System Size Resonance in Coupled Noisy Systems and in the Ising Model

Abstract

We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles -lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance. DOI: 10.1103/PhysRevLett.88.050601 PACS numbers: 05.40.Ca, 05.45. -a, 05.50. +q Stochastic resonance has attracted much interest recently [1]. As was demonstrated in [2], a response of a noisy nonlinear system to a periodic forcing can exhibit a resonancelike dependence on the noise intensity. In other words, there exists a "resonant" noise intensity at which the response to a periodic force is maximally ordered. Stochastic resonance has been observed in numerous experiments [3]. Noteworthy, the order in a noise-driven system can have a maximum at a certain noise level even in the absence of periodic forcing, this phenomenon being called coherence resonance [4].Being first discussed in the context of a simple bistable model, stochastic resonance has been also studied in complex systems consisting of many elementary bistable cells [5]; moreover, the resonance may be enhanced due to coupling [6]. In this paper we discuss another type of resonance in such systems, namely, the system size resonance, when the dynamics is maximally ordered at a certain number of interacting subsystems. Contrary to previous reports of array-enhanced stochastic resonance (cf. also [7]), here we fix the noise strength, coupling, and other parameters; only the size of the ensemble changes.The basic model to be considered below is the ensemble of noise-driven bistable overdamped oscillators, governed by the Langevin equations,Here j i ͑t͒ is a Gaussian white noise with zero mean: ͗j i ͑t͒j j ͑t 0 ͒͘ d ij d͑t 2 t 0 ͒;´is the coupling constant; N is the number of elements in the ensemble, and f͑t͒ is a periodic force to be specified later. In the absence of periodic force, the model (1) has been extensively studied in the thermodynamic limit N !`. It demonstrates an Isingtype phase transition at´´c from the disordered state with vanishing mean field X N 21 P i x i to the "ferromagnetic" state with a nonzero mean field X 6X 0 [8].While in the thermodynamic limit the full description of the dynamics is possible, for finite system sizes we have mainly a qualitative picture: In the ordered phase the mean field X switches between the values 6X 0 and its average vanishes for all couplings. The rate of switchings depends on the system size and tends to zero as N !`.For us, the main importance is the fact that qualitatively the behavior of the mean...

System size stochastic resonance: General nonequilibrium potential framework

“…A related result [18] shows that even in the absence of external forcing, the regularity of the collective firing of a set of coupled excitable FitzHughNagumo units results optimal for a given value of the number of elements. From a physical system point of view, the same phenomenon -that has been called system size stochastic resonance (SSSR)-has also been found in an Ising model as well as in a set of globally coupled units described by a φ 4 theory [19]. It was even shown to arise in opinion formation models [20].…”

“…Even those cases discussed in [19] could be described within this same framework, and the ("effective") scaling of the noise with the system's size could be clearly seen. Here, we discuss in more detail the cases analyzed in [21] and present a new interesting one, corresponding to the study of SSSR in a system that also shows noise induced patterns, the same one studied in [15].…”

“…In this section we consider one of the models discussed in [19] from the point of view of the NEP approach. The model we refer to is described by a set of (globally) coupled nonlinear bistable oscillatorṡ…”

“…In Section II we focus on a simple reactiondiffusion model with a known form of the NEP, that presents SSSR associated to a NEP's symmetry breaking. In Section III we analyze the model of globally coupled nonlinear oscillators discussed in [19], and show that it can also be described within the NEP framework, with SSSR arising due to a deepening of the potential wells, or through an "effective" scaling of the noise intensity with the system size. We start Section IV by briefly reviewing the model and the formalism to be used for the case of multiplicative noise.…”

Optimizing periodicity and polymodality in noise-induced genetic oscillators

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