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Progress of Theoretical Physics Supplement

Non-Equilibrium Relaxation of Fluctuation

Nobuyasu Ito, Keita Ogawa, Koji Hukushima, Yukiyasu Ozeki

Published Jan 1, 2000• 10 citations
Publisher

Abstract

It is shown that the non-equilibrium relaxation (NER) process of the order-parameter and energy is useful to study the phase transition.By observing the log-derivatives of those decay functions, the critical point is located finely and the critical exponents are estimated accurately.The major advantage of NER method is it's simplicity; not only the behavior in the large-size limit can be observed easily, but also it is not necessary to pursue the equilibrium of the system to get information of the final equilibrium state.Applications are expanding from basic systems like Ising ferromagnets, to spin-glass, quantal system, chiral and KT transitions, and so on.Now the NER process of fluctuations is also expected to share those advantages.It is found that the combinations of the NER functions of some fluctuations provide the way to determine static and dynamics critical exponents directly.

DOI
10.1143/ptps.138.555

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It is shown that the non-equilibrium relaxation (NER) process of the order-parameter and energy is useful to study the phase transition.By observing the log-derivatives of those decay functions, the critical point is located finely and the critical exponents are estimated accurately.The major advantage of NER method is it's simplicity; not only the behavior in the large-size limit can be observed easily, but also it is not necessary to pursue the equilibrium of the system to get information of the final equilibrium state.Applications are expanding from basic systems like Ising ferromagnets, to spin-glass, quantal system, chiral and KT transitions, and so on.Now the NER process of fluctuations is also expected to share those advantages.It is found that the combinations of the NER functions of some fluctuations provide the way to determine static and dynamics critical exponents directly.
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.
The nonequilibrium relaxation (NER) process of fluctuations of physical quantities is studied simulationally. It is shown that the NER method, which is a convenient technique for studying the phase and transition, is useful not only for identifying the phase and locating the transition point, but also for estimating both static and dynamical exponents. As an example, the cubic-lattice ferromagnetic Ising model is analyzed. The transition inverse-temperature is estimated to be K c =0.2216595(15). The exponents of correlation length, magnetization, and specific heat are estimated to be ν=0.635(5), β= 0.325(5) and α= 0.14(2), respectively. The dynamical exponent z is estimated to be 2.055(10).
The non-equilibrium relaxation method (NER) is applied to a frustrated spin system with competitive interactions. The critical point and critical exponents are estimated accurately for the Blume-Emery-Griffiths model in three dimensions. It is applied to the region where successive transitions of PM-FM-PM-SQ phases occur; we denote paramagnetic (PM), ferromagnetic (FM) and staggered quadrupole (SQ) phases. With the analysis for relaxation of fluctuations, we can estimate various critical exponents. The result indicates that all transitions belong to the Ising universality class in three dimensions. It is found that the NER method is useful to analyze the critical phenomena for complex systems in which equilibrium simulation is even difficult.
The nonequilibrium relaxation (NER) method, which has been used to investigate equilibrium systems via their nonequilibrium behavior, has been widely applied to various models to estimate critical temperatures and critical exponents. Although the estimation of critical temperatures has become more reliable and reproducible, that of critical exponents raises concerns about the method's reliability. Therefore, we propose a more reliable and reproducible approach using Gaussian process regression. In addition, the present approach introduces statistical errors through the bootstrap method by combining them using the extrapolation method. Our estimation for the two-dimensional Ising model yielded β=0.12504(6)β= 0.12504(6)β=0.12504(6), γ=1.7505(10)γ= 1.7505(10)γ=1.7505(10), and ν=1.0003(6)ν= 1.0003(6)ν=1.0003(6), consistent with the exact values. The value z=2.1669(9)z = 2.1669(9)z=2.1669(9) is reliable because of the high accuracy of these exponents. We also obtained the critical exponents for the three-dimensional Ising model and found that they are close to those reported in a previous study. Thus, for systems undergoing second-order transitions, our approach improves the accuracy, reliability, and reproducibility of the NER analysis. Because the proposed approach requires the relaxation of some observables from Monte Carlo simulations, its simplicity imparts it with significant potential.
We present a dynamic off-equilibrium method for the study of continuous transitions, which represents a dynamic generalization of the usual equilibrium cumulant method. Its main advantage is that critical parameters are derived from numerical data obtained much before equilibrium has been attained. Therefore, the method is particularly useful for systems with long equilibration times, like spin glasses. We apply it to the three-dimensional Ising spin-glass model, obtaining accurate estimates of the critical exponents and of the critical temperature with a limited computational effort.