Diffusion in a sparsely connected space: A model for glassy relaxation
1988; American Physical Society; Volume: 38; Issue: 16 Linguagem: Inglês
10.1103/physrevb.38.11461
ISSN1095-3795
Autores Tópico(s)Complex Network Analysis Techniques
ResumoA model for diffusion in configuration space is proposed which combines the features of infinite dimensionality and low connectivity thought to be important for glassy relaxation. Specifically, a random walk amongst a set of N points, with each of the N(N-1)/2 pairs connected independently with probability p/N (and the mean connectivity p finite for N\ensuremath{\rightarrow}\ensuremath{\infty}), is considered. The model can be solved exactly by the replica method, but the behavior in the long-time regime is difficult to extract. From, instead, intuitive arguments based on the dominance for t\ensuremath{\rightarrow}\ensuremath{\infty} of a particular type of statistical fluctuation in the network connectivity, the mean probability f(t) of return to the origin after time t is predicted to approach its infinite-time limit according to a ``stretched-exponential'' law, f(t)-f(\ensuremath{\infty})\ensuremath{\sim}exp[-(t/\ensuremath{\tau}${)}^{1/3}$] for all finite p, with \ensuremath{\tau}\ensuremath{\sim}\ensuremath{\Vert}p-1${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}3}$ near the percolation threshold ${p}_{c}$=1.
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