Order-Parameter Relaxation Times of Finite Three-Dimensional Ising-like Systems

Koch, W.; Dohm, V.; Stauffer, Dietrich

doi:10.1103/physrevlett.77.1789

Cited by 32 publications

(40 citation statements)

References 25 publications

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“…This calculation is parallel to that performed previously [3] and is expected to be as reliable as the previous results [3]. It is based on the Fokker-Planck equa- [3,7,19]. In terms of the eigenvalues µ 3 (κ) and µ 5 (κ) of the equivalent Schrödinger equation [17] we determine the relaxation times τ 3 and τ 5 as…”

Section: −T/τ1(l)mentioning

confidence: 66%

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

Koch

Dohm

1998

Phys. Rev. E

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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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Section: −T/τ1(l)mentioning

confidence: 66%

“…The relaxation times τ i in Fig. 2 are normalized to the bulk amplitude A [3,15,16], no MC data are presently available for τ 3 .…”

Section: −T/τ1(l)mentioning

confidence: 99%

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

Koch

Dohm

1998

Phys. Rev. E

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“…2) are also very different: below T c with decrease of temperature, τ e exponentially increases, while τ i has a small rounded peak close to the phase transition point (implying some relation of this peak to T c (L)). It should be noted that usually this peak is obtained at a bit higher values of temperature than the peak of susceptibility [12,13]. In the inset to Fig.…”

Section: Introductionmentioning

confidence: 99%

“…However, the most interesting results are expected to occur below the phase transition point, where the response functions and the relaxation times of the magnetization [12,13] are extremely sensitive to finite-size effects. Thus, the problems dealing with occurence of the so-called "ergodic relaxation time" [4], which is related to magnetization reversals of a spin cluster below the phase transition point, might still be of considerable interest.…”

Section: Introductionmentioning

confidence: 99%

On ergodic relaxation time in the three-dimensional Ising model

Grigalaits¹,

Lapinskas²,

Banys³

et al. 2013

Lith. J. Phys.

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We have studied the dynamical decay of the autocorrelation function of the 3D Ising model for different sizes L = 20-52 of spin cluster-cubes. The behaviour of the longest, ergodic relaxation time, τ e , of a finite domain below the phase transition temperature T c was mostly considered for two types of phase transition dynamics. A study of the scaling properties of τ e demonstrates a negligible difference between the types of dynamics used, but a considerable difference for different boundary conditions. In contrast to the known result for periodic boundary conditions (τ e ~ L z exp [const(Lє

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“…This view seemed to be supported by Monte Carlo calculations (Stoll et al, 1973) and high-temperature series expansion of the two-dimensional one-spin flip kinetic Ising model. Later, Koch et al (Koch et al, 1996)  of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. For the ferromagnetic interaction, a short range order parameter as well as the long range order is introduced (Tanaka et al, 1962;Barry, 1966) while there are two long range sublattice magnetic orders and a short range order in the Ising antiferromagnets (Barry & Harrington, 1971).…”

Section: Thermodynamic Description Of the Kinetic Modelmentioning

confidence: 99%