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

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 $N\to \infty$ and $t\to \infty$ ($t$ is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

Figure 1

Isolated oscillator. Random transitions occur at rate ${\gamma }_0$ from state 0 to state 1 and at rate ${\gamma }_1$ from state 1 to state 0.

Figure 2

(a) Generic function $F$ in Eq. (27) undergoing a pitchfork bifurcation. Closed circles denote stable steady states while open circles denote unstable fixed points. (b) Fixed points of Eq. (27) as a function of the control parameter $\varepsilon$, where solid lines denote stable fixed points while dashed lines denote unstable fixed points.

Figure 3

Schematic of the effective potential $\mathcal{U}$ vs $n_1$ for the evolution of $n_1$ in the mean field limit for a particular value of the control parameter $\varepsilon >0$.

Figure 4

Monomodal distributions for $\varepsilon =0.001$. The dots are the results of direct numerical simulations as in Eq. (7) and the curves are the analytic solutions of the Fokker-Planck equation.

Figure 5

Bimodal distributions for $\varepsilon =0.1$. The dots are the results of direct numerical simulations as in Eq. (7) and the curves are the analytic solutions of the Fokker-Planck equation.

Figure 6

Transition of $P\left(N_1\right)$ from a single- to a double-peak structure for $N=50$ as $\varepsilon$ increases, calculated using the master equation (24). This is indicative of a transition from a single-well effective potential to an asymmetric double-well potential.

Figure 7

Fluctuations at the stable fixed points. The fluctuation term in Eq. (18) for the larger fixed point at $n_1^{*}=1/2+\sqrt{\varepsilon }$ (solid line) is always larger than for the smaller fixed point at $n_1^{*}=1/2-\sqrt{\varepsilon }$ (dashed line).

Figure 8

Closed circles show the $(\varepsilon ,N)$ pairs of values at which the transition occurs from a unimodal (left side) to a bimodal (right side) distribution obtained from the numerical integration of the master equation (24). The line is the prediction of the transition according to the Fokker-Planck equation, which is a large-$N$ prediction. The agreement between the two is excellent even for very small values of $N$.

Figure 9

Plot of $log\left|\varepsilon \right|$ (or $log\varepsilon$ since we only need to consider positive $\varepsilon$) vs $logN$. Finite-size scaling theory predicts that the critical value of the control parameter ${\varepsilon }_c\left(N\right)$ converges to its mean field value ${\varepsilon }_c\left(\infty \right)=0$ following a power law $|{\varepsilon }_c\left(N\right)-\left|{\varepsilon }_c\left(\infty \right)\right|\propto N^{-1/\alpha }$. We find $\alpha =1/0.64=1.56$.

Figure 10

Time evolution of globally coupled finite arrays of Markovian oscillators with $\varepsilon =0.1$ (bistable regime).

Figure 11

Distribution of first passage times from the upper to the lower metastable region ($T_{u-l}$) and from the lower to the upper ($T_{l-u}$) for two values of $N$ as indicated in the figure. We see that transitions from the upper to the lower metastable regions occur more rapidly than those from the lower to the upper.

Figure 12

Mean first passage times from (a) state $n_1=0$ to the upper attractor and (b) state $n_1=1$ to the lower attractor. The dots result from the integration of the microscopic evolution equations and the lines are the analytic results obtained from the Fokker-Planck equation.

Figure 13

Steady-state probabilities as a function of the coupling parameter $\varepsilon$ for $N=2$ coupled oscillators. The probability that both oscillators are in the same state is the same for either state.