Evolution of the System with Singular Multiplicative Noise

TL;DR

This paper investigates the evolution of systems influenced by multiplicative white noise with a power-law amplitude, revealing phase-dependent behaviors and long-term asymptotics of autocorrelators and order parameters.

Contribution

It derives governing equations for order parameters under multiplicative noise and analyzes phase space behavior, including disordered and ordered regimes, with novel fractional power-law and exponential asymptotics.

Findings

Disordered phase shows non-monotonic autocorrelator evolution.

Ordered phase attracts to finite autocorrelator and order parameter values.

Long-time asymptotics include hyperbolic decay and exponential behavior.

Abstract

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 $0<a<1$ ($x$ being a stochastic variable). It turns out that equation for autocorrelator includes an anomalous average of the power-law function with the fractional exponent $2a$. Determination of this average for the stochastic system with a self-similar phase space is performed. It is shown that at $a>1/2$, 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 $-t\ln t$.