Non-self-averaging in random trapping transport: The diffusion coefficient in the fluctuation regime.

Abstract

The nonstationary diffusion of particles in a medium with static random traps or sinks is considered. The question of the self-averaging of the diffusion coefficient (or, equivalently, of the mean-square displacement) is addressed for the fluctuation regime in the long-time limit. The property of self-averaging is needed for the result of a single measurement to be representative and reproducible. It is demonstrated that the diffusion coefficient of the surviving particles is a strongly non-self-averaging quantity: In a d-dimensional system its reciprocal standard deviation grows with time exponentially ≈exp[const_{d,1}t^{d/(d+2)}]. The same result is reproduced in the "normalized" formulation "per one survivor on average." The case when all the particles, both the survivors and the trapped ones, are contributing to the diffusion coefficient and its variance is considered also. Non-self-averaging is demonstrated for this case as well, the fluctuations of the diffusion coefficient being of the same order as its average value. The critical dimension, above which the mean-field result becomes exact, is infinite-due to the drastic difference between the classes of trajectories, upon which the corresponding results are being built. In high dimensions the strong non-self-averaging of survivors is preserved. For the case of all the particles taken into account, the nonstrong non-self-averaging is retained for any finite dimension. However, for d→∞ the limiting value of the reciprocal standard deviation, calculated for all the particles, decreases to zero. This signifies restoration of the self-averaging in some sense. In all the cases, the time evolution of the average characteristics and of their variances is governed by the decaying concentration of the survivors in fluctuational cavities.