Approach to equilibrium in adiabatically evolving potentials

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 equilibrium. 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^-λt. 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, this results in a decrease in the Kramers time characteristic 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 characterizing the potential variation and the Kramers time. Both these features are illustrated through representative examples.