Persistence in q-state Potts model: a mean-field approach.
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) approximately t(-theta(q)), where 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)(2)f(r/t(1/2)) for all values of the persistence exponent theta(q). The scaling function has the limiting behavior f(x) approximately x(-2theta) (x<<1) and f(x)-->1 (x>>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(-2phi)g(t,k/t(phi)), where the dynamical exponent has the form phi=max(1/2,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< >t(phi), g(t,x) decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations.