# Glassy dynamics: effective temperatures and intermittencies from a two-state model

###### Abstract

We show the existence of intermittent dynamics in one of the simplest model of a glassy system: the two-state model, which has been used Perez&Reguera&Rubi03 to explain the origin of the violation of the fluctuation-dissipation theorem. The dynamics is analyzed through a Langevin equation for the evolution of the state of the system through its energy landscape. The results obtained concerning the violation factor and the non-Gaussian nature of the fluctuations are in good qualitative agreement with experiments measuring the effective temperature and the voltage fluctuations in gels and in polymer glasses. The method proposed can be useful to study the dynamics of other slow relaxation systems in which non-Gaussian fluctuations have been observed.

, , ,

## I Introduction

Complex systems are often distinguished by the existence of a very intricate free energy landscape consisting of many barriers which the system has to overcome to evolve. It is precisely the presence of these barriers the responsible for the slow relaxation dynamics which manifests in the appearance of peculiar phenomena as aging, lack of a fluctuation-dissipation theorem and intermittencies caused by the presence of large fluctuations. These features, predicted and observed in systems of different nature as glasses, granular flows, foams, crumpled materials and in the dynamics of the disordered systems Rubi&Conrad97 ; Angel95 ; Ben-Naim et al98 ; Sollich97 ; Matan02 ; Oliveira02 , have attracted the interest of many researchers during the last years with the purpose of describing the main features of slow relaxation dynamics Cugliandoloprl .

In a previous paper Perez&Reguera&Rubi03 , we have proposed a minimal relaxation model aimed at characterizing the dynamics of a system relaxing in two very different time scales, which are related to inter-well and intra-well relaxation processes. Two scenarios were analyzed for this purpose. In the first of them, the system may explore the whole reaction coordinate space undergoing a diffusion process described by a Fokker-Planck equation Mohanty94 , which accounts for the intra-well and inter-well relaxations. In the other, obtained from the first one by eliminating the fast variable, the system undergoes an activated process. In spite of its simplicity the model shows some of the peculiar features of the dynamics of slow relaxation systems and proposes an explanation of why and how the fluctuation-dissipation is violated. It was found that the violation factor or effective temperature depends on the observable and on the initial populations in the wells. This result shows that the effective temperature does not univocally characterize the thermal state of a glassy system undergoing activated dynamics Vilar01 .

Our purpose in this paper is to use that model to explain the presence of intermittencies in the dynamics of a system in a glassy phase and the non-Gaussian nature of the probability distribution function, which have recently been observed in measurements of the dielectric properties of gels and polymer glasses Ciliberto1 ; Ciliberto2 , and in some theoretical studies of spin-glass models Felix04 ; Sibani04 .

The paper is organized as follows. In Section 2, we analyze some of the main traits of glassy dynamics from a two-state model. Section 3 is devoted to present the results concerning the intermittent behavior and the non-Gaussian nature of the fluctuations. Some conclusions and perspectives are presented in the final section.

## Ii Glassy dynamics from a two-state model

In a two-state model, the minimal relaxation model for glassy systems Huse86 ; Langer88 ; Langer90 , one assumes that the process consists of two main steps: a slow relaxation, in which the coordinate characterizing the state of the system jumps from a potential well to the next one, and a fast equilibration process in the well. It has been shown Perez&Reguera&Rubi03 that the dynamics of the system can be analyzed in terms of a Fokker-Planck equation describing a diffusion process through the free energy landscape Ignacio97 ,

(1) |

In the simplest case, would be just a bistable potential. In the previous equation, is the probability distribution function which depends on the order parameter or reaction coordinate , is the diffusion coefficient, is the temperature of the bath and the Boltzmann constant. When the height of the barrier separating the two minima of the potential is large enough compared to thermal energy the systems achieves a state of quasi-equilibrium in each well. The evolution of the system then proceeds by jumps from one well to the other undergoing an activated process. The dynamics corresponding to this situation can be obtained by eliminating the fast degrees of freedom in such a way that it can be characterized simply by the populations at each well. The Fokker-Planck equation then reduces to the following kinetic equations governing the population dynamics at both wells Perez&Reguera&Rubi03 ; Schmid01

(2) |

where the current is defined as:

(3) |

Here and are the populations at each well and are the forward and backward reaction rates given by

(4) |

where and denote the position of the two minima and the position of the top at the barrier. The noise term results from the coarsening of the corresponding random term in the Langevin equation related to the Fokker-Planck equation (1), Perez&Reguera&Rubi03 . Whereas the latter satisfies the fluctuation-dissipation theorem inherent to the diffusion process along the reaction coordinate formulated as Landau

(5) |

the former does not obey a similar expression. Its correlation is given by Ignacio97 :

(6) |

where the last term is the value of the correlation at equilibrium, and is the equilibrium population in the first well.

This result clearly indicates that the violation of the fluctuation- dissipation theorem is precisely due to the coarsening of the description. When one eliminates the fast variables reducing the dynamics to that of an activated process the system cannot progressively pass through local equilibrium states from one well to the other but proceeds by jumps and consequently remains outside equilibrium. This explains why the fluctuation- dissipation theorem, a result strictly valid when the fluctuations take place around an equilibrium state Callen&Greene , is not fulfilled.

From the model proposed we can analyze the non-equilibrium response of the system. Let us assume an external perturbation that is plugged in at the waiting time , defined as the time elapsed after quenching ( is the observable considered and is a generelized force that is coupled to the observable). The response is then given by

(7) |

where the average of the observable is defined as

(8) |

whereas the values and are the values of the observable at the minima 1 and 2, respectively. On the other hand, the correlation is

(9) |

Both quantities satisfy the relation

(10) |

in which the quantity plays the role of an “effective” temperature given by

(11) |

Here is the relaxation time of the process and the parameter has the form

(12) |

Since the equilibrium state is achieved for , one can easily verify that the effective temperature coincides with that of the bath and the relation (10) becomes the fluctuation-dissipation theorem.

An important consequence of our previous analysis is that the effective temperature is not a robust quantity since it depends on the observable considered as well as on the initial populations at the wells. We then conclude that in the case of an activated process that quantity is a parameter measuring the distance of the system to the equilibrium state but it is not universal and consequently can hardly be considered as a thermodynamic temperature Vilar01 .

Our expression Eq.(11) can be used to reproduce the dependence of the violation factor on the frequency observed in experiments Ciliberto1 ; Ciliberto2 . If we assume that the whole relaxation process takes place through consecutive activated processes for which Eq.(2) applies, we can infer a global behavior of that quantity by identifying the inverse of the relaxation time with a frequency. By considering the case , in which the violation factor does not depend on the observable , we then conclude that the effective temperature at low frequencies behaves as: . In the experiments one finds that the effective temperature decreases when increasing the frequency following a power law whose exponent is in between and , Ciliberto2 . As in the experiments, it is found that the effective temperature tends to the bath temperature for very large waiting times (see Fig. 1).

## Iii Intermittent dynamics

We will analyze in this section the intermittent behavior of the fluctuations taking place when the system is quenched below the glass transition. This type of behavior has been reported in experiments measuring the fluctuation spectrum and the response of a Laponite solution and of a polymer glass Ciliberto1 ; Ciliberto2 . To this purpose, we will particularize the model discussed in Sec. 2 to the case of the quartic potential plotted in Fig. 2. The dynamics of this model is governed by the Fokker-Planck equation (1) or through the equivalent Langevin equation

(13) |

where has been considered a Gaussian white noise process having, as a first approximation, a constant amplitude: . We will see that this approximation is enough to explain the experimental results.

By numerical simulation of the Langevin equation (13), we have computed the value of the following observable

(14) |

which has a piece-wise dependence on , having a zero value at and . In our case we have taken and and the parameters and have been chosen arbitrarily as and , ensuring the continuity of the observable. The location of the maximum of the observable has been placed at , to show in a clearer way the occurrence of a transition between the two wells. When the state is at the top of the barrier the probability to go back to the minimum is 50%.

The value of should be consistent with the non-stationary nature of the process and has to be intrinsically related to the waiting time. Therefore, we have use different values of this parameter to mimic different values of the waiting time.

In Fig. 3 we have represented the value of the observable corresponding to a single trajectory obtained from the simulations of Eq.(13), with . We observe the presence of an intermittent event, which is the signature of a jump from one well to another. Therefore, the presence of intermittencies is a consequence of the activated nature of the relaxation process.

Averaging over many trajectories we have obtained the probability distribution function for the observable, which has been plotted in Fig. 4, for different values of . The non-Gaussian form of the curves is, in our model, a consequence of the non-parabolic form of the potential. Intermittencies add more weight to the tails of the distribution, thus stressing its non-Gaussian nature.

## Iv Conclusions

In this paper we have used the activation over a barrier to model the evolution of a system when is quenched below the glass transition. Using as a model a simple quartic potential, we have proved the existence of an intermittent dynamics in the aging process caused by the presence of large fluctuations. Similar behavior has been observed in recent experiments. We have shown that the probability distribution function corresponding to those events deviates from its Gaussian form observed for systems close to equilibrium and exhibits exponential tails as those encountered experimentally.

The model proposed and its possible generalizations could, with the help of the stochastic processes theory Hanggi90 ; Talkner04 , constitute a useful tool to characterize the dynamics of systems far from equilibrium for which a common phenomenology in the behavior of the fluctuations is being found.

## Acknowledgments

This work was partially supported by the DGICYT and FEDER under Grant No. BFM2002-01267. D. R. acknowledges support by the Ministerio de Ciencia y Tecnología of Spain through the “Ramón y Cajal” program.

## References

- (1) A. Pérez-Madrid, D. Reguera, J.M.Rubí, Physica A 329 (2003) 357-364.
- (2) J.M. Rubí, C. Pérez-Vicente (Eds.), Complex Behaviour of Glassy Systems, Springer, Berlin, 1997.
- (3) C.A. Angell, Science 267 (1995) 1924.
- (4) E. Ben-Naim, J.B. Knight, E.R. Nowak, H.M. Jaeger, S.R. Nagel, Physica D 123 (1998) 380.
- (5) P. Sollich, F. Lequeux, P. Hébreud, M.E. Cates, Phys. Rev Lett. 78 (1997) 2020.
- (6) K. Matan, R.B. Williams, T.A. Witten, S.R. Nagel, Phys. Rev. Lett. 88 (2002) 076101.
- (7) R. Morgado, F. Oliveira, G.G. Batrouni, A. Hansen, Phys. Rev. Lett. 89 (2002) 100601.
- (8) L.F. Cugliandolo, J. Kurchan, Phys. Rev. Lett. 71 (1997) 3898.
- (9) U. Mohanty, I. Oppenheim, C.H. Taubes, Science 266 (1994)425.
- (10) J.M. Vilar, J.M. Rubí, Proc. Nat. Acad. Sci. 98 (2001) 11081.
- (11) L. Buisson, L. Bellon and S. Ciliberto, J. Phys.: Condens. Matter 15, pp. S1163-S1179, 2003.
- (12) L.Buisson, M. Ciccotti, L. Bellon and S. Ciliberto, Proc. SPIE Int. Soc. Opt. Eng. 5469, 150 (2004).
- (13) A. Crisanti and F. Ritort, Europhys. Lett. 66 (2), pp. 253-259 (2004).
- (14) P. Sibani and H.J. Jensen, arXiv: cond-mat/0403212 v2 (25 May 2004).
- (15) D.A. Huse, D.S. Fisher, Phys. Rev. Lett. 57 (1986) 2203.
- (16) S.A. Langer, J.P. Sethna, Phys. Rev. Lett. 61 (1988) 570.
- (17) S.A. Langer, J.P. Sethna, R. Grannan, Phys. Rev. B 41 (1990) 2261.
- (18) I. Pagonabarraga, A. Pérez-Madrid, J.M. Rubí, Physica A 237 (1997) 205.
- (19) In order to solve the master equation (2) we need to impose reflective boundary conditions, to confine the value of the population between zero and one. See the art. G. Schmid, I. Goychuk, and P. Hänggi, Europhys. Lett. 56 (2001) 22.
- (20) L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, vol. 9 (Pergamon Press, New York 1980).
- (21) H.B. Callen and R.F. Greene, Phys. Rev. 86 (1952) 702; R.F. Greene and H.B. Callen, Phys. Rev. 88 (1952)1387.
- (22) P. Hänggi, P. Talkner, M. Borkevec, Rev. Mod. Phys. 62 (1990) 251.
- (23) P. Talkner and J. Łuczka, Phys. Rev. E 69 (2004) 046109.