Approach to equilibrium in adiabatically evolving potentials

TL;DR

This paper investigates how a one-dimensional potential evolving adiabatically affects the relaxation of probability densities, revealing that time dependence can enhance diffusion over barriers and induce resonance effects.

Contribution

It introduces a model for adiabatic potential evolution and demonstrates how it influences relaxation times and resonance phenomena in overdamped dynamics.

Findings

Adiabatic evolution decreases Kramers time, speeding up barrier crossing.

Time dependence can induce resonance effects between potential scales.

Diffusion over barriers becomes more efficient with potential evolution.

Abstract

For a potential function (in one dimension) which evolves from a specified initial form $V_{i}(x)$ to a different $V_{f}(x)$ asymptotically, we study the evolution, in an overdamped dynamics, of an initial probability density to its final equilibeium.There can be unexpected effects that can arise from the time dependence. We choose a time variation of the form $V(x,t)=V_{f}(x)+(V_{i}-V_{f})e^{-\lambda t}$. For a $V_{f}(x)$, which is double welled and a $V_{i}(x)$ which is simple harmonic, we show that, in particular, if the evolution is adiabatic, the results in a decrease in the Kramers time characteristics of $V_{f}(x)$. Thus the time dependence makes diffusion over a barrier more efficient. There can also be interesting resonance effects when $V_{i}(x)$ and $V_{f}(x)$ are two harmonic potentials displaced with respect to each other that arise from the coincidence of the intrinsic time scale characterising the potential variation and the Kramers time.