Dynamic crossover in the global persistence at criticality

Abstract

We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter (e.g., the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m0 of the order parameter and the typical time at which it occurs diverges as m0 vanishes. Monte-Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.

Persistence and first-passage properties in nonequilibrium systems

“…For the pure Ising model in d = 2, the most precise estimate yields θ G = 0.237(3) [198] in agreement with other Monte-Carlo estimates [192,197,199], which is still slightly larger than the perturbative result (278) with = 4 − d = 2, yielding θ G = 0.218 (where we have used λ G = 1.585 [200], η = 1/4 (exact) and z = 2.166 [201]). Note that in this case, it was checked that Metropolis and HeatBath algorithms both yield the same value of θ G [198].…”

“…(199) for θ can be written as a real-time integral as follows. We first write A(T ) = A 0 (T ) + a(T ), and we note that, since G(T ) is the inverse function of…”

“…The exponent θ G has also been measured in numerical simulations of coarsening ferromagnets at T = T c evolving via Model A dynamics both for the pure system [192,[197][198][199] in dimension d = 2, 3 and for the diluted Ising model in d = 3 [195] (in d = 2 the random dilution yields only logarithmic corrections to the pure case). For the pure Ising model in d = 2, the most precise estimate yields θ G = 0.237(3) [198] in agreement with other Monte-Carlo estimates [192,197,199], which is still slightly larger than the perturbative result (278) with = 4 − d = 2, yielding θ G = 0.218 (where we have used λ G = 1.585 [200], η = 1/4 (exact) and z = 2.166 [201]).…”

Relaxation phenomena at criticality

“…(1) and (2) In summary, field-theoretical methods are a viable approach to investigate non-equilibrium and ageing phenomena during critical relaxation in the bulk. The long-time properties of the non-equilibrium relaxation turn out to depend on the initial condition (e.g., T 0 ) via the corresponding magnetization M 0 and novel crossovers in the response, correlation functions and persistence properties of M (t) [16] occur for finite M 0 , as confirmed by Monte Carlo simulations. Within this approach one can also investigate the dependence of X ∞ -and therefore of the effective temperature T eff = T /X ∞ -on the observable used for its definition.…”

“…This approach allows a systematic analysis of several aspects of these relaxation phenomena and yields analytic predictions for scaling functions, exponents, and amplitude ratios which characterize the scaling behaviour of correlation and response functions at large times, within different universality classes and dynamics. In addition, it is possible to highlight dynamical crossovers in C(t, s), R(t, s) [14,15] and in the persistence properties of M (t) [16], due to different initial conditions. Within this field-theoretical approach, for example, instead of studying on a lattice Z d the non-equilibrium dynamics of the Ising model with spin-flip Glauber dynamics (which captures the behaviour of some anisotropic ferromagnets and alloys), one studies the Landau-Ginzburg effective Hamiltonian H[ϕ] [1] for the order parameter ϕ(x, t) (x ∈ R d ) with a suitable relaxational dynamics, known as Model A [2]:…”