Detailed balance for diffusion in a potential with trapping and forward-backward symmetry of trapping time distributions.

Alexander M Berezhkovskii,Dmitrii E Makarov

doi:10.1063/1.5142566

Abstract

For particles diffusing in a potential, detailed balance guarantees the absence of net fluxes at equilibrium. Here, we show that the conventional detailed balance condition is a special case of a more general relation that works when the diffusion occurs in the presence of a distributed sink that eventually traps the particle. We use this relation to study the lifetime distribution of particles that start and are trapped at specified initial and final points. It turns out that when the sink strength at the initial point is nonzero, the initial and final points are interchangeable, i.e., the distribution is independent of which of the two points is initial and which is final. In other words, this conditional trapping time distribution possesses forward–backward symmetry.

Highlights

Consider a particle diffusing along a one-dimensional coordinate x in a constraining potential U(x), U(x)∣∣x∣→∞ → ∞
In Eq (1), D is the particle diffusivity, and β = 1/(kBT), where kB and T are the Boltzmann constant and absolute temperature. This propagator describes the relaxation of the initial δ-distribution to the equilibrium one, lim t→∞
There is no equilibrium in this case, the propagator satisfies the detailed balance condition [Eq (3)]

Summary

Introduction

Consider a particle diffusing along a one-dimensional coordinate x in a constraining potential U(x), U(x)∣∣x∣→∞ → ∞. We show that the conventional detailed balance condition is a special case of a more general relation that works when the diffusion occurs in the presence of a distributed sink that eventually traps the particle. This conditional trapping time distribution possesses forward–backward symmetry.

Results

Conclusion