Stochastic dynamics in a two-level model of disorder: comparison of mean-field and exact solutions |
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Authors: | V Balakrishnan S Lakshmibala |
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Institution: | (1) Department of Physics, Indian Institute of Technology, 600 036 Madras, India |
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Abstract: | 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 valuesx
a
andx
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. |
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Keywords: | Stochastic dynamics disorder mean-field approximation correlation function |
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