Slow dynamics and aging



We do not usually worry about the age of a physical system: a Helium atom which was formed in the Big Bang cannot be distinguished from one synthesized in the Sun a few days ago. However, we do know that some everyday materials, like plastics, change their properties with age: older samples are less resilient than younger ones when a steady tension is applied to them. It is remarkable that the difference only becomes apparent if one waits long enough: experiments of short duration are unable to distinguish between samples of different age [1]. This distinguishes physical aging from mere metastability, in which the ``equilibrium'' properties of the sample change with time. A subtle consequence of physical aging was noticed some time ago by L. F. Cugliandolo and J. Kurchan [2]. At equilibrium, there is a universal relation between the linear response function of any observable O, $R(t,t')\equiv \delta \langle O(t)\rangle/\delta h_O(t')$, and the corresponding correlation function $C(t,t')\equiv \langle O(t)O(t')\rangle-\langle O\rangle^2$:

\begin{displaymath} R(t,t')=\frac{X}{k_{\rm B}T}\,\frac{\partial C(t,t')}{\partial t'}. \end{displaymath} (1)

The factor X is equal to 1 at equilibrium, but turns out to depend only on C(t,t') (in a suitable long-time limit) for a class of solvable models of aging systems, which encompasses mean-field models of spin glasses and schematic models of structural glasses. In all these models, X appears to be smaller than 1. The form of eq. (1) suggests to interpret $T_{\rm eff}=T/X(C)$ as an effective temperature: indeed, it was shown in ref. [3] that a thermometric thought experiment would yield a temperature equal to $T_{\rm eff}$, provided that the response time of the thermometer is such that the value of correlation function C(t,t') is equal to C on that time scale. The details of the thermometric process have been analyzed further in refs. [4,5] in two exactly solvable models. A very simple lattice-gas model with kinetic constraints exhibits aging properties which can be investigated by mean-field arguments [6]. If one modifies the evolution of the system by a small non-conservative perturbation, aging effects disappear in general and the system reaches a steady non-equilibrium state. However, even in this state, the aging form of the relation (1) between response and fluctuation remains true, as shown in Fig. 1. Therefore the introduction of a small non-conservative perturbation is a powerful heuristic method to investigate the aging properties of complex systems, and has been applied [7] to the behavior of manifolds in random potentials. These researches have shown a remarkably universal phenomenology of aging, and have lead to intriguing predictions--like the possibility of measuring effective temperatures higher than the ambient one in aging systems--which are currently under active experimental investigation.

More recent investigations [8,9] have shown the intimate and quite general relationship between the factor X which describes the violation of the fluctuation-dissipation relation in aging and driven systems, and the functional order parameter q(x) which describes replica-symmetry breaking in the static properties of disordered systems and spin glasses.

Figure: Top: Integrated response $\chi(t,t_w)=\int_{t_w}^t dt'\, R(t,t')$ vs. the correlation function C(t,tw) for different values of tw in a schematic model of a structural glass. Bottom: The corresponding plot in the presence of a nonconservative perturbation. Different lines correspond to different intensities of the perturbation. At equilibrium one observes a single straight line of slope $-1/k_{\rm B}T$. From ref. [3].
\includegraphics[width=7cm]{p3chitfinite.ps}
\includegraphics[width=7cm]{p3chialpha.ps}

Figure 2: Scheme of a thought experiment for measuring effective temperatures in an aging magnetic system. The coil is wound around the sample, which is in contact with the heat bath. Coil and capacitor have zero resistance. From ref. [3].
\includegraphics[width=6cm]{dibu.ps}



[1] L. C. E. Struik, Physical Aging in Amorphous Polymers and Other Materials (Elsevier, Houston, 1978).
[2] See, e.g., L. F. Cugliandolo and J. Kurchan, Physica A 263, 242 (1999).
[3] L. F. Cugliandolo, J. Kurchan and L. Peliti, Phys. Rev. E55, 3898 (1997).
[4] R. Exartier and L. Peliti, Physics Letters A 261, 94 (1999).
[5] R. Exartier and L. Peliti, Eur. Phys. J. B 16, 119 (2000).
[6] L. Peliti and M. Sellitto, J. Phys. France IV 8, Pr6, 49 (1998).
[7] P. Le Doussal, L. F. Cugliandolo and L. Peliti, Europhys. Lett. 39, 111 (1997).
[8] S. Franz, M. Mézard, G. Parisi and L. Peliti, Phys. Rev. Lett. 81, 1758 (1998).
[9] S. Franz, M. Mézard, G. Parisi and L. Peliti, J. Stat. Phys. 97, 459 (1999).

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