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.

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases

“…Discrete stochastic models for synchronization phenomena have been increasing in popularity as a simple paradigm of synchronization [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Of course, this simplicity is related precisely to the relative ease of dealing with only a few states.…”

“…If N is finite, we need to work with a set of coupled Langevin equations, as in Ref. [12,14]. However, except for our numerical simulations, finite size effects are beyond the scope of this paper; here we focus on the mean-field theory.…”

“…However, except for our numerical simulations, finite size effects are beyond the scope of this paper; here we focus on the mean-field theory. We note that first taking the limit N → ∞, and then the limit t → ∞ to study the steady states of (30) in general does not commute with taking these two limits in the opposite order [12]. The continuous-time Markov-chain modeling scheme rests on the assumption that the rates (31) are positive, which is not trivially satisfied.…”

Tristable and multiple bistable activity in complex random binary networks of two-state units

“…Most of the referenced works consider Markovianthereby memoryless -discrete-state models [9][10][11]17]. A disordered environment [18] or the reduction of models with a high number of discrete states to a model with fewer states generically demands a non-Markovian description.…”

“…The structure of the network is given by the node distribution p(k). A big number of nodes is assumed, it is known that finite size effects [17] have a strong effect on the dynamic of the stochastic process as well as on the network influence. Complex networks are of top interest in statistical physics because it allows deviation from global coupling without specification of a spatial structure as well as providing a framework to map complex spatial structures to an abstract space.…”