Effective diffusion of Brownian motion in spatially quasi-periodic noise

Abstract

The effective diffusion of Brownian particles in periodic potential has been a central topic in nonequilibrium statistical physcis. A classical result is the Lifson formula which provides the effective diffusion constant in periodic potentials. Extending beyong periodicity, our recent work [arXiv:2504.16527] has demonstrated that a modified Lifson expression remains valid for Brownian motion in quasi-periodic potentials. In this work, we extend our previous results by incorporating spatial quasi-periodic noise and examining different stochastic interpretations, $α\in[0,1]$. The proposed framework is simple, computationally efficient, and unifies the treatment of diffusion in both periodic and quasi-periodic systems.

Figures (1)

• Figure 1: The effective diffusion with different intepreter factor. (a) The probability density function $p(x,t)$ with $\alpha=1/2$ and $D(x)=1+\sin(x)/4+\sin(x/\sqrt{2})/4$. (b) Gussian-like function $g(x,t)$ with same parameter as (a). (c) The effective diffusion cosntant with different diffusion coefficients and potential functions (eqn. \ref{['Dstar-alpha-1']} and \ref{['Dstar-alpha-2']}). The periodic and quasi-periodic potential are $\sin(x)$ and $\sin(x)/2+\sin(\sqrt{3}x)/2$, respectively. The periodic and quasi-periodic diffusion coefficient are $1+\sin(x)/2$ and $1+\sin(x)/4+\sin(\sqrt{2})/4$, respectively.