title: Globally coupled stochastic two-state oscillators: fluctuations due to finite numbers.
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 Itô 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 → ∞ and t → ∞ (t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.
authors.affiliation: Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-900 João Pessoa, Brazil.
authors.forename: Italo'Ivo Lima Dias
authors.initials: IL
authors.lastname: Pinto
authors.affiliation: Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile.
authors.forename: Daniel
authors.initials: D
authors.lastname: Escaff
authors.affiliation: Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore, Karnataka 560012, India.
authors.forename: Upendra
authors.initials: U
authors.lastname: Harbola
authors.affiliation: Departamento de Física, CCEN, Universidade Federal da Paraíba, Caixa Postal 5008, 58059-900 João Pessoa, Brazil.
authors.forename: Alexandre
authors.initials: A
authors.lastname: Rosas
authors.affiliation: Department of Chemistry and Biochemistry and BioCircuits Institute, University of California San Diego, La Jolla, California 92093-0340, USA.
authors.forename: Katja
authors.initials: K
authors.lastname: Lindenberg
mesh_terms: D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D003198:Computer Simulation; D005246:Feedback; D006801:Humans; D008954:Models, Biological; D015233:Models, Statistical; D009991:Oscillometry / Q000379:methods*; D012815:Signal Processing, Computer-Assisted