Application of Monte Carlo Methods to Phase Transition Dynamics

蒙特卡罗方法在相变动力学中的应用

• 批准号: 12640379
• 负责人: OKABE Yutaka
• 金额: $ 1.98万
• 依托单位: TOKYO METROPOLITAN UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640379/
• 关键词: simulation algorithm; Monte Carlo method; finite-size scaling; Potts model; Ising model; XY model; clock model; quantum simulation; ランダムスピン系; Tsallis統計

In the study of random systems, or complex systems, we often suffer from the problems of long time scales, or the problem of slow dynamics, which make computer simulation difficult. It is an urgent task to elucidate the nature of the slow dynamics, and to develop a novel simulation algorithm to conquer the problem of slow dynamics. The purpose of the present research is to study the dynamics in phase transitions by means of simulation methods such as Monte Carlo simulations.In the present research project, we developed a new cluster algorithm to determine the critical point automatically, which is called the probability-changing cluster (PCC) algorithm. Applying this algorithm, we elucidated the crossover and self-averaging properties of two-dimensional (2D) random spin systems. We also applied the PCC algorithm to the 2D XY model, which shows the Kosterlitz-Thouless (KT) transition. Moreover, we generalized the PCC algorithms based on the finite-size scaling properties of the ratio of the correlation functions. With this generalized scheme, we studied the 2D quantum XY model of spin 1/2 to determine the KT transition point accurately.We also proposed a new Monte Carlo dynamics, which combines the cluster algorithm and the extended ensemble method. This algorithm, which is based on the broad histogram relation, can be used as a Monte Carlo dynamics as well as the precise calculation of density of states. We showed the efficiency of this method for the Potts models.

在研究随机系统或复杂系统时，我们经常会遇到长时间尺度的问题，或缓慢动态的问题，这使得计算机模拟变得困难。阐明慢动力学的本质并开发一种新颖的仿真算法来克服慢动力学问题是一项紧迫的任务。本研究的目的是通过蒙特卡罗模拟等模拟方法来研究相变的动力学。在本研究项目中，我们开发了一种新的聚类算法来自动确定临界点，称为概率-改变簇（PCC）算法。应用该算法，我们阐明了二维（2D）随机自旋系统的交叉和自平均特性。我们还将 PCC 算法应用于 2D XY 模型，该模型显示了 Kosterlitz-Thouless (KT) 转变。此外，我们还基于相关函数之比的有限尺寸缩放特性推广了 PCC 算法。利用这个广义方案，我们研究了自旋1/2的二维量子XY模型，以准确确定KT转变点。我们还提出了一种新的蒙特卡罗动力学，它将聚类算法和扩展系综方法相结合。该算法基于广泛的直方图关系，可用作蒙特卡罗动力学以及状态密度的精确计算。我们展示了这种方法对于 Potts 模型的效率。

项目成果

Y. Tomita et al.: "Crossover and self-averaging in the two-dimensional site-diluted Ising model: Application of probability-changing cluster algorithm"Phys. Rev. E. 64・3. 036114-1-036114-6 (2001)

Y. Tomita 等：“二维位点稀释伊辛模型中的交叉和自平均：概率变化聚类算法的应用”Phys Rev. E. 036114-1-036114-6（ 2001）

K.Kaneda et al.: "Finite-size scaling for the Ising model on the Mobius strip and the Klein bottle"Phys.Rev.Lett.. 86・10. 2134-2137 (2001)

K. Kaneda 等：“莫比乌斯带和克莱因瓶上伊辛模型的有限尺寸缩放”Phys.Rev.Lett.. 86・10 (2001)。

Y.Okabe et al.: "Universal relations in the finite-size correction terms of two-dimensional Ising models"Phys.Rev.B. 64・3. 035103-1-035103-4 (2001)

Y.Okabe 等：“二维 Ising 模型的有限尺寸校正项中的通用关系”Phys.Rev.B. 035103-1-035103-4 (2001)

C. Yamaguchi et al.: "Broad histogram relation for the bond number and its applications"Phys. Rev. B. 66・3. 036704-1-036704-7 (2002)

C. Yamaguchi 等：“键数的广义直方图关系及其应用”Phys. Rev. B. 036704-1-036704-7 (2002)

Y.Okabe et al.: "Novel Monte Carlo algorithms and their applications"Physica A. (accepted). (2003)

Y.Okabe 等人：“新颖的蒙特卡洛算法及其应用”Physica A.（已接受）。