Fraction of uninfected walkers in the one-dimensional Potts model

Abstract

The dynamics of the one-dimensional q-state Potts model, in the zero temperature limit, can be formulated through the motion of random walkers which either annihilate (A + A -> 0) or coalesce (A + A -> A) with a q-dependent probability. We consider all of the walkers in this model to be mutually infectious. Whenever two walkers meet, they experience mutual contamination. Walkers which avoid an encounter with another random walker up to time t remain uninfected. The fraction of uninfected walkers is investigated numerically and found to decay algebraically, U(t) \sim t^{-ϕ(q)}, with a nontrivial exponent ϕ(q). Our study is extended to include the coupled diffusion-limited reaction A+A -> B, B+B -> A in one dimension with equal initial densities of A and B particles. We find that the density of walkers decays in this model as ρ(t) \sim t^{-1/2}. The fraction of sites unvisited by either an A or a B particle is found to obey a power law, P(t) \sim t^{-θ} with θ\simeq 1.33. We discuss these exponents within the context of the q-state Potts model and present numerical evidence that the fraction of walkers which remain uninfected decays as U(t) \sim t^{-ϕ}, where ϕ\simeq 1.13 when infection occurs between like particles only, and ϕ\simeq 1.93 when we also include cross-species contamination.

Expanded introduction with more discussion of related work

References in corpus (3)

• Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model
• Persistence in One-dimensional Ising Models with Parallel Dynamics
• Uninfected random walkers in one dimension

Cited by in corpus (10)

• Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps
• Formal Solution of a Class of Reaction-Diffusion Models: Reduction to a Single-Particle Problem
• Persistence properties of a system of coagulating and annihilating random walkers
• Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers
• Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps
• Segregation in diffusion-limited multispecies pair annihilation
• Perturbation theory for the one-dimensional trapping reaction
• Persistence in Cluster--Cluster Aggregation
• Uninfected random walkers in one dimension
• Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation