Satya N. Majumdar

In this paper, we present a detailed calculation of the persistence exponent $\theta$ for a nearly Markovian Gaussian process $X\left(t\right),$ a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for $\theta$ are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of $\theta$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent $\theta {(X}_0),$ describing the probability that the process remains larger than $X_0\sqrt{〈X^2\mathrm{}\left(t\right)\mathrm{}〉}.$

We study the decay of the probability for a non-Markovian stationary Gaussian walker not to cross the origin up to time $t$. This result is then used to evaluate the fraction of spins that do not flip up to time $t$ in the zero temperature Monte-Carlo spin flip dynamics of the Ising model. Our results are compared to extensive numerical simulations.

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 $L\left(t\right)\sim t^{1/2}$, while the autocorrelation function decays as $L(t{)}^{-\lambda \left(q\right)}$. We also establish a correspondence of this model to the Ising model evolving with a fixed magnetization $(2/q-1)$. Extensive numerical simulations of the Potts model in two dimensions show good agreement with our theory.

We study the nonequilibrium dynamics of the $q$-state Potts model following a quench from the high temperature disordered phase to zero temperature. The time dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale $L(t)\sim t^{1/2}$. In particular, the autocorrelation function decays as $L(t)^{-\lambda(q)}$. We illustrate these properties by solving exactly the kinetic Potts model in $d=1$. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general $q$ and $d$. We establish a correspondence between the two-point correlations of the $q$-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization $(2/q-1)$. The dynamics of this Ising model is solved exactly in the large q limit, and in the limit of a large number of components $n$ for the order parameter. For general $q$ and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent $\lambda (q)$. These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.