Critical Exponents in Zero Dimensions
In the vicinity of the onset of an instability, we investigate the effect of colored multiplicative noise on the scaling of the moments of the unstable mode amplitude. We introduce a family of zero dimensional models for which we can calculate the exact value of the critical exponents β m for all the moments. The results are obtained through asymptotic expansions that use the distance to onset as a small parameter. The examined family displays a variety of behaviors of the critical exponents that includes anomalous exponents: exponents that differ from the deterministic (mean-field) prediction, and multiscaling: non-linear dependence of the exponents on the order of the moment.
KeywordsStochastic processes Nonlinear physics Instabilities Multiplicative noise Critical exponents
The authors would like to thank Stephan Fauve for his support and suggestions. Numerical computations were performed using the MESOPSL parallel cluster, the new supercomputing center of Paris Science and Letters, and their computational support is acknowledged.
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