Globally coupled stochastic two-state oscillators: Fluctuations due to finite numbers

Dias Pinto, Italo'Ivo Lima and Escaff, Daniel and Harbola, Upendra and Rosas, Alexandre and Lindenberg, Katja (2014) Globally coupled stochastic two-state oscillators: Fluctuations due to finite numbers. In: PHYSICAL REVIEW E, 89 (5).

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Abstract

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a 1/N expansion and the Langevin equation via standard methods of Ito calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits N -> infinity and t -> infinity(t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

Item Type:	Journal Article
Publication:	PHYSICAL REVIEW E
Publisher:	AMER PHYSICAL SOC
Additional Information:	Copyright for this article belongs to AMER PHYSICAL SOC, USA.
Department/Centre:	Division of Chemical Sciences > Inorganic & Physical Chemistry
Date Deposited:	18 Jul 2014 05:44
Last Modified:	18 Jul 2014 05:44
URI:	http://eprints.iisc.ac.in/id/eprint/49432

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