the superposition of two exponentials.Summarizing, we have shown the existence of two step relaxational processes (alphaand beta relaxation) in the Backgammon model, which is a simple model where the slowdynamics is a consequence of pure entropic barriers. We have closed the dynamicalL364 Letter to the EditorFigure 3. The fluctuation–dissipation ratiox(t, s) for different values ofs = 10, 102, 103, 104(from bottom to top) as a function of 1−Cnorm(t, s) at T = 0.05. The two plateaux correspondto the alpha and beta relaxations.equations for the density–density correlation function and the response of the system toa staggered field coupled to the density. Our results for those quantities and the fluctuationdissipation ratio allow for a clear identification of the fast (beta) and slow (alpha) relaxationprocesses in this system. The physical interpretation of these processes leads naturally to theconcept of an entropic trap. In the case of entropic traps, the entropy barrier associated withthe trap itself depends on the number of available configurations within the trap. Because theheight of the entropic barriers varies in a continuous way, the dynamics proceeds slowly butwithout macroscopic jumps in the energy. The results presented here for the Backgammonmodel are expected to apply for slowly relaxing systems where the dynamics are mainlydriven by entropy barriers (as for instance, Bose–Einstein condensation). A detailed accountof our work will be given elsewhere.FR acknowledges FOM in the Netherlands for financial support and ICTP in Trieste for itskind hospitality in the final stages of this work.References[1] For a review see Gotze W 1989Liquid, Freezing and the Glass transition (Les Houches)d J P Hansen,D Levesque and J Zinn-Justin (Amsterdam: North-Holland)Angell C A 1995 Science267 1924[2] Gotze W and Sjogren L 1992Rep. Prog. Phys.55 241 and references therein[3] Odagaki T and Hiwatari Y 1990Phys. Rev.A 41 929Odagaki T, Matsui J and Hiwatari Y 1994PhysicaA 204Odagaki T 1995Phys. Rev. Lett.75 3701[4] Bouchaud J P 1994J. PhysiqueI 4 139Monthus C and Bouchaud J P 1996J. Phys. A: Math. Gen.29 3847[5] Ritort F 1995Phys. Rev. Lett.75 1190[6] Franz S and Ritort F 1995Europhys. Lett.31 507Letter to the Editor L365[7] Franz S and Ritort F 1996J. Stat. Phys.81 131[8] Godreche C, Bouchaud J P and Mezard M 1995J. Phys. A: Math. Gen.28 L603[9] Godreche C and Luck J M 1996J. Phys. A: Math. Gen.29 1915[10] Cugliandolo L F and Kurchan J 1993Phys. Rev. Lett.71 173Cugliandolo L F and Kurchan J 1994J. Phys. A: Math. Gen.27 5649[11] Franz S and Ḿezard M 1994PhysicaA 210 48[12] Franz S and Hertz J 1995Phys. Rev. Lett.74 2114Franz S and Parisi G 1995 Mode-coupling approximations, glass theory and disordered systemsProc. Conf.‘Workshop on Nonequilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials’ (Pisa,25–29 September, 1995)ed J P Bouchaud, L F Cugliandolo, J Kurchan and M Mezard 1996Physica226A243

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Relaxational processes and entropic traps in the Backgammon modelFranz, S.; Ritort, F.DOI10.1088/0305-4470/30/11/003Publication date1997Published inJournal of Physics. A, Mathematical and GeneralLink to publicationCitation for published version (APA):Franz, S., & Ritort, F. (1997). Relaxational processes and entropic traps in the Backgammonmodel. Journal of Physics. A, Mathematical and General, 30(L), L359-L365.https://doi.org/10.1088/0305-4470/30/11/003General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:10 Nov 2022J. Phys. A: Math. Gen.30 (1997) L359–L365. Printed in the UK PII: S0305-4470(97)80216-8LETTER TO THE EDITORRelaxation processes and entropic traps in theBackgammon modelSilvio Franz†§ and Felix Ritort‡‖† International Center for Theoretical Physics, Strada Costiera 11, PO Box 563, 34100 Trieste,Italy‡ Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65,1018 XE Amsterdam, The NetherlandsReceived 9 December 1996, in final form 5 March 1997Abstract. We examine the density–density correlation function in a model recently proposedto study the effect of entropy barriers in glassy dynamics. We find that the relaxation proceedsin two steps with a fast beta process followed by alpha relaxation. The results are physicallyinterpreted in the context of an adiabatic approximation which allows one to separate the twoprocesses and define an effective temperature in the off-equilibrium dynamics of the model. Weinvestigate the behaviour of the response function associated with the density and find violationsof the fluctuation dissipation theorem.The relaxation in supercooled liquids near the glass transition has a characteristic two-step form. Experiments on very different materials reveal the existence of a first fastrelaxation process, called beta relaxation, followed by a much slower one, called alpharelaxation [1]. One of the most striking successes of mode coupling theory [2] is its abilityto capture this phenomenon and to give a correct prediction for the relation among theexponents characterizing the two relaxations. However, we believe that comprehensionof the basic mechanisms underlying the relaxation in glasses is missing. Experiments inglasses have been recently interpreted in terms of traps models [3, 4]. In these models, thesystem evolves among traps—or metastable states—which have a lifetime that grows withdecreasing temperature, and finally diverges at the glass transition. The two-step relaxationfollows naturally from the hypothesis that equilibration inside a trap occurs much faster than‘jumps’ among different traps. How a trap may be defined and described for real systemsor microscopic models is an interesting open problem. If the traps have to be interpretedas the result of energy barriers in a rough energy landscape, one finds the difficulty thatthe relaxation should appear as a random process even on a large scale. A ‘jump’ amongtwo different traps should imply a discontinuity in various quantities as the energy or thecorrelation function. This problem was already noted in [4], where it was proposed, asa way out, that real systems could be composed of a large number of quasi-independentsubsystems leading to the observed self-averaging properties for the different quantities. Inthis direction, it can be instructive to investigate a different mechanism for slow relaxationand, in particular, the role of entropy barriers.§ E-mail address: franz@ictp.trieste.it‖ E-mail address: ritort@phys.uva.nl0305-4470/97/110359+07$19.50c© 1997 IOP Publishing Ltd L359L360 Letter to the EditorIn this letter we study the nature of density fluctuations in the so-called ‘Backgammonmodel’, a microscopic model that has proved useful in studying some mechanismsunderlying the glassy relaxation and, in particular, the role of entropy barriers. The model[5] is a Boltzmann gas withN particles in anN site space with Hamiltonian given by thetotal number of empty sites,H = −N∑r=1δnr ,0 (1)where r = 1, . . . , N denotes the sites of the space andnr the occupation number of thesite r. The system evolves following a single-particle Metropolis dynamics: at each sweepa particle to move and an arrival site are chosen at random. The particle is moved withprobability 1 if the energy does not increase and with probability exp(−β) if the move costsone unit of energy. The relaxation of the energy has been studied in detail in [5–9]; webriefly re-assume here the main results. At zero temperature the dynamics are slower andslower as time goes by: the average density of particles in the occupied states increasesand as a consequence the dynamics slows down. This observation has allowed for theidentification of fast and slow degrees of freedom: the relaxation within the occupied sitesat a given time proceeds much faster than the variation of the energy and the diffusionof ‘towers of particles’. This allows a self-consistent treatment of the dynamics which isin very good agreement with the exact solution. At large times the occupation probabilityP(n, t) mimics the equilibrium probability,P(n, t) = eβ(t)δn,0−ζ(t) ζ(t)n−1n!(2)with an effective time-dependent temperatureT (t) = 1/β(t) larger than the real one, andeffective fugacityζ(t) related toβ(t) by the condition of constant density〈nr(t)〉 = 1,i.e. eβ(t) + eζ(t) − 1 = ζ(t) eζ(t). At zero temperature, the decay of the energy follows thelaw E(t) ∼ −1+ O(1/ log(t)), whereas at small but finite temperatures this behaviour iscut off for times of the order of the relaxation timeτ ∼ exp(β)/β2 when the relaxationbecomes exponential. Probing the system on finite time scales, for example mimickingheating–cooling experiments, one finds a characteristic glassy behaviour as hysteresis loopsfor the energy [6]. Additional information on the off-equilibrium dynamics is gained bystudying the energy–energy autocorrelation function [5]CE(t, s) = 〈δnr (t),0δnr (s),0〉 − E(t)E(s)E(s)(1− E(s)) .This quantity shows aging at zero temperature with a scaling behaviourCE(t, s) ∼(s/t)12 (log(s)/ log(t)) [9]. Again, at finite temperature this scaling is observed up to timesof the order of the relaxation time.Here we concentrate our attention on the relaxation of the density–density correlationfunction, a quantity that is measured in experiments on real systems [1]. This is a betterquantity to use when studying fast processes in systems since on short timescales localdensities vary while the energy stays essentially constant.The fluctuations of the local density around its average〈nr(t)〉 = 1 are studied byintroducing the density–density correlation functionC(r)(t, s) = 〈nr(t)nr(s)〉 =∑n,mnmP (r)(n, t |m, s)P (r)(m, s) (3)and its associated responseR(r)(t, s) = δ〈nr(t)〉δhr(s)(4)Letter to the Editor L361having denoted asP (r)(n, t) the probability that stater is occupied byn particles at timet ,P (r)(n, t |m, s) the same probability conditioned to havem particles at times andhr as aninfinitesimal inhomogeneous ‘pressure field’ coupled linearly with the local density in theHamiltonian†. We will present data for the normalized correlationCnorm(t, s) = 〈nr(t)nr(s)〉 − 1〈nr(s)2〉 − 1 .Choosing a site-independent initial distribution, the correlation function remains siteindependent while the response at finite time depends on the distribution ofhr ′ n thedifferent sites in the form of a dependence onµ(r) = Prob(hr ′ > hr) andν(r) = Prob(hr ′ <hr). Despite this, in the following we will drop the indexr from the various quantities.At equilibrium, correlation and response are time translation invariant and related by thefluctuation–dissipation theorem relationT R(t − s) = ∂C(t − s)/∂s. In off-equilibriumconditions it has been often proved useful to characterize the violation of equilibrium bythe ‘fluctuation dissipation ratio’ [10, 11]x(t, s) = T R(t, s)(∂C(t, s)/∂s). (5)The evolution of the previously defined functions can be studied starting from thehierarchy of equations forP(n, t), P(n, t |m, s) and related quantities following a proceduresimilar to the one used in [7]. One can then derive closed integral equations in terms of afew functions that can be integrated numerically [7] or analytically in the long time limit [9].Details of this analysis, and in particular the full set of hierarchies for the density–densitycorrelation (3) and the associated response function (4), will be presented elsewhere. Inthis paper we integrate directly, truncating the hierarchy to some large order. In practicewe have found that for low enough temperature and not too large times the truncation atn = 100 yields excellent results.Let us now discuss the form of the density correlation function in equilibrium. Startingfrom a random initial configuration, after times of the order ofτα ≈ eβ/β2 the systemeventually reaches equilibrium. The correlation function is then time translation invariantand can be studied exactly as a Laplace transform. It turns out that for temperatures smallenough the dominant contribution to the equilibrium correlation function for all times isgiven by the superposition of two Debye processes,Ceqnorm(t) =ζ − 1+ e−t/ζζe−t/τα (6)with τα = 2((ζ −1) eζ +1)/ζ 2 ≈ 2 eβ/β2 ζ . The fast and slow processes at equilibriumare related to the relaxation within occupied states and to diffusion, respectively. We notethat the value of the plateau at equilibrium is(ζ − 1)/ζ , which corresponds to completedecorrelation within occupied states but no diffusion. The equilibrium curve forT = 0.05 isthe dashed curve in figure 1. In the time window shown in figure 1 the correlation functionstays essentially constant and equal to the plateau value(ζ −1)/ζ ' 0.94, decaying to zeroafterwards. Here the alpha relaxation time isτα ' 3× 106.Regarding the off-equilibrium correlation function, we show in figure 1 the data fortheCnorm(t, s) for T = 0.05 as a function oft − s for various values ofs. Over the timewindow we explore, the system is far from being thermalized. The integration atT = 0 onthe same time window leads to almost identical results. Theα andβ processes are well† A homogenous pressure would not have any effect in the system due to the constraint on the global density∑Nr=1 nr = N .L362 Letter to the EditorFigure 1. The density–density correlation functionCnorm(t, s) as a function oft−s for differentvalues ofs = 1, 10, 102, 103, 104 (from left to right) atT = 0.05 (continuous curves). Theexistence of a plateau separates the beta and alpha regimes. The dashed curve corresponds tothe equilibriumCeqnorm(t − s). The curves withs = 103, 104 are excellently fitted by the form(9) with b(s) = 2.20, 3.85, respectively. For the same times,ζ(s) = logP(1, s) = 8.93, 11.51.separated for large enoughs. In this case, the shape of the off-equilibrium relaxation curvecan be understood qualitatively within the framework of the adiabatic approximation. Thedecorrelation time for the density among filled states is much smaller that the time neededto diffuse and/or change sensibly the energy; there must then be a timescale such that wecan approximateP(n, t |m, s) ≈ δm,0δn,0+ (1− δm,0)(1− δn,0) P (n, s)1− P0(s) . (7)This equation reflects the simple fact that empty states never become occupied. Thesecond factor is determined by the closure condition∑n P (n, t |m, s) = 1. ConsequentlyCnorm(t, s) = ζ(s)− 1ζ(s). (8)For short times (t − s  s), ζ(s) = − log(P1(s)) does not vary too much; it is then naturalto describe the beta relaxation by form (6) just substitutingζ by ζ(s). In this way we havechecked that we correctly describe even the beginning of the alpha relaxation. However,the best combined description of slowα relaxation together withβ relaxation is obtainedby functions of the aging form:Cnorm(t, s) ≈ ζ(s)− 1+ e−(t−s)/ζ(s)ζ(s)(1+ b(s))(1+ (b(s)√t log(t)/√s log(s))) (9)a form inspired by that of the energy–energy correlation function found in [9]. At finitetemperature,b(s) is a crossover function to form (6).In figure 2 we see that the two relaxation processes are also manifest in the responsefunction. In order to characterize better the two processes in off-equilibrium conditionswe study the relation between the response and correlation during the dynamics. The firstLetter to the Editor L363Figure 2. The response function for the values ofs = 1, 10, 102, 103, 104 (from top to bottom)at T = 0.05.quantity of interest is the value of the fluctuation–dissipation ratio at equal times. We seethatx(t, t) reaches values close to one long before total equilibrium sets in. This is a furtherindication that the system is in local equilibrium and yields in a natural way the notion ofa trap in this system. Note that the traps in the Backgammon model are purely entropic.This means that the system escapes from the trap even at zero temperature when thermalexcitations are absent. In other words, the dynamics are slowed down by entropic trapswhich can be considered as metastable states with a finite (energy-dependent) lifetime.Most interestingly, we find that for large times (large values of) x(t, s) remains nearlyconstant in each of the two processes. This is shown in figure 3 where we plotx(t, s) asa function of 1− Cnorm(t, s) for different values ofs. We find thatx(t, s) is close to 1in the beta relaxational process and jumps to as-dependent smaller valuexα(s) at a valueCnorm(t, s) = qEA which remains constant in the alpha process.It is rather natural, with the adiabatic approximation in mind, to look for a possibleinterpretation ofx as the ratio between the effective and actual temperature. Unfortunatelywe found negative evidence for such interpretation of the data. For low enough temperatureand times smaller thanτα the actual value ofx(t, s) is nearly temperature independent.The step behaviour ofx(t, s) is quite reminiscent of the dynamics in spin glasses with onestep of replica symmetry breaking [10]. This in turn represents the suitable off-equilibriumgeneralization of mode coupling theory [12] in the case of a Whitney fold glass singularity,i.e. precisely the case where there are well separatedα andβ relaxations. We do not knowat this stage if this corresponds to a deep analogy or is a mere coincidence. We stress thatat equilibrium, the ending of the beta relaxation, as well as the onset of theα relaxation,are described by power laws in mode coupling theory. As is seen in (6), in our case theequilibrium relaxation is the superposition of two exponentials.Summarizing, we have shown the existence of two step relaxational processes (alphaand beta relaxation) in the Backgammon model, which is a simple model where the slowdynamics is a consequence of pure entropic barriers. We have closed the dynamicalL364 Letter to the EditorFigure 3. The fluctuation–dissipation ratiox(t, s) for different values ofs = 10, 102, 103, 104(from bottom to top) as a function of 1−Cnorm(t, s) at T = 0.05. The two plateaux correspondto the alpha and beta relaxations.equations for the density–density correlation function and the response of the system toa staggered field coupled to the density. Our results for those quantities and the fluctuationdissipation ratio allow for a clear identification of the fast (beta) and slow (alpha) relaxationprocesses in this system. The physical interpretation of these processes leads naturally to theconcept of an entropic trap. In the case of entropic traps, the entropy barrier associated withthe trap itself depends on the number of available configurations within the trap. Because theheight of the entropic barriers varies in a continuous way, the dynamics proceeds slowly butwithout macroscopic jumps in the energy. The results presented here for the Backgammonmodel are expected to apply for slowly relaxing systems where the dynamics are mainlydriven by entropy barriers (as for instance, Bose–Einstein condensation). A detailed accountof our work will be given elsewhere.FR acknowledges FOM in the Netherlands for financial support and ICTP in Trieste for itskind hospitality in the final stages of this work.References[1] For a review see Gotze W 1989Liquid, Freezing and the Glass transition (Les Houches)d J P Hansen,D Levesque and J Zinn-Justin (Amsterdam: North-Holland)Angell C A 1995 Science267 1924[2] Gotze W and Sjogren L 1992Rep. Prog. Phys.55 241 and references therein[3] Odagaki T and Hiwatari Y 1990Phys. Rev.A 41 929Odagaki T, Matsui J and Hiwatari Y 1994PhysicaA 204Odagaki T 1995Phys. Rev. Lett.75 3701[4] Bouchaud J P 1994J. PhysiqueI 4 139Monthus C and Bouchaud J P 1996J. Phys. A: Math. Gen.29 3847[5] Ritort F 1995Phys. Rev. Lett.75 1190[6] Franz S and Ritort F 1995Europhys. Lett.31 507Letter to the Editor L365[7] Franz S and Ritort F 1996J. Stat. Phys.81 131[8] Godreche C, Bouchaud J P and Mezard M 1995J. Phys. A: Math. Gen.28 L603[9] Godreche C and Luck J M 1996J. Phys. A: Math. Gen.29 1915[10] Cugliandolo L F and Kurchan J 1993Phys. Rev. Lett.71 173Cugliandolo L F and Kurchan J 1994J. Phys. A: Math. Gen.27 5649[11] Franz S and Ḿezard M 1994PhysicaA 210 48[12] Franz S and Hertz J 1995Phys. Rev. Lett.74 2114Franz S and Parisi G 1995 Mode-coupling approximations, glass theory and disordered systemsProc. Conf.‘Workshop on Nonequilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials’ (Pisa,25–29 September, 1995)ed J P Bouchaud, L F Cugliandolo, J Kurchan and M Mezard 1996Physica226A243

Relaxation processes and entropic traps in the Backgammon model

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