古典蒙地卡羅方法對相變化之研究

Classical Monte Carlo Study on Phase Transitions

謝昀達(Yun-Da Hsieh)

指導教授 ： 高英哲

國立臺灣大學/理學院/物理研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.00784

摘要

在相變化的研究上,晶格模型扮演著重要的角色。 例如,三 維的XY 模型和4He 的 λ 相變有關; 二維的XY 模型中,可以發現 Kosterlitz-Thouless(KT) 相變。 特別是在連續相變裡,可以用不同的臨 界現象來分類相變, 利用這個特性,我們可以提出適當的晶格模型, 此模型與某自然界的相變有同樣的臨界現象,如此,我們就可以藉由 理論模型來研究相變。 然而,很多模型無法用解析的方式解出,所以 近年來數值模擬變得愈來愈重要。 再加上顯示卡(GPU)的發展,人們 開始利用GPU 的平行運算資源來加速模擬的速度, 平行運算的發展, 讓我們能夠處理原先需要大量運算的模型,並且減少統計上的誤差。
這個論文主要討論數值方法為古典蒙地卡羅。蒙地卡羅在晶格模型 的模擬上,是一個強而有力的工具, 例如在三維Ising 模型或是XY 模 型。 但數值模擬不可必免的問題是,模擬的系統大小是有限的;系統 大小對有限大小修正(finite-size scaling) 有很大的影響,這個影響也是 主要的統計誤差來源。 要得到好的模擬結果,在很多情況下,增大系 統大小是不可避免的。 但要增加系統大小,又要維持好的統計品質, 用傳統的處理器(CPU)做運算需要很長的時間。 有鑑於此,我們運用 了NVIDIA CUDA 的架構,把蒙地卡羅模擬實作在顯示卡上。 和我們 自己在CPU 的程式相比,大概快了100倍左右,其結果和CPU 是一致 的。
在論文中,第一章介紹相變現象和文中使用的理論模型。 第二章 前半段介紹蒙地卡羅方法,從它的理論基礎-馬可夫鏈過程, 到應用上 的Metropolis 演算法以及物理期望值的計算,都有基本但完整的說明。 後半段則詳述了GPU 的架構並且循序漸進地說明在GPU 上的實做方 法。 學會了模擬的方法後,第三章介紹在統計上如何處理由模擬中 得到的數據, 如何做有限大小修正去找出臨界溫度和臨界指數(critical exponents)。 文中以二維和三維的Ising 模型為例,說明如何從模擬數 據中找出臨界現象, 同時,也以這個例子,展示GPU 版本的蒙地卡羅 的正確性。
第四章中,我們展示了二維和三維的XY 模型的結果。 在三維的 模型中,我們最大的模擬系統大小是 L = 160,以期減少有限系統 大小對結果的影響, 我們做出來的結果和之前的實驗和模擬結果一致,且擁有較小的統計誤差。 在二維的部份,我們利用了KT 相變 和spin-stiffness 的關係,找出臨界溫度。 在這模型中,要找到相變點 是十分困難的,因為它的有限系統大小修正是一個對數的形式, 我們 藉由將系統大小增加至 L = 512和更多的蒙地卡羅量測來克服這個困 難。
此論文是為了以下三種類型讀者設計的:
• 對蒙地卡羅有興趣,需要基本但完整的蒙地卡羅方法介紹。
• 熟悉蒙地卡羅模擬,想學習如果實做在GPU 上。文中有詳細的指 導和一些清楚的例子。 如果需要,可以提供測試過的GPU 版本的 程式。
• 已經有CPU 或 GPU 上的模擬結果,想要檢查自己數據的正確 性。在論文中, 所有的結果都和之前其它的模擬結果或是現有的 解析解一致,但擁有比之前更好的統計結果。

關鍵字

蒙地卡羅 ； 顯示卡 ； Ising ； XY ； CUDA

並列摘要

Lattice models are of significance in studying phase transitions and crit- ical phenomena. For example, the three-dimensional (3D) XY or O(2) universality class has relevance to the nature of the λ-transition in 4He and the exhibition of the Kosterlitz-Thouless(KT) transition in two-dimensional (2D) XY model. In particular, the second-order phase transitions in nature can be classified into different universality classes by its critical behaviors. Using the universality properties, we could investigate the critical phenomena by proposing a suitable lattice model belonging to the same universality class of the problem in which we are interested. Due to the difficulty of solving these models analytically, recently numerical research has played an important role in investigating model Hamiltonians with the help of the development of high performance computers. Moreover, with the advantage of the graphics pro- cessing unit (GPU), computation can be accelerated by parallelization of pro- grams. As a result, it is now possible to perform numerical studies on some difficult cases and boost the quality of the statistical results.
This thesis is devoted to classical Monte Carlo simulations on lattice mod- els. The Monte Carlo method is a powerful tool for exploring the properties of physical models which have not yet been solved analytically, such as the 3D Ising model, XY model in two and three dimensions and so on. How- ever, in numerical studies, we can only simulate systems with finite lattice sizes, which is also an inevitable limit for other numerical methods. In the finite-size scaling hypothesis, the statistical estimates are sensitive to the size, which may be the main source of the statistical bias. In the quest for the true physics behind statistical error, increasing the lattice size in simulations is indispensible in most cases. Nevertheless, in order to increase the lattice size while keeping the quality of the statistical data, the amount of computations may become unaffordable for the traditional central processing unit (CPU). In light of the situation and also the parallel nature of the Monte Carlo method, we implement a GPU version of Monte Carlo simulations with NVIDIA CUDA frameworks. The performance of the GPU versions of var- ious models are satisfying, running approximately 100 times faster than our CPU versions while generating outcomes consistent with our CPU studies and other works.
Here is an overview of the thesis. In the first chapter, we introduce the phase transitions occuring in nature and the model Hamiltonians we will dis- cuss in the following chapters. The second chapter, which is the most impor- tant one, describes the basic idea of performing Monte Carlo simulations on lattice systems. It provides both the theoretical basis, such as Markov chain processes and practical guidance, such as the Metropolis algorithm and evaluation of expectation values. In the latter half of the chapter, we cover the step-by-step procedure of the GPU implementation from the point of view of its architecture. The third chapter contains the techniques of processing statistical data from the Monte Carlo simulations and the procedures for find- ing the critical exponents with finite-size data. We provide two- and three- dimensional Ising models as an example to demonstrate the procedures of fitting the finite-size scaling functions and to benchmark the GPU versions of the simulation programs.
In the fourth chapter, we show our simulation results of the 2D and 3D XY models. For the 3D case, we perform the simulations up to lattice size L = 160 to reduce the finite-size effects and have consistent results but better statistics compared to previous numerical and experimental works. In th 2D XY model, we adopt a method related to the spin-stiffness in order to locate the critical temperature of the KT transition. The difficulty of locating the KT transition point due the logarithmic size correction is a well-known fact. However, we can overcome the difficulties by increasing the lattice size up to L = 512 and boosting the statistical quality with longer Monte Carlo runs.
This thesis is intended for the reader who:
• is interested in Monte Carlo simulations of lattice models and needs a basic but complete introduction to the Monte Carlo method.
• wants to migrate his or her Monte Carlo programs to GPU. Detailed guidance and examples are included. Also the benchmarked GPU pro- grams can be provided if needed.
• already has CPU or GPU simulation results, and would like to to check the accuracy of his or her implementation. In the thesis, all the results are consistent with previous works (or exact solutions if they exist) and even have better statistical quality.

並列關鍵字

Monte Carlo ； GPU ； Ising ； XY ； CUDA

參考文獻

[1] E. A. Guggenheim, “The principle of corresponding states,” J. Chem. Phys., vol. 13, no. 7, pp. 253–261, 1945.

[2] J. D. Cohen and T. R. Carver, “New measurement of the critical exponent β for nickel by microwave transmission,” Phys. Rev. B, vol. 15, pp. 5350–5366, Jun 1977.

[6] L. Onsager, “Crystal statistics. i. a two-dimensional model with an order-disorder transition,” Phys. Rev., vol. 65, pp. 117–149, Feb 1944.

[7] B. Kaufman, “Crystal statistics. ii. partition function evaluated by spinor analysis,” Phys. Rev., vol. 76, pp. 1232–1243, Oct 1949.

[8] A. E. Ferdinand and M. E. Fisher, “Bounded and inhomogeneous ising models. i. specific-heat anomaly of a finite lattice,” Phys. Rev., vol. 185, pp. 832–846, Sep 1969.