José Manuel Casado

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. DOI: http://dx.doi.org/10.1103/PhysRevE.85.051121 ©2012 American Physical Society

We analyze the stochastic response of a finite set of globally coupled noisy bistable units driven by rather weak time-periodic forces. We focus on the stochastic resonance and phase frequency synchronization of the collective variable, defined as the arithmetic mean of the variable characterizing each element of the array. For single-unit systems, stochastic resonance can be understood with the powerful tools of linear response theory. Proper noise-induced phase frequency synchronization for a single-unit system in this linear response regime does not exist. For coupled arrays, our numerical simulations indicate an enhancement of the stochastic resonance effects leading to gains larger than unity as well as genuine phase frequency synchronization. The nonmonotonicity of the response with the strength of the coupling strength is investigated. Comparison with simplifying schemes proposed in the literature to describe the random response of the collective variable is carried out.

We analyze the stochastic response of a finite set of globally coupled noisy bistable units driven by rather weak time-periodic forces. We focus on the stochastic resonance and phase frequency synchronization of the collective variable, defined as the arithmetic mean of the variable characterizing each element of the array. For single unit systems, stochastic resonance can be understood with the powerful tools of linear response theory. Proper noise induced phase frequency synchronization for a single unit system in this linear response regime does not exist. For coupled arrays, our numerical simulations indicate an enhancement of the stochastic resonance effects leading to gains larger than unity as well as genuine phase frequency synchronization. The non-monotonicity of the response with the strength of the coupling strength is investigated. Comparison with simplifying schemes proposed in the literature to describe the random response of the collective variable is carried out.

We analyze the noise-induced synchronization between a collective variable characterizing a complex system with a finite number of interacting bistable units and time-periodic driving forces. A random phase process associated to the collective stochastic variable is defined. Its average phase frequency and average phase diffusion are used to characterize the phenomenon. Our analysis is based on numerical solutions of the corresponding set of Langevin equations.