Short-time dynamics of finite-size mean-field systems

Abstract

We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie-Weiss with Glauber dynamics) by solving the associated Fokker-Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.

Nonsteady relaxation and critical exponents at the depinning transition

“…This relation was analytically shown to be valid in the limit of small initial order parameter m 0 ≪ 1, with t larger that some non-universal microscopic time and smaller than the equilibration time τ eq ∼ L z . Nevertheless, in agreement with numerical simulations [46][47][48] and analytical approximations in mean-field models [49], it was shown that the following homogeneity relation is valid in the short (but macroscopic) time regime when starting from an ordered condition (commonly m 0 = 1).…”

Analysis of finite-size effect of infinite-range Ising model under Glauber dynamics

“…However, when N is large, the time range in which we can use these equations is lengthened because v is inversely proportional to N . Time development at the critical temperature was already discussed by Anteneodo et al, and they found a linear increase in the fluctuation of magnetization similar to (47) [2]. Note that the discussion on δm at T = T c is more complex than that of v, because v increases with time.…”

“…However, its consideration in nonequilibrium systems is generally more difficult than in equilibrium ones. Hence, an infinite-range Ising model under the Glauber dynamics, one of the simplest models of such systems, has been studied as an example [1,2,3,4]. Most of these studies considered the probability density function of magnetization, P whole (m ′ ), and used the Fokker-Planck equation describing the time development of this function.…”

“…As previous studies pointed out [2], these equations themselves need information of higher-order moments. However, if we assume that the probability distribution of m ′ has a Gaussian form with a sharp peak at m ′ like the discussions of our main text, we can ignore the contributions of higher-order moments.…”

Stability limits for the supercooled liquid and superheated crystal of Lennard-Jones particles

“…26 So STD was used to distinguish first-from second-order phase transitions, by measuring the difference T (+) T () , which vanishes in the second-order case. Building on this idea, and based on the fact that the developing of an instability makes the spinodal similar to a second-order critical point, the temperatures T (+) and T () were interpreted as thermodynamic spinodals and shown to coincide with mean field spinodals 9,27 or pseudospinodals in lattice 9,12 and more recently off-lattice 13 systems. Then, in the specific case of the liquid-solid transition, we identify T () with the liquid spinodal (T * Liq sp ) and T (+) with the solid spinodal (T * Sol sp ).…”