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Journal of the Physical Society of Japan

Nonequilibrium Relaxation of Fluctuations of Physical Quantities

Nobuyasu Ito, Koji Hukushima, Keita Ogawa, Yukiyasu Ozeki

Published Jul 15, 2000• 71 citations
Publisher

Abstract

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).

DOI
10.1143/jpsj.69.1931

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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).
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.
Ferromagnetic (FM) transition of the ± J Ising model is studied for square and simple-cubic lattices by Monte Carlo simulations. Non-equilibrium relaxation (NER) process of magnetization from the all-aligned spin state is analyzed. The transition temperature is estimated at several values of FM bond concentrations to provide the boundary of the FM phase. The estimated NER exponent of magnetization shows non-universal behavior and it becomes larger smoothly when the transition temperature becomes higher.
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.
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.