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arXiv:cond-mat/0211453 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Persistence in q -state Potts model: A mean-field approach

G. Manoj

Published Feb 18, 2003• 5 citations
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Abstract

We study the persistence properties of the $T=0$ coarsening dynamics of one-dimensional q-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density $P(t)$ of persistent spins is imposed. For this model, it is known that $P(t)$ follows a power-law decay with time, $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}(q)},$ where $\ensuremath{\theta}(q)$ is the q-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function ${P}_{2}(r,t)$ has the scaling form ${P}_{2}{(r,t)=P(t)}^{2}{f(r/t}^{1/2})$ for all values of the persistence exponent $\ensuremath{\theta}(q).$ The scaling function has the limiting behavior $f(x)\ensuremath{\sim}{x}^{\ensuremath{-}2\ensuremath{\theta}}$ $(x\ensuremath{\ll}1)$ and $f(x)\ensuremath{\rightarrow}1$ $(x\ensuremath{\gg}1).$ We then show within the independent interval approximation (IIA) that the distribution $n(k,t)$ of separation k between two consecutive persistent spins at time t has the asymptotic scaling form ${n(k,t)=t}^{\ensuremath{-}2\ensuremath{\varphi}}{g(t,k/t}^{\ensuremath{\varphi}}),$ where the dynamical exponent has the form $\ensuremath{\varphi}=\mathrm{max}(1/2,\ensuremath{\theta}).$ The behavior of the scaling function for large and small values of the arguments is found analytically. We find that for small separations $k\ensuremath{\ll}{t}^{\ensuremath{\varphi}},n(k,t)\ensuremath{\sim}{P(t)k}^{\ensuremath{-}\ensuremath{\tau}},$ where $\ensuremath{\tau}=\mathrm{max}[2(1\ensuremath{-}\ensuremath{\theta}),2\ensuremath{\theta}],$ while for large separations $k\ensuremath{\gg}{t}^{\ensuremath{\varphi}},$ $g(t,x)$ decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations.

DOI
10.1103/physreve.67.026115

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We study the persistence properties of the T=0T=0T=0 coarsening dynamics of one-dimensional q-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density P(t)P(t)P(t) of persistent spins is imposed. For this model, it is known that P(t)P(t)P(t) follows a power-law decay with time, P(t)\ensuremath∼t\ensuremath−\ensuremathθ(q),P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}(q)},P(t)\ensuremath∼t\ensuremath−\ensuremathθ(q), where \ensuremathθ(q)\ensuremath{\theta}(q)\ensuremathθ(q) is the q-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function P2(r,t){P}_{2}(r,t)P2​(r,t) has the scaling form P2(r,t)=P(t)2f(r/t1/2){P}_{2}{(r,t)=P(t)}^{2}{f(r/t}^{1/2})P2​(r,t)=P(t)2f(r/t1/2) for all values of the persistence exponent \ensuremathθ(q).\ensuremath{\theta}(q).\ensuremathθ(q). The scaling function has the limiting behavior f(x)\ensuremath∼x\ensuremath−2\ensuremathθf(x)\ensuremath{\sim}{x}^{\ensuremath{-}2\ensuremath{\theta}}f(x)\ensuremath∼x\ensuremath−2\ensuremathθ (x\ensuremath≪1)(x\ensuremath{\ll}1)(x\ensuremath≪1) and f(x)\ensuremath→1f(x)\ensuremath{\rightarrow}1f(x)\ensuremath→1 (x\ensuremath≫1).(x\ensuremath{\gg}1).(x\ensuremath≫1). We then show within the independent interval approximation (IIA) that the distribution n(k,t)n(k,t)n(k,t) of separation k between two consecutive persistent spins at time t has the asymptotic scaling form n(k,t)=t\ensuremath−2\ensuremathφg(t,k/t\ensuremathφ),{n(k,t)=t}^{\ensuremath{-}2\ensuremath{\varphi}}{g(t,k/t}^{\ensuremath{\varphi}}),n(k,t)=t\ensuremath−2\ensuremathφg(t,k/t\ensuremathφ), where the dynamical exponent has the form \ensuremathφ=max(1/2,\ensuremathθ).\ensuremath{\varphi}=\mathrm{max}(1/2,\ensuremath{\theta}).\ensuremathφ=max(1/2,\ensuremathθ). The behavior of the scaling function for large and small values of the arguments is found analytically. We find that for small separations k\ensuremath≪t\ensuremathφ,n(k,t)\ensuremath∼P(t)k\ensuremath−\ensuremathτ,k\ensuremath{\ll}{t}^{\ensuremath{\varphi}},n(k,t)\ensuremath{\sim}{P(t)k}^{\ensuremath{-}\ensuremath{\tau}},k\ensuremath≪t\ensuremathφ,n(k,t)\ensuremath∼P(t)k\ensuremath−\ensuremathτ, where \ensuremathτ=max[2(1\ensuremath−\ensuremathθ),2\ensuremathθ],\ensuremath{\tau}=\mathrm{max}[2(1\ensuremath{-}\ensuremath{\theta}),2\ensuremath{\theta}],\ensuremathτ=max[2(1\ensuremath−\ensuremathθ),2\ensuremathθ], while for large separations k\ensuremath≫t\ensuremathφ,k\ensuremath{\gg}{t}^{\ensuremath{\varphi}},k\ensuremath≫t\ensuremathφ, g(t,x)g(t,x)g(t,x) decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations.
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The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.