Discontinuous transitions in globally coupled potential systems with additive noise

Abstract

An infinite array of globally coupled overdamped constituents moving in a double-well potential with $n$-th order saturation term under the influence of additive Gaussian white noise is investigated. The system exhibits a continuous phase transition from a symmetric phase to a symmetry-broken phase. The qualitative behavior is independent on $n$. The critical point is calculated for strong and for weak noise, these limits are also bounds for the critical point. Introducing an additional nonlinearity, such that the potential can have up to three minima, leads to richer behavior. There the parameter space divides in three regions, a region with a symmetric phase, a region with a phase of broken symmetry and a region where both phases coexist. The region of coexistence collapses into one of the others via a discontinuous phase transition whereas the transition between the symmetric phase and the phase of broken symmetry is continuous. The tricritical point where the three regions intersect, can be calculated for strong and for weak noise. These limiting values form optimal bounds on the tricritical point. In the region of coexistence simulations of finite systems are performed. One finds that the stationary distribution of finite but large systems differs qualitatively from the one of the infinite system. Hence the limits of stationarity and large system size do not commute.