2025/04/23 by Sang H. Yang, Yang, Sang, Sun, Juyuan +4 · 1 citation
Mathematics · #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.16527
openalex publication_date 2025/04/23 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
Brownian motion in periodic potentials has been widely investigated in statistical physics and related interdisciplinary fields. In the overdamped regime, it has been well-known that the diffusion constant D^* is given by the Lifson-Jackson (LJ) formula. With a tilted potential, this model can exhibit giant diffusion. In this work, we start from the basic argument that since any quasi-periodic potential can be approximated accurately using a periodic potential, this formula and the associated physics should also apply to the quasi-periodic potential after some proper redefinition. We derive D^* from the Smoluchowski equation using the fact that its asymptotic solution is a product of a Boltzmann weight and a Gaussian envelope function. Then we analytically calculate D^* in terms of Bessel functions. Finally, we study the giant diffusion with quasi-periodic potentials, generalize the corresponding formula to the condition with tilted potential under the same argument, and calculate D^* analytically. This work generalizes the Brownian motion from periodic potentials to the much broader quasi-periodic potentials, which should have applications in interdisciplinary fields in physics, chemistry, engineering, and life sciences.