长程相互作用系统的热力学性质

论文总字数：29876字

摘 要

长程相互作用系统广泛存在，如：重力系统，二维流体动力学，二维弹性，带电和偶极系统。本文通过微正则方法研究了置于交错磁场中的长程相互作用自旋链。在第一章中，介绍了长程相互作用系统独特的属性，比如长程相互作用的本质是不可加性：宏观子系统的能量总和不等于整个系统的能量。而且，宏观热力学参数可访问的空间也可能是非凸的。同时还介绍了前人研究的理论成果以及模型。在第二章中，首先构建了全新的哈密顿量，从哈密顿量出发，计算了系统的微观状态数并推导了单个旋子的能量和系统的熵。其次，研究了系统在热力学极限下的微正则熵，发现该系统是非遍历的，通过改变外部磁场强度可以表现出一级相变、二级相变或同时出现一级与二级相变；第三章中，分析了系统的温度函数，热容函数，发现了系统可能出现温度跳跃，以及热容跳跃，某些情况下还可能出现负热容。第四章中，构造了系统的全局相图，我们发现这个模型表现出由三临界点分开的一级相变和二级相变，可以在一级相变中观察到温度跳变。并且，随着系统哈密顿参量的变化，系统相图会发生显著的改变。

关键词：长程相互作用；非遍历性；负热容；一级相变；二级相变；微正则；自旋链。

Abstract

Long-range interaction systems exist widely, such as gravity systems, two-dimensional fluid dynamics, two-dimensional elasticity, electrification and dipole systems. In this paper, the long-range interacting spin-1 chain placed in a staggered magnetic field is studied by means of micro-canonical approach. In Chapter 1, we introduce the unique properties of the long-range interaction system. For example, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. At the same time, the theoretical results and models of previous studies are also introduced. In Chapter 2, we first construct a new Hamiltonian. Starting from the Hamiltonian, we calculate the number of microscopic states of the system and derive the energy of a single spin and the entropy of the system. Secondly, we study the micro-canonical entropy of the system in the thermodynamic limit and find the system is non-ergodic and can exhibit either first-order phase transition, second-order phase transition or both by shifting the external magnetic field strength. In Chapter 3, we analyze the temperature function, heat capacity function, in some cases the system may also have negative heat capacity. In Chapter 4, the global phase diagram of the system is constructed. We find that this model exhibits both first order phase transition and second order phase transition separated by a tricritical point, and temperature jump can be observed in the first order phase transition. Plus, as the system Hamiltonian parameters change, the system phase diagram will change significantly.

Keywords: long-range interaction; Non-ergodicity; Negative heat capacity; First-order phase transition; Second-order phase transition; Micro-canonical approach; Spin chain.

目录

摘要 I

Abstract II

第一章 序言 4

• 1. 广延性（Extensivity）与可加性（Additivity） 5
  2. 长程相互作用系统的定义 6
  3. 热力学参量的凸性（Convexity） 7
  4. 不可加性（Non additivity） 8

第二章 长程相互作用系统热力学参量理论计算及系统的熵 10

• 1. 哈密顿量 10
  2. 单个自旋的能量 11
  3. 微观状态数 11
  4. 系统的熵 12
     1. I = 0 特殊情况下系统的熵 13
     2. I gt; 0 时系统的熵 15
     3. I lt; 0 时系统的熵 17

第三章 长程相互作用系统温度及热容 20

• 1. 系统的温度 20
  2. 系统的热容 24

第四章 长程相互作用系统相图 29

第五章 结论 34

• 1. 总结 34
  2. 展望 35

致谢 36

参考文献 36

附录 38

1. 自旋为的长程相互作用系统理论推导 38

第一章 序言

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