Block persistence

TL;DR

This paper introduces a scaling form for block persistence probability in phase ordering, demonstrating how it can be used to determine the global persistence exponent and its independence from temperature, supported by numerical results.

Contribution

It proposes a new scaling relation for block persistence probability and shows how to extract the global persistence exponent from simulations, applicable at finite temperatures.

Findings

Scaling form p_l(t) ~ l^{-zθ_0}f(t/l^z) verified at T=0

Global persistence exponent θ_0 can be obtained from simulations

Persistence probability decays exponentially at finite temperature due to thermal fluctuations

Abstract

We define a block persistence probability $p_l(t)$ as the probability that the order parameter integrated on a block of linear size $l$ has never changed sign since the initial time in a phase ordering process at finite temperature T<T_c. We argue that p_l(t)\sim l^{-z\theta_0}f(t/l^z) in the scaling limit of large blocks, where \theta_0 is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent \theta for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large n model, and generically it can be used to determine easily \theta_0 from simulations of coarsening models. We also argue that \theta_0 and the scaling function do not depend on temperature, leading to a definition of \theta at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. We also discuss conserved models for which different scaling are shown to arise depending on the value of the autocorrelation exponent \lambda. We illustrate our discussion by extensive numerical results. We also comment on the relation between this method and an alternative definition of \theta at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)].