Mean-field limit of systems with multiplicative noise

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 strongnoise) 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.

Critical wetting of a class of nonequilibrium interfaces: A mean-field picture

“…All moments m k starting from 2D/σ 2 + 1 scale with the same exponent, while the usual scaling m k ∼ m k obtains for k ≤ 2D/σ 2 + 1. This same behavior was observed in a mean-field study of nonequilibrium, MN1 complete wetting as reported in [14], and seems to be a common feature of multiplicative-noise controlled transitions.…”

“…This represents a strong form of multiscaling, very similar to the one reported for the moments in the self-consistent solution for MN1 [14].…”

Stability in a system subject to noise with regulated periodicity

“…In particular, in these systems the random variations of temperature, food resources, and other environmental parameters can be described using models in which multiplicative noise sources are present [8,16,17]. In fact, the models that can effectively describe experimental data in population dynamics are those with multiplicative noise, which can give rise to behavior characterized by a power law [9,18,19].…”

Predator population depending on lemming cycles

“…In population dynamics, random fluctuations are always modeled as a multiplicative noise source, because the dynamics of the population density is driven by statedependent fluctuation amplitude, 4,[14][15][16][17][18][19][20][21][22][23][24][25][26][27] which allows to reproduce experimental data. This means that, the effects of fluctuations have to be proportional to the activity density, which is the predator density in our system.…”