Application of Monte Carlo Methods to Phase Transition Dynamics

在随机系统或复杂系统的研究中，我们经常遇到长时间尺度的问题，或慢动力学的问题，这给计算机模拟带来了困难。阐明慢动力学的本质，开发一种新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过程中的动力学。在本研究项目中，我们开发了一种新的自动确定临界点的聚类算法，称为概率变化聚类（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性质。我们还将PCC算法应用于2D XY模型，该模型显示Kosterlitz-Thouless （KT）转换。此外，基于相关函数比例的有限尺度特性，对PCC算法进行了推广。利用这一广义格式，我们研究了自旋为1/2的二维量子XY模型，以准确地确定KT跃迁点。我们还提出了一种新的蒙特卡罗动力学方法，它结合了聚类算法和扩展集成方法。该算法基于广义直方图关系，既可以作为蒙特卡罗动力学，又可以作为状态密度的精确计算。我们证明了这种方法对波茨模型的有效性。

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.