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=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
P2(r,t) has the scaling form
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) 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.
The spatial distribution of persistent (unvisited) sites in one dimensional
A+A→∅ 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)∼t−θk−τf(k/tz). The new exponents
τ and
z change with the initial particle density
n0, 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
n0,
1<τ<2, which is confirmed by numerical results.
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 spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ( t )~ t Z , is identified such that for length scales r <<ξ( t ) the persistent spins form a fractal with dimension d f ; for length scales r >>ξ( t ) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z (2- d f ) with θ = 0.209±0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and d f ~1.58.
We study the nonequilibrium dynamics of the
q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. We calculate, within a Gaussian closure approximation, the time-dependent two-point correlation functions for general
q. These correlation functions obey dynamic scaling with a length scale $L(t)\ensuremath{\sim}{t}^{1/2}$, while the autocorrelation function decays as $L(t{)}^{\ensuremath{-}\ensuremath{\lambda}(q)}$. We also establish a correspondence of this model to the Ising model evolving with a fixed magnetization $(2/q\ensuremath{-}1)$. Extensive numerical simulations of the Potts model in two dimensions show good agreement with our theory.