Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes

TL;DR

This paper investigates the decay of persistent spins in a phase ordering process using a linear diffusion approximation, linking the persistence exponent to zero-crossings of stationary Gaussian processes, supported by simulations and an approximation method.

Contribution

It introduces a novel approach to compute the persistence exponent by relating it to zero-crossings of Gaussian processes and proposes an approximation method that aligns well with simulation results.

Findings

Persistence fraction decays as a power-law with a dimension-dependent exponent.

The exponent is linked to the asymptotic behavior of zero-crossing intervals.

The proposed approximation accurately predicts the exponent values.

Abstract

The fraction r(t) of spins which have never flipped up to time t is studied within a linear diffusion approximation to phase ordering. Numerical simulations show that, even in this simple context, r(t) decays with time like a power-law with a non-trival exponent $\theta$ which depends on the space dimension. The local dynamics at a given point is a special case of a stationary gaussian process of known correlation function and the exponent $\theta$ is shown to be determined by the asymptotic behavior of the probability distribution of intervals between consecutive zero-crossings of this process. An approximate way of computing this distribution is proposed, by taking the lengths of the intervals between successive zero-crossings as independent random variables. The approximation gives values of the exponent $\theta$ in close agreement with the results of simulations.