Spatial distribution of persistent sites

Abstract

We study the distribution of persistent sites (sites unvisited by particles $A$) in one dimensional $A+A\to \varnothing$ reaction-diffusion model. We define the {\it empty intervals} as the separations between adjacent persistent sites, and study their size distribution $n(k,t)$ as a function of interval length $k$ and time $t$. The decay of persistence is the process of irreversible coalescence of these empty intervals, which we study analytically under the Independent Interval Approximation (IIA). Physical considerations suggest that the asymptotic solution is given by the dynamic scaling form $n(k,t)=s^{-2}f(k/s)$ with the average interval size $s\sim t^{1/2}$. We show under the IIA that the scaling function $f(x)\sim x^{-\tau }$ as $x\to 0$ and decays exponentially at large $x$. The exponent $\tau$ is related to the persistence exponent $\theta$ through the scaling relation $\tau =2(1-\theta )$. We compare these predictions with the results of numerical simulations. We determine the two-point correlation function $C(r,t)$ under the IIA. We find that for $r\ll s$, $C(r,t)\sim r^{-\alpha }$ where $\alpha =2-\tau$, in agreement with our earlier numerical results.

15 pages in RevTeX, 5 postscript figures

References in corpus (1)

• Persistence of Kardar-Parisi-Zhang Interfaces

Cited by in corpus (19)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• A new model of binary opinion dynamics: coarsening and effect of disorder
• Persistence in higher dimensions : a finite size scaling study
• Persistence in the two dimensional ferromagnetic Ising model
• Opinion dynamics model with weighted influence: Exit probability and dynamics
• A stochastic opinion dynamics model with domain size dependent dynamic evolution
• Critical scaling in standard biased random walks
• Fractality in Persistence Decay and Domain Growth during Ferromagnetic Ordering: Dependence upon initial correlation
• Fraction of uninfected walkers in the one-dimensional Potts model
• Opinion dynamics model with domain size dependent dynamics: novel features and new universality class
• Persistence in Cluster--Cluster Aggregation
• Zero temperature dynamics in two dimensional ANNNI model
• Noise driven dynamic phase transition in a a one dimensional Ising-like model
• Persistence in q-state Potts model: A Mean-Field approach
• Persistence in systems with conserved order parameter
• Novel ballistic to diffusive crossover in the dynamics of a one dimensional Ising model with variable range of interaction
• Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation
• Aging Exponents for Nonequilibrium Dynamics following Quenches from Critical Point
• Zero temperature ordering dynamics in two dimensional BNNNI model