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arXiv:cond-mat/9901130

Dynamic scaling in the spatial distribution of persistent sites

G. Manoj, Purusattam Ray

Published Jan 14, 1999• 1 citation
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Abstract

The spatial distribution of persistent (unvisited) sites in one dimensional $A+A\to\emptyset$ model is studied. The `empty interval distribution' $n(k,t)$, which is the probability that two consecutive persistent sites are separated by distance $k$ at time $t$ is investigated in detail. It is found that at late times this distribution has the dynamical scaling form $n(k,t)\sim t^{-θ}k^{-τ}f(k/t^{z})$. The new exponents $τ$ and $z$ change with the initial particle density $n_{0}$, and are related to the persistence exponent $θ$ through the scaling relation $z(2-τ)=θ$. We show by rigorous analytic arguments that for all $n_{0}$, $1< τ< 2$, which is confirmed by numerical results.

DOI
10.48550/arxiv.cond-mat/9901130

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The spatial distribution of persistent (unvisited) sites in one dimensional A+A→∅A+A\to\emptysetA+A→∅ model is studied. The `empty interval distribution' n(k,t)n(k,t)n(k,t), which is the probability that two consecutive persistent sites are separated by distance kkk at time ttt is investigated in detail. It is found that at late times this distribution has the dynamical scaling form n(k,t)∼t−θk−τf(k/tz)n(k,t)\sim t^{-θ}k^{-τ}f(k/t^{z})n(k,t)∼t−θk−τf(k/tz). The new exponents τττ and zzz change with the initial particle density n0n_{0}n0​, and are related to the persistence exponent θθθ through the scaling relation z(2−τ)=θz(2-τ)=θz(2−τ)=θ. We show by rigorous analytic arguments that for all n0n_{0}n0​, 1<τ<21< τ< 21<τ<2, which is confirmed by numerical results.
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 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, $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}(q)},$ 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 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 $\ensuremath{\theta}(q).$ 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)n(k,t)n(k,t) of separation k between two consecutive persistent spins at time t has the asymptotic scaling form ${n(k,t)=t}^{\ensuremath{-}2\ensuremath{\varphi}}{g(t,k/t}^{\ensuremath{\varphi}}),$ 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)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.
The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).
A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at x=0x=0x=0 at time t=0t=0t=0, the persistence probability P(x,t)P(x,t)P(x,t) that a site at xxx has not been visited till time ttt has been calculated. P(x,t)P(x,t)P(x,t) behaves as (t/∣x∣−1)−α(t/|x|-1)^{-α}(t/∣x∣−1)−α with α∼0.3α\sim 0.3α∼0.3 while the global fraction P(t)=∑xP(x,t)/2t{\cal{P}}(t) = \sum_xP(x,t)/2tP(t)=∑x​P(x,t)/2t of sites remaining unvisited at time ttt attains a constant value. F(x,t)F(x,t)F(x,t), the probability that the site at xxx is visited for the first time at ttt behaves as (t/∣x∣−1)−β/∣x∣(t/|x|-1)^{-β}/|x|(t/∣x∣−1)−β/∣x∣ where β=1+αβ= 1+ αβ=1+α for t/∣x∣>>1t/|x|>> 1t/∣x∣>>1,and F(t)=∑xF(x,t)/2t∼1/t{\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/tF(t)=∑x​F(x,t)/2t∼1/t. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.
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