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Journal of Physics A Mathematical and General

Exponents appearing in the zero-temperature dynamics of the 1D Potts model

Bernard Derrida

Published Mar 23, 1995• 63 citations
Publisher

Abstract

For the Glauber dynamics of the one-dimensional q-state Potts model at zero temperature, the fraction of spins which have never flipped up to time t decreases as t - theta (q ) where the exponent theta (q) varies continuously with q. The calculation of the q dependence of this exponent can be reduced to the study of a simple reaction-diffusion model. Using finite-size scaling, this allows one to obtain accurate estimates of B(q) for all values of q. One can also show that for 1=1+ epsilon , the expansion of theta (y) is given by theta (1)=(2/ pi ) epsilon +O( epsilon 2 ).

DOI
10.1088/0305-4470/28/6/006

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For the Glauber dynamics of the one-dimensional q-state Potts model at zero temperature, the fraction of spins which have never flipped up to time t decreases as t - theta (q ) where the exponent theta (q) varies continuously with q. The calculation of the q dependence of this exponent can be reduced to the study of a simple reaction-diffusion model. Using finite-size scaling, this allows one to obtain accurate estimates of B(q) for all values of q. One can also show that for 1=1+ epsilon , the expansion of theta (y) is given by theta (1)=(2/ pi ) epsilon +O( epsilon 2 ).
We obtain θ_p(q) = 2θ_s(q) for one-dimensional q-state ferromagnetic Potts models evolving under parallel dynamics at zero temperature from an initially disordered state, where θ_p(q) is the persistence exponent for parallel dynamics and θ_s(q) = -{1/8}+ \frac{2}{π^2}[cos^{-1}{(2-q)/q\sqrt{2}}]^2 [PRL, {\bf 75}, 751, (1995)], the persistence exponent under serial dynamics. This result is a consequence of an exact, albeit non-trivial, mapping of the evolution of configurations of Potts spins under parallel dynamics to the dynamics of two decoupled reaction diffusion systems.
The distribution n(k,t) of the interval sizes k between clusters of persistent sites in the dynamical evolution of the one-dimensional q-state Potts model is studied using a combination of numerical simulations, scaling arguments, and exact analysis. It is shown to have the scaling form n(k,t)=t(-2z)f(k/t(z)), with z=max(1/2, straight theta), where straight theta(q) is the persistence exponent which describes the fraction P(t) approximately t(-straight theta) of sites which have not changed their state up to time t. When straight theta>1/2, the scaling length t(straight theta) for the interval-size distribution is larger than the coarsening length scale t(1/2) that characterizes spatial correlations of the Potts variables.
We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins $0\ensuremath{\le}x\ensuremath{\le}1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for $x\ensuremath{\le}0.5$, persistence for the up (minority) spins shows the behavior ${P}_{\text{min}}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}exp[\ensuremath{-}{(t/\ensuremath{\tau})}^{\ensuremath{\delta}}]$ with time ttt, while for the down (majority) spins, Pmaj(t){P}_{\text{maj}}(t)Pmaj​(t) approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where xc=0.5{x}_{c}=0.5xc​=0.5 and $\ensuremath{\lambda}\ensuremath{\simeq}2.31$. The exponent $\ensuremath{\gamma}$ varies as ${\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}$ where ${\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215$ is very close to the persistence exponent in two dimensions; $\ensuremath{\beta}\ensuremath{\simeq}1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]$, with $\ensuremath{\nu}\ensuremath{\approx}1.47$. This result suggests that $\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11$ is an exponent not explored earlier.