臺灣博碩士論文加值系統

換尺不變 (scale invariance) 與冪次定律 (power law) 同為臨界現象與自組臨界現象系統之特徵，然而，由於自組臨界現象系統缺乏臨界點之概念，使得所謂的相變之刻劃在自組臨界現象系統有其困難。本論文嘗試以平衡相變理論中的李楊定理 (Lee-Yang zeros)，輔以有限尺度理論(finite-size scaling) 對自組臨界現象系統之沙堆模型作分析。我們發現當系統大小逼近於無限大時， 由沙堆崩落面積的機率分佈所建構之李楊圓根，的確會趨近實數1，此方法對臨界指數與普適性 (universality) 的研究，提供另一有效的路徑，更重要的是建立了臨界現象與自組臨界現象系統的連結。

A self-organized criticality system (SOC) exhibits scale invariance and power-law behavior which are the features of critical phenomena. In this thesis, a SOC system is studied by the methods of the traditional critical phenomena.
Following the steps of B.Cessac and J.L.Meunier, we use Lee-Yang theorem to form the generating function for the probability distributions of BTW sandpile model proposed by Bak, Tang and Wiesenfeld. Here, we also use the concepts of finite-size effects and scaling law to investigate the critical exponents and universality. We find that some zeros defined by Lee-Yang theorem will pinch on the real axis of a complex plane as the system size approaches infinity. It establishes a new link between the classical theory of critical phenomena and SOC systems.

1 Phase Transitions and Critical Phenomena
1.1 Phase Transitions
1.1.1 The First-Order Phase Transitions
1.1.2 The Second-Order Phase Transitions
1.2 Critical Phenomena
1.2.1 Correlation Length
1.2.2 Scaling Law
1.3 Self-Organized Criticality
2 Sandpile Model
2.1 BTW Sandpile Model
2.1.1 The Cellular Automata of BTW Sandpile
2.2 Wave of Topplings
3 Probability Distribution and Lee-Yang Zeros
3.1 Introduction to Lee-Yang Zeros
3.1.1 Partition Function
3.2 Probability Distribution Function
3.2.1 Finite-Size Effect
3.2.2 Finite-Size Scaling
3.2.3 Theoretical Result
4 Numerical Simulations
5 Conclusion

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