Langevin dynamics with a tilted periodic potential

TL;DR

This paper investigates how small stochastic noise influences the long-term behavior of a particle in a tilted periodic potential, showing that noise suppresses unbounded orbits and stabilizes confined ones over time.

Contribution

It provides a rigorous analysis of the effect of small noise on Langevin dynamics with a tilted periodic potential, highlighting the survival of confined orbits.

Findings

Small noise eliminates unbounded orbits over sub-exponential timescales.

Confined orbits persist in the presence of arbitrarily small stochastic perturbations.

Deterministic heteroclinic orbits are destabilized by noise, leading to confinement.

Abstract

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity $\gamma$ and subject to a further external field $\alpha$. For a suitable choice of the parameters $\alpha$ and $\gamma$ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale.