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Behavior of a single element in a finite stochastic array.

We describe statistical properties of a single element in a nonlinear stochastic array with a finite number of elements with mean-field-like (global) coupling. Desai and Zwanzig [J. Stat. Phys. 19, 1 (1978)] made use of a self-consistent dynamic mean-field ansatz to derive, in the infinite-size limit, a nonlinear Fokker-Planck equation (NLFPE) for the single-particle distribution function. In this work we explore the reliability of this approximation to describe a finite system with a small number of elements. We carry out a numerical analysis of the NLFPE as well as numerical simulations of the full set of Langevin equations governing the dynamics of the coupled elements. We find that for parameter values such that the NLFPE has a single stationary solution, it provides a reliable asymptotic approximation to the time evolution and stationary properties of a finite system of even modest size. In contrast, for parameter values such that the NLFPE has two stable stationary coexisting solutions, the NLFPE leads to results that are greatly at variance with those obtained from the numerical simulations of the Langevin equations for small systems.