Nonequilibrium phase transitions in finite arrays of globally coupled Stratonovich models: Strong coupling limit

TL;DR

This paper analyzes the nonequilibrium phase transition in finite arrays of coupled Stratonovich models, revealing a strong coupling limit where the array behaves as a single entity and identifying the scaling behavior of moments near the critical point.

Contribution

It provides an analytical characterization of the stationary distribution and moments in the strong coupling limit, extending previous results to finite arrays and exploring the scaling crossover behavior.

Findings

Fast relaxation of relative coordinates in strong coupling limit

Scaling crossover from linear to square root near critical point

Agreement between analytical results and numerical simulations

Abstract

A finite array of $N$ globally coupled Stratonovich models exhibits a continuous nonequilibrium phase transition. In the limit of strong coupling there is a clear separation of time scales of center of mass and relative coordinates. The latter relax very fast to zero and the array behaves as a single entity described by the center of mass coordinate. We compute analytically the stationary probability and the moments of the center of mass coordinate. The scaling behaviour of the moments near the critical value of the control parameter $a_c(N)$ is determined. We identify a crossover from linear to square root scaling with increasing distance from $a_c$. The crossover point approaches $a_c$ in the limit $N \to \infty$ which reproduces previous results for infinite arrays. The results are obtained in both the Fokker-Planck and the Langevin approach and are corroborated by numerical simulations. For a general class of models we show that the transition manifold in the parameter space depends on $N$ and is determined by the scaling behaviour near a fixed point of the stochastic flow.