Dynamical systems with multiplicative noise: Time-scale competition, delayed feedback and effective drifts

TL;DR

This paper reviews the theoretical, experimental, and numerical aspects of noise-induced drifts in stochastic differential equations with multiplicative noise, highlighting recent advances and future directions in the field.

Contribution

It provides an extensive review of the state of the art in noise-induced drifts, including case studies, mathematical methods, and experimental results, with insights into future research directions.

Findings

Experimental evidence of noise-induced drift in Brownian motion with diffusion gradients

Mathematical techniques for analyzing systems with multiplicative noise

Numerical methods for simulating noise-driven dynamical systems

Abstract

Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales became very small. In the presence of multiplicative noise (i.e., noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient, and the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback. This allows us to present the experimental results, as well as mathematical methods and numerical techniques that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment.