Mean-field limit of systems with multiplicative noise

TL;DR

This paper investigates the mean-field behavior of Langevin systems with multiplicative noise, revealing regimes of stability and instability, and highlighting the limitations of mean-field approaches in finite systems.

Contribution

It provides an analytical and numerical analysis of the mean-field limit for systems with multiplicative noise, emphasizing the instability of the strong-noise regime and the finite-size effects.

Findings

Strong-noise regime is intrinsically unstable due to fluctuations.

Mean-field approach is valid only in the thermodynamic limit.

Finite systems exhibit broad distributions of the self-consistent field.

Abstract

A detailed study of the mean-field solution of Langevin equations with multiplicative noise is presented. Three different regimes depending on noise-intensity (weak, intermediate, and strong-noise) are identified by performing a self-consistent calculation on a fully connected lattice. The most interesting, strong-noise, regime is shown to be intrinsically unstable with respect to the inclusion of fluctuations, as a Ginzburg criterion shows. On the other hand, the self-consistent approach is shown to be valid only in the thermodynamic limit, while for finite systems the critical behavior is found to be different. In this last case, the self-consistent field itself is broadly distributed rather than taking a well defined mean value; its fluctuations, described by an effective zero-dimensional multiplicative noise equation, govern the critical properties. These findings are obtained analytically for a fully connected graph, and verified numerically both on fully connected graphs and on random regular networks. The results presented here shed some doubt on what is the validity and meaning of a standard mean-field approach in systems with multiplicative noise in finite dimensions, where each site does not see an infinite number of neighbors, but a finite one. The implications of all this on the existence of a finite upper critical dimension for multiplicative noise and Kardar-Parisi-Zhang problems are briefly discussed.