Memory-dependent noise-induced resonance and diffusion in non-markovian systems

TL;DR

This paper investigates non-markovian systems with memory effects, revealing conditions for noise-induced resonance and diffusion, and providing new solutions to the Mori-Zwanzig equation that describe their dynamics.

Contribution

It introduces new analytical solutions for non-markovian processes and characterizes the boundary conditions separating different dynamic regimes, including noise-induced resonance without external forcing.

Findings

Existence of two types of boundary conditions with distinct dynamics.

Noise-induced resonance occurs without external periodic forcing.

Behavior of variance differs across stationary and non-stationary regimes.

Abstract

We study the random processes with non-local memory and obtain new solutions of the Mori-Zwanzig equation describing non-markovian systems. We analyze the system dynamics depending on the amplitudes $\nu$ and $\mu_0$ of the local and non-local memory and pay attention to the line in the ($\nu$, $\mu_0$)-plane separating the regions with asymptotically stationary and non-stationary behavior. We obtain general equations for such boundaries and consider them for three examples of the non-local memory functions. We show that there exist two types of the boundaries with fundamentally different system dynamics. On the boundaries of the first type, the diffusion with memory takes place, whereas on borderlines of the second type, the phenomenon of noise-induced resonance can be observed. A distinctive feature of noise-induced resonance in the systems under consideration is that it occurs in the absence of an external regular periodic force. It takes place due to the presence of frequencies in the noise spectrum, which are close to the self-frequency of the system. We analyze also the variance of the process and compare its behavior for regions of asymptotic stationarity and non-stationarity, as well as for diffusive and noise-induced-resonance borderlines between them.