Slow logarithmic relaxation in models with hierarchically constrained dynamics

Slow logarithmic relaxation in models with hierarchically constrained dynamics J. J. Brey and A. Prados Fı ´sica Teo ´rica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 共Received 14 September 2000; published 24 January 2001兲 A general kind of model with hierarchically constrained dynamics is shown to exhibit logarithmic anomalous relaxation similar to a variety of complex strongly interacting materials. The logarithmic behavior describes most of the decay of the response function. DOI: 10.1103/PhysRevE.63.021108 PACS number共s兲: 82.40.Bj. 05.40.⫺a, 31.70.Hq In past years, an anomalous slow relaxation that can be accurately described by a logarithmic law has been found in the time evolution of a variety of complex strongly interacting materials. This includes spin-glasses 关1,2兴, granular materials 关3–6兴, structural glasses 关7–9兴, and protein models 关11,10兴. In all the above cases, the decay of some perturbation ␾ (t) can be fitted at intermediate times by an expression of the form ␾ 共t兲⫽C1⫺C2lnt,共1兲 where C1and C2are constants, and may depend on the macroscopic variables 共temperature, compactivity, etc.兲characterizing the state of the system. The time range in which the logarithmic law accurately describes the relaxation covers most of the change of ␾ (t) from its initial value to the final one. Palmer, Stein, Abrahams, and Anderson 共PSAA兲in a seminal paper 关12兴described some dynamical models for relaxation in complex systems with strong interactions. The models were based in a hierarchically constrained dynamics so that the relaxation of the system involves a sequential series of correlated processes. Although PSAA concentrated on those physical systems showing ‘‘stretched exponential’’ relaxation, it is clear that the general picture emerging from hierarchically constrained dynamics can be translated into very different physical situations. As a consequence, the above models provide a very useful tool to investigate many kinds of ‘‘anomalous relaxation’’ in complex systems. In particular, the picture of hierarchical constrained dynamics seems to be physically appropriate to describe many of the complex systems showing logarithmic relaxation. Let us consider, as a particular example the densification of powders and structural glasses at high hydrostatic pressure studied in Refs. 关4,5,7,9兴. External pressure causes some selected mesoscopic regions of the system to reach a more compact state very quickly. The nearest neighbor regions are influenced by this rearrangement, and can also reorganize themselves. Of course, these new structural changes will affect the regions outside the nearest neighbors, allowing them to relax, and so on. The above discussion suggests that the relaxation in this kind of system is slow because it consists of a large number of correlated reorganization steps with an increasing characteristic time. In other words, it has a hierarchical structure, with the faster degrees of freedom 共associated with the ‘‘inner’’ regions兲controlling the relaxation of the slower degrees of freedom 共associated with the ‘‘outer’’ regions兲. A similar picture has been already suggested in Refs. 关4,5兴. The aim of this paper is to introduce a quite general class of hierarchical models, which seem appropriate to describe this kind of system, in order to derive the logarithmic relaxation of Eq. 共1兲. Like PSAA, we will formulate our model in the most general way, hoping it characterizes the behavior of a specific class of systems. We assume that the state of the system can be given by means of Nmodes. They do not correspond to a particle description of the system, but to some mesoscopic level of description. In this sense, Ndoes not scale with the size of the system, but it is assumed to have a definite value, independent of the size, for a large enough 共macroscopic兲system. Of course, the specific value depends on the system and the physical situation we are dealing with. The Nmodes can be grouped into a discrete series of levels, n⫽0,1,2,.... The number of modes belonging to level nwill be denoted by Nn, so that N⫽兺nNn. Moreover, modes in level nwill be indicated by the variables Si (n),i ⫽1,2,...,Nn. For the densification experiments of Refs. 关4,5,7,9兴, we can regard Si (n)as a pseudospin variable, with its two possible values corresponding to different volumes of the mesoscopic regions. The dynamics of the system is defined in a hierarchical way as follows. A mode in level n ⫹1 can only modify its state if the configuration of modes in level nbelongs to a well-defined subset of all the possible configurations of the modes in that level. Clearly, this slows down the modes in a given level, as compared with those in the previous level. More specifically, the average transition rate Wn⫹1for a mode in level n⫹1 is assumed to have the form 关12兴 Wn⫹1⫽WnPn,共2兲 where Pnis the probability that the modes in level nare in any of the configurations allowing us to relax level n⫹1. Our choice for Pnwill be Pn⫽exp共⫺Nn⌬ ␮ 兲,共3兲 with ⌬ ␮ being a dimensionless free energy defining the activation barrier per mode. Equation 共3兲appears quite naturally if Pnis evaluated from the equilibrium distribution of the system under consideration, provided that the subset of configurations of level nallowing us to relax level n⫹1 involves all the modes 共or an extensive part of them兲in level n. Of course, the free energy ⌬ ␮ must be appropriately in- PHYSICAL REVIEW E, VOLUME 63, 021108 1063-651X/2001/63共2兲/021108共4兲/$15.00 ©2001 The American Physical Society63 021108-1 terpreted in each particular situation. For instance, in the densification experiments at high pressure 关4,5,7,9兴it would be an activation enthalpy divided by the temperature of the sample. On the other hand, in a granular system, it could be understood as an ‘‘effective volume’’ divided by the compactivity 关13–16兴. Thus, the corresponding average relaxation time of the mode is ␶ n⫹1⫽ ␶ nexp共Nn⌬ ␮ 兲.共4兲 For the sake of simplicity, the barrier ⌬ ␮ is assumed to be independent of the level nof the mode and also of the number of modes in the level Nn. Iteration of Eq. 共4兲yields ␶ n⫽ ␶ 0exp 冉 ⌬ ␮ 兺 m⫽0 n⫺1 Nm 冊 .共5兲 The range of variation of nand the values of the populations Nncharacterize the complexity of the time evolution of the system. In principle, there is no need for a maximum value nmax of n, i.e., nmax may be infinite. On the other hand, the systems we are interested in show slow relaxation, which requires a large relaxation time. In this way, it is necessary that ␶ max ⫽lim n→nmax ␶ nⰇ ␶ 0.共6兲 This condition for slow relaxation was already pointed out by PSAA 关12兴. We are interested in the relaxation of the quantity Q共t兲⫽1 N兺 n兺 i⫽1 Nn 具 Si (n)共t兲 典 ,共7兲 where the angular brackets denote configuration average. Note that in the context of the densification experiments, 关4,5,7,9兴,Q(t) would be proportional to the actual volume of the sample at time t, with the physical meaning given above to the variables Si (n). The corresponding relaxation function is defined in terms of the asymptotic long-time value Q(⬁) as ␾ 共t兲⫽Q共t兲⫺Q共⬁兲 Q共0兲⫺Q共⬁兲⫽兺 n ␻ nexp 冉 ⫺t ␶ n 冊 ,共8兲 with ␻ n⫽Nn/Nbeing the fraction of modes in level n. Combination of Eqs. 共5兲and 共8兲gives ␾ 共t兲⫽兺 n⫽0 n⫽nmax ␻ nexp 冋 ⫺t ␶ 0exp 冉 ⫺ ␨ 兺 m⫽0 n⫺1 ␻ m 冊 册 ,共9兲 where ␨ ⫽N⌬ ␮ .共10兲 Next we will assume that ␻ nchanges very smoothly with n. This allows us to introduce a continuum limit for the distribution of relaxation times, which is expected to be closer to real systems than the discrete level model considered up to now. We define a function ␻ ˜ (x)by ␻ ˜ (n ⑀ )⫽ ␻ n, with ⑀ Ⰶ1, so that the variable xn⫽n ⑀ is almost continuum and the sum over ncan be replaced by integrals. Let us introduce a new variable xmax ⫽nmax ⑀ u⫽ 冕 0 xdx⬘ ␻ ˜ 共x⬘兲 冕 0 xmax dx⬘ ␻ ˜ 共x⬘兲 ,共11兲 0⭐u⭐1, which measures the fraction of the total number of modes Nthat are in levels up to a value n⫽x/ ⑀ . It is then a simple task to transform Eq. 共9兲into ␾ 共t兲⫽ 冕 0 1duexp 冉 ⫺t ␶ 0e⫺ ␨ u 冊 ,共12兲 showing that the decay of ␾ can be characterized by only two parameters, namely, ␶ 0and ␨ . The former fixes the time scale, while the latter determines the actual shape of the relaxation. The elementary relaxation times given by Eq. 共5兲 are ␶ (u)⫽ ␶ 0exp( ␨ u) and, in particular, ␶ max ⫽ ␶ 0exp( ␨ ). Thus, the condition for slow relaxation, Eq. 共6兲, implies that exp( ␨ )Ⰷ1. The analysis of Eq. 共12兲is greatly simplified by realizing that it is equivalent to ␾ 共t兲⫽1 ␨ 冋 E1 冉 t ␶ 0e⫺ ␨ 冊 ⫺E1 冉 t ␶ 0 冊 册 ,共13兲 where E1(z) is the integral exponential function 关17兴. There are two natural time scales in Eq. 共13兲. One is defined by t/ ␶ 0, while the other is a slow time scale defined by s ⫽(t/ ␶ 0)e⫺ ␨ Ⰶ(t/ ␶ 0). In the following, we will use the asymptotic expansions of the exponential integral function from Ref. 关17兴. For very short times such that t/ ␶ 0Ⰶ1, Eq. 共13兲is equal to ␾ 共t兲⫽1⫺1⫺e⫺ ␨ ␨ t t0 ⫹O 冉 t ␶ 0 冊 2 ,共14兲 i.e., ␾ (t) decays very little from its initial unity value in this region. In the opposite limit of very large times for which sⰇ1, it is ␾ 共t兲⬃1 ␨ e⫺s s,共15兲 the response function has already decayed to very small values. One main conclusion of the above discussion is that the relevant part of the relaxation, in which ␾ (t) changes from values very close to unity to very close to zero, takes place between the two asymptotic time regimes we have analyzed. In this way, we are led to consider the distinguished limit t/ ␶ 0Ⰷ1, sⰆ1. In this intermediate time window it is found that J. J. BREY AND A. PRADOS PHYSICAL REVIEW E 63 021108 021108-2 ␾ 共t兲⬃1⫺1 ␨ 冉 ␥ ⫹ln t ␶ 0 冊 .共16兲 This is precisely the logarithmic decay of Eq. 共1兲. Moreover, the previous discussion of the short- and large-time limits clearly indicates that most of relaxation of the system is expected to be accurately described by Eq. 共16兲.Asanexample, we present in Fig. 1 the relaxation function ␾ (t) for exp( ␨ )⫽103and exp( ␨ )⫽105. The integral expression given by Eq. 共12兲共circles兲is well fitted by the logarithmic asymptotic form of Eq. 共16兲共solid lines兲. The agreement extends over several decades of intermediate times, measured in units of the basic time ␶ 0. The fit improves as the value of ␨ increases, consistently with the condition for time scales separation. It is also interesting to compute an average relaxation time ␶ ¯ , given by ␶ ¯ ⫽ 冕 0 ⬁dt ␾ 共t兲⫽ ␶ 0 e ␨ ⫺1 ␨ ⬃ ␶ 0 e ␨ ␨ ,共17兲 showing an exponential increase with ␨ . Taking into account the definition of ␨ 关Eq. 共10兲兴 the above result can be understood as corresponding to an activation energy barrier whose height is proportional to the complexity of the system, measured by the total number of modes Nactively involved in the relaxation. As already mentioned, for a given specific situation, the dimensionless parameter ⌬ ␮ might have a well-defined physical interpretation, being, for instance, the ratio between the Gibbs free energy barrier per mode and the thermal energy. In this way, it is also interesting to note that the coefficient of lntin Eq. 共16兲is ␨ ⫺1, which is then proportional to the temperature T. This would lead to the Tlnt behavior observed in many physical systems, for instance, in Refs. 关2,9兴. In summary, we have studied a simple class of models that exhibit an anomalous logarithmic relaxation. The dynamics is hierarchically constrained and is adequate to study slow relaxation because there is a clear separation of time scales. The slow logarithmic relaxation is similar to what is experimentally observed in a variety of complex systems; fitting the data in most of the relevant part of the relaxation, i.e., the time window where the relaxation function is neither too small nor too close to its initial value. Perhaps the main assumption of this work is the ‘‘extensiveness’’ with Nnof the effective barrier for modes in level n⫹1, as expressed by Eq. 共3兲. This is quite a general condition and, therefore, the class of hierarchical models presented here may provide an explanation of the appearance of linear logarithmic relaxation in very different physical situations. In this context, the role of hierarchically constrained dynamics may also be relevant in order to understand other kinds of anomalous relaxation, as the inverse logarithmic behavior found in compaction experiments of granular systems 关18,19兴. Also, it seems interesting to investigate the behavior of hierarchical models in the temperature cycling experiments usually carried out in glassy systems 关20兴. This research was partially supported by Grant No. PB98- 1124 from the Direccio ´n General de Investigacio ´n Cientı ´fica yTe ´cnica 共Spain兲. 关1兴P. Nordblad, P. Svedlindh, L. Lundgren, and L. Sandlund, Phys. Rev. B 33, 645 共1986兲. 关2兴E. I. Shakhnovich and A. M. Gutin, Europhys. Lett. 9, 569 共1989兲. 关3兴H. M. Jaeger, C.-h. Liu, and S. R. Nagel, Phys. Rev. Lett. 62, 40 共1989兲. 关4兴V. V. Bredikhin, V. A. Sidorov, O. B. Tsiok, E. V. Gurovich, and L. G. Khvostantsev, Europhys. Lett. 18, 111 共1992兲. 关5兴O. B. Tsiok, V. A. Sidorov, V. V. Bredikhin, L. G. Khvostantsev, V. N. Troitskiy, and L. I. Trusov, Phys. Rev. B 51,12127 共1995兲. 关6兴A. Mehta, R. J. Needs, and S. Dattagupta, J. Stat. Phys. 68, 1131 共1992兲. 关7兴O. B. Tsiok, V. Brazhkin, A. G. Lyapin, and L. G. Khovstantsev, Phys. Rev. Lett. 80, 999 共1998兲. 关8兴L. Fabbian, W. Go ¨tze, F. 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Sa ´nchez-Rey, Physica A 284, 277 共2000兲. 关17兴Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共Dover, New York, 1965兲. 关18兴J. B. Knight, C. G. Frandich, C. N. Lau, H. M. Jaeger, and S. R. Nagel, Phys. Rev. E 51, 3957 共1995兲. 关19兴E. R. Nowak, J. B. Knight, E. Ben-Naim, H. M. Jaeger, and S. R. Nagel, Phys. Rev. E 57, 1971 共1998兲. 关20兴A. Prados and J. J. Brey 共unpublished兲. J. J. BREY AND A. PRADOS PHYSICAL REVIEW E 63 021108 021108-4