Finite-size effects on critical diffusion and relaxation towards metastable equilibrium

Abstract

We present the first analytic study of finite-size effects on critical diffusion above and below Tc of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below Tc. Two new universal dynamic amplitude ratios at Tc are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte-Carlo simulations.PACS numbers: 64.60. Ht, 75.40.Gb, 75.40.Mg The dissipative critical dynamics of bulk systems with a non-conserved order parameter are fairly well understood. Depending on whether the order parameter is governed by purely relaxational dynamics or whether it is coupled to a hydrodynamic (conserved) density such systems belong to the universality classes of models A or C [1,2]. The fundamental dynamic quantities of these systems are the relaxation and diffusion times which diverge as the critical temperature T c is approached.For finite systems, these times are expected to become smooth and finite throughout the critical region and to depend sensitively on the geometry and boundary conditions. These finite-size effects are particularly large in Monte Carlo (MC) simulations of small systems. On a qualitative level, they can be interpreted on the basis of phenomenological finite-size scaling assumptions. For a more stringent analysis the knowledge of the shape of universal finite-size scaling functions is necessary. So far there exist reliable theoretical predictions on finitesize dynamics in three dimensions only on two relaxation times τ 1 and τ 2 determining the long-time behavior of the order parameter and the square of the order parameter [3,4]. No analytic work exists, to the best of our knowledge, on the important universality class [1] of diffusive finite-size behavior near T c . This is of relevance, e.g., to magnetic systems with mobile impurities [1], to binary alloys with an order-disorder transition [5,6], to uniaxial antiferromagnets [1] or to systems whose order parameter is coupled to the conserved energy density [1,2].In this Letter we present the first prediction of the finite-size scaling function for the critical diffusion time of three-dimensional systems above and below T c . Furthermore we shall present the analytic identification and quantitative calculation of a new leading relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state of finite systems below T c . Our predictions contain no adjustable parameters other than two amplitudes of the bulk system. We start from model C [2], i.e., from the relaxational and diffusive Langevin equations for the onecomponent order-parameter field ϕ(x, t) and for the density ρ(x, t) = ρ + m(x, t) in a finite volume V ,∂m(x, t) ∂twhere Θ ϕ and Θ m are Gaussian δ-correlated random forces. We consider an equilibrium ensemble near. This corresponds to the experimental situation of ...

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