Spatial fluctuations of a surviving particle in the trapping reaction

TL;DR

This paper analyzes the spatial fluctuations of a surviving particle in a trapping reaction, showing that in one dimension, the fluctuation grows as a power law with exponent 1/4 for unconstrained paths, supported by numerical evidence.

Contribution

It provides a theoretical calculation of the asymptotic spatial fluctuation growth for a surviving particle in a trapping reaction in one dimension, using an effective dynamics approach.

Findings

Fluctuation growth exponent for surviving particle: 1/4

Path-constrained fluctuations grow with exponent 1/3

Numerical results support the theoretical predictions

Abstract

We consider the trapping reaction, $A+B\to B$, where $A$ and $B$ particles have a diffusive dynamics characterized by diffusion constants $D_A$ and $D_B$. The interaction with $B$ particles can be formally incorporated in an effective dynamics for one $A$ particle as was recently shown by Bray {\it et al}. [Phys. Rev. E {\bf 67}, 060102 (2003)]. We use this method to compute, in space dimension $d=1$, the asymptotic behaviour of the spatial fluctuation, $<z^2(t)>^{1/2}$, for a surviving $A$ particle in the perturbative regime, $D_A/D_B\ll 1$, for the case of an initially uniform distribution of $B$ particles. We show that, for $t\gg 1$, $<z^2(t)>^{1/2} \propto t^{\phi}$ with $\phi=1/4$. By contrast, the fluctuations of paths constrained to return to their starting point at time $t$ grow with the larger exponent 1/3. Numerical tests are consistent with these predictions.