Asymptotic form of the approach to equilibrium in reversible recombination reactions

TL;DR

This paper analyzes the long-time approach to equilibrium in reversible recombination reactions, deriving a universal power-law decay for particle density and exploring effects of initial conditions and diffusion differences.

Contribution

It provides a systematic approximation scheme for the approach to equilibrium and reveals the universality of the power-law decay across dimensions.

Findings

Approach to equilibrium follows a power law At^(-d/2) for long times.

Amplitude of decay is computed exactly and is model dependent.

Reaction front width follows mean-field exponent regardless of dimension.

Abstract

The reversible reactions A+A<=>C and A+B<=>C are investigated. From the exact Langevin equations describing our model, we set up a systematic approximation scheme to compute the approach of the density of C particles to its equilibrium value. We show that for sufficiently long time t, this approach takes the form of a power law At^(-d/2), for any dimension d. The amplitude A is also computed exactly, but is expected to be model dependent. For uncorrelated initial conditions, the C density turns out to be a monotonic time function. The cases of correlated initial conditions and unequal diffusion constants are investigated as well. In the former, correlations may break the monotonicity of the density or in some special cases they may change the long time behavior. For the latter, the power law remains valid, only the amplitude changes, even in the extreme case of immobile C particles. We also consider the case of segregated initial condition for which a reaction front is observed, and confirm that its width is governed by mean-field exponent in any dimension.