Nonequilibrium relaxation method

Ito, Nobuyasu

doi:10.1007/bf02704149

Cited by 10 publications

(7 citation statements)

References 40 publications

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“…For Metropolis dynamics, 29 in which N single-spin flip attempts define a unit of time in updating the system configurations, many works have been devoted to extracting the value of z (which in the case of Metropolis dynamics we will often call z M ) for the 2D Ising model, e.g., Refs. [34][35][36][37][38][39][40][41]. The values obtained are typically close to 2.2, with z M = 2.1667 (5), obtained in Ref.…”

Section: B Dynamic Exponentssupporting

confidence: 79%

Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

2014

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We use Monte Carlo simulations to demonstrate generic scaling aspects of classical phase transitions approached through a quench (or annealing) protocol where the temperature changes as a function of time with velocity v. Using a generalized Kibble-Zurek ansatz, we demonstrate dynamic scaling for different types of stochastic dynamics (Metropolis, Swendsen-Wang, and Wolff) on Ising models in two and higher dimensions. We show that there are dual scaling functions governing the dynamic scaling, which together describe the scaling behavior in the entire velocity range v ∈ [0, ∞). These functions have asymptotics corresponding to the adiabatic and diabatic limit, and close to these limits they are perturbative in v and 1/v, respectively. Away from their perturbative domains, both functions cross over into the same universal power-law scaling form governed by the static and dynamic critical exponents (as well as an exponent characterizing the quench protocol). As a by-product of the scaling studies, we obtain high-precision estimates of the dynamic exponent z for the two-dimensional Ising model subject to the three variants of Monte Carlo dynamics; for single-spin Metropolis updates zM = 2.1767(5), for Swendsen-Wang multi-cluster updates zSW = 0.297(3), and for Wolff singlecluster updates zW = 0.30(2). For Wolff dynamics, we find an interesting behavior with a non-analytic breakdown of the quasi-adiabatic and diabatic scaling, instead of the generic smooth cross-over described by a power law. We interpret this disconnect between the two scaling regimes as a dynamic phase transition of the Wolff algorithm, caused by an effective sudden loss of ergodicity at high velocity.

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Section: B Dynamic Exponentssupporting

confidence: 79%

Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

2014

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“…The range of the temperature shown in Fig. 5 is consistent with an estimate of the critical temperature based on the conventional NER analysis using the 6648 × 6648 × 6656 cluster, J/k B T c = 0.2216547 (5) or T c = 4.511522(10)J/k B [11] .…”

Section: Numerical Results In the Three-dimensional Ising Modelsupporting

confidence: 82%

“…8(a)-(f), relaxation exponent σ(t) estimated with the above procedure (each data point is obtained from the data from t 0 (t)-to t-MCS, where t 0 (t) gives the minimum residue) is plotted versus t based on Eqs. (10) (circles) and (11) 5, respectively. Apparently, behavior of σ obtained from |m| in Figs.…”

Section: B Three-dimensional Ising Model Around Tcmentioning

confidence: 99%

Cluster nonequilibrium relaxation in Ising models observed with the Binder ratio

Nonomura,

Tomita

2018

Preprint

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The Binder ratios exhibit discrepancy from the Gaussian behavior of the magnetic cumulants, and their size independence at the critical point has been widely utilized in numerical studies of critical phenomena. In the present article we reformulate the nonequilibrium relaxation (NER) analysis in cluster algorithms using the (2, 1)-Binder ratio, and apply this scheme to the two-and threedimensional Ising models. Although the stretched-exponential relaxation behavior at the critical point is not explicitly observed in this quantity, we find that there exists a logarithmic finite-size scaling formula which can be related with a similar formula recently derived in cluster NER of the correlation length, and that the formula enables precise evaluation of the critical point and the stretched-exponential relaxation exponent σ. Physical background of this novel behavior is explained by the simulation-time dependence of the distribution function of magnetization in two dimensions and temperature dependence of σ obtained from magnetization in three dimensions.

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“…As an example, let us see the NER analysis for the FM Ising model in three dimensions [37][38][39][40]. The Hamiltonian is given by…”

Section: Estimation Of Transition Temperaturementioning

confidence: 99%

Ozeki¹,

Ito²

2007

J. Phys. A: Math. Theor.

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The nonequilibrium relaxation (NER) method is a numerical technique to analyse equilibrium phase transitions. One can estimate the transition point and critical exponents calculating relaxations of order parameter and fluctuations in NER processes from nonequilibrium initial states to the equilibrium ones. It is applied to the second-order transition as well as the first-order one; the precise estimation of transition temperature for the latter case is achieved with the mixed phase initialization technique. Since the equilibration is not necessary in the simulation, one can analyse systems with large sizes which can be recognized as in the thermodynamic limit up to the observation time. This leads to the analysis being more efficient for slowly relaxing systems with frustrations and randomness. It can be extended to the Kosterlitz-Thouless transition and the spin glass transition cases by the use of the finite-time scaling analysis. It is also extended to quantum systems.

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Nonequilibrium relaxation method — an alternative simulation strategy

Cited by 10 publications

References 40 publications

Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

Cluster nonequilibrium relaxation in Ising models observed with the Binder ratio

Nonequilibrium relaxation method

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