Macroscopic Finite Size Effects in Relaxational Processes

We present results on dynamical processes that exhibit a stretched exponential relaxation. When the relaxation is a result of two competing exponential processes, the size of the system, although macroscopic, play a dominant role. There exist a crossover time t× that depends logarithmically on the size of the system, above which, the relaxation changes from a stretched exponential to a simple exponential decay. The decay rate also depends logarithmically on the size of the system. The results are relevant to large-scale Monte-Carlo simulations and should be amenable to experiments in low-dimensional macroscopic systems and mesoscopic systems. Many relaxational processes in macroscopic systems are characterized by a relaxation function Q(t) that exhibits a stretched exponential behavior, Q(t) ∼ Q(0) exp[−(t/τ) ], (1) where 0 < β < 1. Examples include viscoelastic relaxation [1], dielectric relaxation [2], glassy relaxations [3,4,5], relaxation in polymers [6,7] and long-time decay in trapping processes [8]. Many more examples [9,10,11,12,13,14,15] suggest that (1) is common to a very wide range of phenomena and macroscopic materials. The origin of the stretched exponential is not always clear. In many cases it is assumed to be the result of a competition between two exponential processes. In some cases, e.g., trapping processes at long times, this assumption is well established, while in others, such as relaxation in glassy materials, this assumption has been controversially discussed [16,17] and alternative models have been also suggested [10,18,19,20]. We have recently investigated the occurence of stretched exponential behavior in finite systems, in cases where the relaxation arises due to two competing exponential processes [21] We have found that: (a) the size of the system, although macroscopic, plays a dominant role in the relaxation time pattern, leading to an exponential decay sufficiently at long times; (b) the crossover time, t×, to the 2 Shlomo Havlin, Armin Bunde, and Joseph Klafter exponential depends logarithmically on the system size; (c) the rate of the exponential decay also depends logarithmically on the system size, and (d) in the special examples of the trapping and the hierarchically constrained dynamics models the exponential relaxation may enter before the stretched exponential is reached. These results are of relevance to experiments in confined systems, mesoscopic systems and to Monte-Carlo simulations. Our theoretical predictions on the finite size effects can serve as an experimental test for identifying the origin of the mechanism leading to stretched exponential decay. We assume that the relaxation function of the whole system can be represented by an integration over all possible states n, namely,