Stochastic dynamics in a two-level model of disorder: comparison of mean-field and exact solutions

Balakrishnan, V. ; Lakshmibala, S. (1991) Stochastic dynamics in a two-level model of disorder: comparison of mean-field and exact solutions Pramana - Journal of Physics, 37 (3). pp. 235-252. ISSN 0304-4289

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Stochastic dynamics in the presence of quenched disorder (e.g., diffusion in a random medium) is generally treated in a suitable mean-field or effective medium approximation. While numerical simulations may help determine the accuracy of such approximations in specific models, there are relatively few instances in which analytic solutions are possible, to enable a precise comparison to be made with the mean-field results. We consider in this paper a simple but general model of quenched disorder in which a system variablex jumps stochastically between two values x a and x b . However, in each level there occurs with a certain probability a branch (or internal) state into which the system may fall, and from which a jump to the other level is possible only after a return to the original (or ‘active') state. Four different configurations of the states of the system are thus possible, and the transitions between the states are governed by Markovian transition probabilities. The moments ofx and its autocorrelation function are computed in each case, and then configuration-averaged over the four realizations. This represents the exact solution. Next, a mean-field theory of the dynamics is developed: this turns out to involve an effective waiting-time density at each of the two levels that is non-exponential in time, so that the mean-field dynamics is a non-Markovian alternating renewal process. The moments and autocorrelation ofx are again computed, and compared with the exact solutions. The extent of the differences at both short and long times is elucidated, and a numerical comparison is presented for the case of maximal disorder.

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Stochastic Dynamics; Disorder; Mean-Field Approximation; Correlation Function
ID Code:1078
Deposited On:27 Sep 2010 04:29
Last Modified:16 May 2016 12:15

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