Exponents appearing in heterogeneous reaction-diffusion models in one dimension

Cécile Monthus

doi:10.1103/PhysRevE.54.4844

paper

Exponents appearing in heterogeneous reaction-diffusion models in one dimension

arXiv:cond-mat/9607030 · doi:10.1103/PhysRevE.54.4844

Abstract

We study the following 1D two-species reaction diffusion model : there is a small concentration of B-particles with diffusion constant $D_{B}$ in an homogenous background of W-particles with diffusion constant $D_{W}$ ; two W-particles of the majority species either coagulate ( $W + W ⟶ W$ ) or annihilate ( $W + W ⟶ \emptyset$ ) with the respective probabilities $p_{c} = (q - 2) / (q - 1)$ and $p_{a} = 1/ (q - 1)$ ; a B-particle and a W-particle annihilate ( $W + B ⟶ \emptyset$ ) with probability 1. The exponent $θ (q, λ = D_{B} / D_{W})$ describing the asymptotic time decay of the minority B-species concentration can be viewed as a generalization of the exponent of persistent spins in the zero-temperature Glauber dynamics of the 1D $q$ -state Potts model starting from a random initial condition : the W-particles represent domain walls, and the exponent $θ (q, λ)$ characterizes the time decay of the probability that a diffusive "spectator" does not meet a domain wall up to time $t$ . We extend the methods introduced by Derrida, Hakim and Pasquier ({\em Phys. Rev. Lett.} {\bf 75} 751 (1995); Saclay preprint T96/013, to appear in {\em J. Stat. Phys.} (1996)) for the problem of persistent spins, to compute the exponent $θ (q, λ)$ in perturbation at first order in $(q - 1)$ for arbitrary $λ$ and at first order in $λ$ for arbitrary $q$ .

29 pages. The three figures are not included, but are available upon request

Exponents appearing in heterogeneous reaction-diffusion models in one dimension

Abstract

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