Evolution of the System with Singular Multiplicative Noise

Details Evolution of the System with Singular Multiplicative Noise Evolution of the System with Singular Multiplicative Noise

1999-08-06

arxiv.org/abs/cond-mat/9908093v1

Physics

Condensed Matter

Statistical Mechanics

12 pages, 7 figures, LaTeX

Scientific paper

10.1134/1.1131244

The governed equations for the order parameter, one-time and two-time correlators are obtained on the basis of the Langevin equation with the white multiplicative noise which amplitude $x^{a}$ is determined by an exponent $01/2$, when the system is disordered, the correlator behaves non-monotonically in the course of time, whereas the autocorrelator is increased monotonically. At $a<1/2$ the phase portrait of the system evolution divides into two domains: at small initial values of the order parameter, the system evolves to a disordered state, as above; within the ordered domain it is attracted to the point having the finite values of the autocorrelator and order parameter. The long-time asymptotes are defined to show that, within the disordered domain, the autocorrelator decays hyperbolically and the order parameter behaves as the power-law function with fractional exponent $-2(1-a)$. Correspondingly, within the ordered domain, the behavior of both dependencies is exponential with an index proportional to $-t\ln t$.

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