Dynamic properties of the Monte Carlo method in statistical mechanics

Müller‐Krumbhaar, H.; Binder, Kurt

doi:10.1007/bf01008440

Cited by 267 publications

(98 citation statements)

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“…Nevertheless, delicate methods have been developed for extracting critical exponents as well as T c (Amit & Martin-Mayor, 2005;Landau & Binder, 2005). A sequence of Monte Carlo updates may be interpreted as a discrete Markov process (Glauber, 1963;Landau & Binder, 2005;Müler-Krumbhaar & Binder, 1973). Consequently, Monte Carlo simulations can also be applied to study time-dependent dynamic behavior, though usually studied is stochastic relaxational dynamics instead of 'true dynamics' in which the dynamics is determined by the equations of motion derived from a Hamiltonian.…”

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confidence: 99%

Finite-time Scaling and its Applications to Continuous Phase Transitions

Zhong¹

2011

Applications of Monte Carlo Method in Science and Engineering

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Finite-time Scaling and its Applications to Continuous Phase Transitions 18www.intechopen.com forms for the magnetization M, the susceptibility χ, and the specific heat C as M(τ, L)=Lχ(τ, L)=L γ/ν f 2 (τL 1/ν ),using the scaling laws or relationswhere α and γ are critical exponents and the f s including those that will appear later are all scaling functions. In terms of the infinite system correlation length ξ ∞ that diverges at T c asthe argument of f s in Equations (3) is proportional to L/ξ ∞ that governs the finite-size behavior; for small L/ξ ∞ , finite-size scaling appears in which L is a relevant length scale, while large L/ξ ∞ is the thermodynamic limit in which equilibrium behavior shows and L is irrelevant. Note that all the critical exponents assume their infinite-lattices values due to the aforemention assumption (Brézin, 1982;Brézin & Zinn-Justin, 1985). Consequently, measuring the observables for a series of L can then determine the corresponding exponent ratios and finally the critical exponents themselves, the pitch of the critical properties, from the pure power laws emerged exactly at T c or τ = 0 at which f s are assumed to be analytic. In fact, for too small systems sizes and temperatures too far away from T c , corrections to scaling (Wegner, 1972) have to be taken into account. Nevertheless, delicate methods have been developed for extracting critical exponents as well as T c (Amit & Martin-Mayor, 2005;Landau & Binder, 2005). A sequence of Monte Carlo updates may be interpreted as a discrete Markov process (Glauber, 1963;Landau & Binder, 2005;Müler-Krumbhaar & Binder, 1973). Consequently, Monte Carlo simulations can also be applied to study time-dependent dynamic behavior, though usually studied is stochastic relaxational dynamics instead of 'true dynamics' in which the dynamics is determined by the equations of motion derived from a Hamiltonian. Yet, the stochastic dynamics for the kinetic Ising model with local spin dynamics as realized in the single-site Metropolis algorithm (Metropolis et al., 1953), for instance, is believed to fall into the same universality class as that governed by the time-dependent Ginzburg-Landau equation (Hohenberg & Halperin, 1977). Dynamic critical phenomena (Cardy, 1996;Ferrell et al., 1967;Folk & Moser, 2006;Halperin & Hohenberg, 1967;Hohenberg & Halperin, 1977;Ma, 1976) are also companying with a divergent correlation time t eq which diverges with the correlation length ξ ∞ aswith a new dynamical critical exponent z dynamic finite-size scaling (Suzuki, 1977) can be obtained by formally incorporating the time argument t in Equation (1), giving rise to which implies a dynamic finite-size scaling form for the correlation timeTherefore, at the criticality,in the asymptotic region of large time, large size, and small τ. This is again a standard method to estimate z, though when the asymptotic region is reached is not easy to determine (Landau & Binder, 2005;Wansleben & Landau, 1991). However, actual simulations can only be performed inevitably in a limited time ...

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Finite-time Scaling and its Applications to Continuous Phase Transitions

Zhong¹

2011

Applications of Monte Carlo Method in Science and Engineering

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“…2. Note that a correction due to the finite simulation time [17,18,19] has been applied to the specific heat calculated from the energy fluctuations. The agreement between the Monte Carlo, the Brownian dynamics, and the two methods of calculating the specific heat is very good except for the derivative of the energy for the Brownian dynamics simulation at the lowest temperature.…”

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Hybrid Monte Carlo simulation of a glass-forming binary mixture

Flenner

Szamel

2006

Phys. Rev. E

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We propose and use a novel, hybrid Monte Carlo algorithm that combines configurational bias particle swaps with parallel tempering. We use this new method to simulate a standard model of a glass forming binary mixture above and below the so-called mode-coupling temperature, TMCT . We find that an ansatz that was used previously to extrapolate thermodynamic quantities to temperatures below TMCT breaks down in the vicinity of the mode-coupling temperature. Thus, previous estimates of the so-called Kauzmann temperature need to be reexamined. Also, we find that the Adam-Gibbs relations D ∝ exp(−a/T Sc) and τ ∝ exp(b/T Sc), which connect the diffusion coefficient D and the relaxation time τ with the configurational entropy Sc, are valid for all temperatures for which the configurational and vibrational contributions to the free energy decouple. Understanding the nature of the glass transition has been of great interest for several decades. One of the earliest paradigms [1,2], that has recently been reformulated [3] and subsequently received considerable attention, assumes the existence of an "ideal" glass transition which occurs when the entropy of the supercooled liquid becomes equal to the entropy of a disordered solid. This paradigm has stimulated several simulational studies [4,5] which have confirmed both the qualitative description of supercooled liquids' dynamics and even the quantitative, numerical prediction of the transition temperature, the so-called Kauzmann temperature T K , for a popular model of a glass forming binary mixture. However, these simulational investigations were restricted to temperatures above the so-called mode-coupling temperature, T MCT . Thus, in these previous studies, in order to investigate the equality of the liquid and disordered solid entropies, one had to extrapolate higher temperature data obtained directly from simulations to significantly lower temperatures (for example, for the binary mixture studied in Refs. [4,5] it was found that T MCT /T K ≈ 1.45). The commonly used extrapolations were questioned in a recent investigation of the configurational entropy [6]. This study is unique in that it accessed directly the low temperature region below T MCT . It showed that the commonly used extrapolations break down below T MCT and the existence of the Kauzmann temperature was put in doubt. Reference [6] used a novel density-of-states Monte Carlo method. Also, it used a binary mixture model that was somewhat different [7] than the model used in most of the previous simulations. Finally, although several system sizes were considered in Ref.[6], the largest system (216 particles) was significantly smaller than systems used in previous studies (the largest ones had 1000 particles).The goal of our investigation was to address the question of the existence of the Kauzmann temperature for the original model studied in Refs. [4,5]. Since the long relaxation times below T MCT makes equilibration of molecular and Brownian dynamics simulations very difficult, we propose and use a novel, speci...

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“…We assume in this paper that τ s is much greater than the correlation time τ of the simulation algorithm used (also measured in Monte Carlo steps) so that the states may be considered to be statistically independent. More generally, if τ s and τ are comparable, then the variance in a measured quantity is increased by a factor of 1 + 2τ /τ s over its value for uncorrelated samples [10]. All the results given in this paper can be generalized to this case in a straightforward manner; see Ref.…”

Section: Finite Sample Size Errorsmentioning

confidence: 75%

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Newman

Palmer

1999

Journal of Statistical Physics

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We examine the sources of error in the histogram reweighting method for Monte Carlo data analysis. We demonstrate that, in addition to the standard statistical error which has been studied elsewhere, there are two other sources of error, one arising through correlations in the reweighted samples, and one arising from the finite range of energies sampled by a simulation of finite length. We demonstrate that while the former correction is usually negligible by comparison with statistical fluctuations, the latter may not be, and give criteria for judging the range of validity of histogram extrapolations based on the size of this latter correction.

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Dynamic properties of the Monte Carlo method in statistical mechanics

Cited by 267 publications

References 41 publications

Finite-time Scaling and its Applications to Continuous Phase Transitions

Finite-time Scaling and its Applications to Continuous Phase Transitions

Hybrid Monte Carlo simulation of a glass-forming binary mixture

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