Spatial fluctuations of a surviving particle in the trapping reaction

Details Spatial fluctuations of a surviving particle in the trapping reaction Spatial fluctuations of a surviving particle in the trapping reaction

2004-08-26

arxiv.org/abs/cond-mat/0408584v1

J. Phys. A: Math. Gen. 38 (2005) 133-144

Physics

Condensed Matter

Statistical Mechanics

10 pages, 5 figures

Scientific paper

10.1088/0305-4470/38/1/009

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, $^{1/2}$, 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 $t\gg 1$, $^{1/2} \propto t^{\phi}$ 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.

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