The European Physical Journal B (EPJ B)

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Eur. Phys. J. B 7, 111-127

Block persistence

S. Cueille - C. Sire

Laboratoire de Physique Quantique, Université Paul Sabatier, 31062 Toulouse Cedex, France
cueille@irsamc2.ups-tlse.fr, clement@irsamc2.ups-tlse.fr

Received: 25 February 1998 / Revised: 24 July 1998 / Accepted: 27 July 1998

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 z is the growth exponent ( $L(t)\sim t^$ ), $\theta_0$ 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. These ideas are applied to the study of persistence for conserved models. We illustrate our discussions 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)].

PACS
02.50.-r Probability theory, stochastic processes, and statistics - 05.40.+j Fluctuation phenomena, random processes, and Brownian motion - 05.20.-y Statistical mechanics

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