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.

在随机系统或复杂系统的研究中，我们经常遇到大时间尺度的问题，或慢动态问题，这给计算机仿真带来了困难。阐明慢动态的本质，并开发一种新的仿真算法来克服慢动态问题是一项紧迫的任务。为了研究相变过程中的动力学行为，采用蒙特卡罗模拟等模拟方法，提出了一种新的自动确定相变临界点的聚类算法，称为概率变化聚类(PCC)算法。应用这个算法，我们阐明了二维随机自旋系统的交叉和自平均性质。我们还将PCC算法应用于显示Kosterlitz-Thouless(KT)转变的二维XY模型。此外，我们还推广了基于相关函数之比的有限尺度性质的PCC算法。在这一推广方案下，我们研究了自旋1/2的二维量子XY模型，以精确地确定KT相变点。我们还提出了一种新的蒙特卡罗动力学，它结合了簇算法和扩展的系综方法。该算法基于广义直方图关系，既可作为蒙特卡罗动力学模型，又可用于精确计算态密度。我们以Potts模型为例说明了该方法的有效性。

项目成果

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. 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）

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.: "Three-dimensional antiferromagnetic q-state Potts models : application of the Wang-Landau algorithm"J.Phys.A. 34・42. 8781-8794 (2001)

C. Yamaguchi 等人：“三维反铁磁 q 态 Potts 模型：Wang-Landau 算法的应用”J.Phys.A. 8781-8794 (2001)

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）。