Annihilating random walks in one-dimensional disordered media

Details Annihilating random walks in one-dimensional disordered media Annihilating random walks in one-dimensional disordered media

1998-01-12

arxiv.org/abs/cond-mat/9801103v1

Physics

Condensed Matter

Statistical Mechanics

LaTeX, 2 eps-figures, to be published in PRE

Scientific paper

10.1103/PhysRevE.57.2563

We study diffusion-limited pair annihilation $A+A\to 0$ on one-dimensional lattices with inhomogeneous nearest neighbour hopping in the limit of infinite reaction rate. We obtain a simple exact expression for the particle concentration $\rho_k(t)$ of the many-particle system in terms of the conditional probabilities $P(m;t|l;0)$ for a single random walker in a dual medium. For some disordered systems with an initially randomly filled lattice this leads asymptotically to $\bar{\rho(t)}=\bar{P(0;2t|0;0)}$ for the disorder-averaged particle density. We also obtain interesting exact relations for single-particle conditional probabilities in random media related by duality, such as random-barrier and random-trap systems. For some specific random barrier systems the Smoluchovsky approach to diffusion-limited annihilation turns out to fail.

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