Dynamic crossover in the persistence probability of manifolds at criticality

Details Dynamic crossover in the persistence probability of manifolds at criticality Dynamic crossover in the persistence probability of manifolds at criticality

2010-10-04

arxiv.org/abs/1010.0560v2

J. Stat. Mech. (2010) P12029

Physics

Condensed Matter

Statistical Mechanics

23 pages, 6 figures; minor changes, added one figure, (old) fig.4 replaced by the correct fig.5

Scientific paper

10.1088/1742-5468/2010/12/P12029

We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter \zeta = (D-2+\eta)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical exponents z and \eta. We find that the asymptotic behavior of P(t) is exponential for \zeta > 1, stretched exponential for 0 <= \zeta <= 1, and algebraic for \zeta < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d'=d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising model, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O(n) model in the limit n->\infty and its subtle connection with the spherical model is also discussed in detail.

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