Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers

TL;DR

This paper analyzes the survival probability decay of a diffusing test particle in a system of coagulating and annihilating particles, using renormalization group methods and simulations to explore universality and corrections.

Contribution

It provides a perturbative renormalization group calculation of the survival exponent in dimensions less than two, including logarithmic corrections in two dimensions, and compares with exact solutions and simulations.

Findings

Survival probability decays algebraically as t^{-theta} in d<2.

Theta depends only on diffusion and reaction rate ratios, not on the test particle's death rate.

Logarithmic corrections in two dimensions are non-universal.

Abstract

We calculate the survival probability of a diffusing test particle in an environment of diffusing particles that undergo coagulation at rate lambda_c and annihilation at rate lambda_a. The test particle dies at rate lambda' on coming into contact with the other particles. The survival probability decays algebraically with time as t^{-theta}. The exponent theta in d<2 is calculated using the perturbative renormalization group formalism as an expansion in epsilon=2-d. It is shown to be universal, independent of lambda', and to depend only on delta, the ratio of the diffusion constant of test particles to that of the other particles, and on the ratio lambda_a/lambda_c. In two dimensions we calculate the logarithmic corrections to the power law decay of the survival probability. Surprisingly, the log corrections are non-universal. The one loop answer for theta in one dimension obtained by setting epsilon=1 is compared with existing exact solutions for special values of delta and lambda_a/lambda_c. The analytical results for the logarithmic corrections are verified by Monte Carlo simulations.