arXiv subjects

S. J. O'Donoghue

Publications and source records attributed to S. J. O'Donoghue.

4 recordsLinked to original sources

arXiv2002

Fraction of uninfected walkers in the one-dimensional Potts model

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.

cond-mat.stat-mech↗

arXiv2001

Uninfected random walkers in one dimension

We consider a system of unbiased diffusing walkers (A0 <-> 0A) in one dimension with random initial conditions. We investigate numerically the relation between the fraction of walkers, U(t), which have never encountered another walker up to time t, calling such walkers ``uninfected'' and the fraction of sites, P(t), which have never been visited by a diffusing particle. We extend our study to include the A + B -> 0 diffusion-limited reaction in one-dimension, with equal initial densities of A and B particles distributed homogeneously at t=0. We find U(t) \simeq [P(t)]^γ, with γ\simeq 1.39, in both models, though there is evidence that a smaller value of γis required for t -> infinity.

cond-mat.stat-mech↗

arXiv2001

Persistence in the One-Dimensional A+B -> 0 Reaction-Diffusion Model

The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t) -> const + t^{-θ}$, where $θ$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $ρ>> 1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $θ\approx 0.1207$. In the case of an initially low density, $ρ_0 << 1$, we find $θ\approx 1/4$ asymptotically. The probability that a site remains unvisited by any random walker (``type II persistence'') is also investigated and found to decay with a stretched exponential form, $P(t) \sim \exp(-const ρ_0^{1/2}t^{1/4})$, provided $ρ_0 << 1$. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.

cond-mat.stat-mech↗

arXiv2000

Unusual Dynamical Scaling in the Spatial Distribution of Persistent Sites in 1D Potts Models

The distribution, n(k,t), of the interval sizes, k, between clusters of persistent sites in the dynamical evolution of the one-dimensional q-state Potts model is studied using a combination of numerical simulations, scaling arguments, and exact analysis. It is shown to have the scaling form n(k,t) = t^{-2z} f(k/t^z), with z= max(1/2,theta), where theta(q) is the persistence exponent which characterizes the fraction of sites which have not changed their state up to time t. When theta > 1/2, the scaling length, t^theta, for the interval-size distribution is larger than the coarsening length scale, t^{1/2}, that characterizes spatial correlations of the Potts variables.

cond-mat.stat-mech↗