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Mean-field limit of systems with multiplicative noise

Miguel A. Muñoz

Francesca Colaiori and Claudio Castellano

  • Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

  • Dipartimento di Fisica, Università di Roma “La Sapienza” and Center for Statistical Mechanics and Complexity, INFM Unità Roma 1, Piazzale A. Moro 2, 00185 Roma, Italy

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Phys. Rev. E 72, 056102 – Published 3 November, 2005

DOI: https://doi.org/10.1103/PhysRevE.72.056102

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.

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Phys. Rev. E 72, 056102– Published 3 November, 2005
  • Received 6 July 2005

© 2005 American Physical Society

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