Decay to the nonequilibrium steady state of the thermal diffusion in a tilted periodic potential

TL;DR

This paper studies how thermal diffusion in a tilted periodic potential approaches a nonequilibrium steady state, revealing how decay rates depend on parameters through analytical and numerical methods involving the Fokker-Planck and Schrödinger equations.

Contribution

It introduces a detailed analysis of decay rates in tilted periodic potentials using WKB methods, covering small, intermediate, and large tilting cases with new analytical insights.

Findings

Decay rate saturates at large tilting, depending only on temperature, potential periodicity, and viscosity.

Eigenvalues of the Fokker-Planck operator are nearly real in small tilting cases.

Analytical results show good agreement with numerical data across various tilting regimes.

Abstract

We investigate asymptotic decay phenomenon towards the nonequilibrium steady state of the thermal diffusion in the presence of a tilted periodic potential. The parameter dependence of the decay rate is revealed by investigating the Fokker-Planck (FP) equation in the low temperature case under the spatially periodic boundary condition (PBC). We apply the WKB method to the associated Schr\"odinger equation. While eigenvalues of the non-Hermitian FP operator are complex in general, in a small tilting case the imaginary parts of the eigenvalues are almost vanishing. Then the Schr\"odinger equation is solved with PBC. The decay rate is analyzed in the context of quantum tunneling through a triple-well effective periodic potential. In a large tilting case, the imaginary parts of the eigenvalues of FP operator are crucial. We apply the complex-valued WKB method to the Schr\"odinger equation with the absorbing boundary condition, finding that the decay rate saturates and depends only on the temperature, the potential periodicity and the viscous constant. The intermediate tilting case is also explored. The analytic results agree well with the numerical data for a wide range of tilting.