Exact Exponent $\lambda$ of the Autocorrelation Function for a Soluble Model of Coarsening

TL;DR

This paper calculates the exact autocorrelation decay exponent for a 1D phase ordering model, revealing precise decay behavior and its relation to magnetization growth, advancing understanding of coarsening dynamics.

Contribution

The paper provides the exact value of the autocorrelation decay exponent for the 1D Ginzburg-Landau model and establishes its equality with the magnetization growth exponent.

Findings

Exact exponent $oxed{ ext{0.3993835}}$ for autocorrelation decay in 1D

Demonstration that $oxed{ ext{$ ext{lambda} = ext{mu}$}}$ in the model

Explicit relation between bias-induced magnetization growth and autocorrelation decay

Abstract

The exponent $\lambda$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t) \sim L^{-(d-\lambda)}$, where $d$ is the dimension and $L$ the characteristic length scale at time $t$, is calculated exactly for the time-dependent Ginzburg-Landau equation in $d=1$. We find $\lambda = 0.399\,383\,5\ldots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t) \sim L^\mu$ and prove that for the $1d$ Ginzburg-Landau equation, $\mu=\lambda$, exemplifying a general result.