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The theory of phase ordering dynamics -the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a brokensymmetry phase -is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time-dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, e.g. vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings, monopoles) provides a unifying framework for discussing coarsening in these different systems.

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

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The diffusion equation \partial_t\phi = \nabla^2\phi is considered, with initial condition \phi( _x_ ,0) a gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p_n(t_1,t_2) that \phi( _x_ ,t) [for a given space point _x_ ] changes sign n times between t_1 and t_2 has the asymptotic form p_n(t_1,t_2) \sim [\ln(t_2/t_1)]^n(t_1/t_2)^{-\theta}. The exponent \theta has predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.Comment: Minor typos corrected, affecting table of exponents. 4 pages, REVTEX, 1 eps figure. Uses epsf.sty and multicol.st

Numerical and analytic results for the exponent θ describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β < 1; for β = 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θS = 1 − β. The exponent θ0 for the flat initial condition appears to be nontrivial. We prove that θ0 → ∞ for β → 0, θ0 ≥ θS for β < 1/2 and θ0 ≤ θS for β > 1/2, and calculate θ0,S perturbatively to first order in an expansion around the Markovian case β = 1/2. Using the exact result θS = 1 − β, accurate upper and lower bounds on θ0 can be derived which show, in particular, that θ0 ≥ (1 − β) 2 /β for small β.

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