Finite-size fluctuations for stochastic coupled oscillators: A general theory

Abstract

Phase transitions, sharp in the thermodynamic limit, get smeared in finite systems where macroscopic order-parameter fluctuations dominate. Achieving a coherent and complete theoretical description of these fluctuations is a central challenge. We develop a general framework to quantify these finite-size effects in synchronization transitions of generic stochastic, globally-coupled nonlinear oscillators. By applying a center-manifold reduction to the nonlinear stochastic PDE for the single-oscillator distribution in finite systems, we derive a mesoscopic description that yields the complete time evolution of the order parameter in the form of a Langevin equation. In particular, this equation provides the first closed-form steady-state distribution of the order parameter, fully capturing finite-size effects. Free from integrability constraints and the celebrated Ott-Antonsen ansatz, our theory shows excellent agreement with simulations across diverse coupling functions and frequency distributions, demonstrating broad applicability. Strikingly, it surpasses recent approaches near criticality and in the incoherent phase, where finite-size fluctuations are most pronounced.

Figures (6)

• Figure 1: For model in Application 1 with $K_2=0, D=1.0$, the figure shows agreement between our theory (lines) and numerical simulation (markers in (a) and histograms in (b) -- (e)) for the average of the steady-state order parameter in (a) and for its distribution in (b) -- (e). Parameters for (b) -- (e) are $K_1 = 1.6, 1.9, 2.1,2.2$, respectively. Simulation details for all plots are in Simulation.
• Figure 2: For model in Application 1, the figure shows the behavior near a first-order transition point with $K_1 = 1.987, K_2 = 2.3, D=1.0, N = 10^4$. For the order parameter $R_1=|Z_1|$, panel (a) shows a numerically-obtained trajectory of $Z_1$ from $t=0.0$ (black marker) to $t=500.0$ (orange marker), displaying that it jumps between two regions that are close to the maxima of the distribution $P(R_1)$ and denoted by the two concentric circles. The effective potential $V(R_1)$ driving the dynamics of $R_1$ is shown in (b), and agreement between theory (line) and numerical simulation (histogram) is shown in (c).
• Figure 3: For model in Application 2, (a) shows the $N\to \infty$ schematic phase diagram containing the tricritical point (green marker), continuous (red dashed line) and first-order (blue solid line) transition lines for fixed $K_2$. (b) Variation of the tricritical point (black line) with $K_2$. Upon varying $K_1$, $R_1$ shows continuous (respectively, first-order) transition for $(\omega_0,K_2)$ in red-shaded region $(\mathrm{e.g.},A\equiv(D,0.2D))$ (respectively, blue-shaded region $(\mathrm{e.g.},B\equiv(2.3D,0.2D))$). For $D=1.0$, agreement between our theory (lines) and numerical simulation (histogram) for first-order transition (point $B$ with $K_1 = 2.05, 2.059, 2.0605, 2.063$ and $N = 10^4$) is shown is (c) -- (f) and for continuous transition (point $A$ with $K_1 = 1.6, 2.059, 2.1, 2.2$ and $N = 2 \times 10^3$) is shown is (g) -- (j).
• Figure 4: Agreement between $P_\approx(R_1)$ (line) and $P(R_1)$ (unfilled markers) for the model in Application 2 ((a) -- (h)) and the one in Application 3 ((i) -- (p)) is shown. Parameters for (a) -- (h) are the same as in Fig. \ref{['fig:3']}, panels (c) -- (j), respectively. Similarly, parameters for (i) -- (p) are the same as in Fig. \ref{['fig: 4']}, panels (c) -- (j), respectively.
• Figure 5: For model in Application 3, (a) and (b) show the $N\to \infty$ phase diagram containing the continuous (red dashed line) and first-order (blue solid line) transition lines in the $\sigma-K_1$ plane for fixed $K_2$. For $\sigma = 1.0, D =1$, agreement between our theory (lines) and numerical simulation (histogram) for first-order transition ($K_2=2.3$ with $K_1 = 3.9, 3.97, 3.98, 3.99$ and $N =10^4$) is shown is (c) -- (f) and for continuous transition ($K_2=1.0$ with $K_1 = 3.5, 3.9, 4.1, 4.2$ and $N = 2\times 10^3$) is shown is (g) -- (j).