非广延统计物理是传统的广延统计物理的一个延伸和拓展，主要用于处理包含长程相互作用和/或有记忆效应的复杂系统。它是一个全新的理论，能够解释广延统计物理无法解释的一些奇异实验现象，是当前国际基础物理研究的一大热点。由于对具有复杂内部作用的系统有不同的处理方法，非广延统计物理已经发展出许多分支，其中有Tsallis统计、非完整统计、q形变统计等等。本学位论文主要研究非广延统计物理中存在的四个基本问题和广义量子气体的热力学性质，主要内容安排如下：第一章简要介绍了非广延统计诞生的背景。首先引用了关于自引力系统的一个例子，具体分析了长程相互作用的判断依据，阐明处理此类系统时需要非广延统计理论。然后通过一个...Nonextensive statistical physics is the extension and deploitation of the conventional (extensive) statistical physics. It’s a new theory to deal with some complex systems which present long-range interactions and/or long-duration memory. It can explain some abnormal experimental phenomena that cannot be explained by the theory of the extensive statistical physics. It is a hot problem in the basic...学位：理学博士院系专业：物理与机电工程学院物理学系_理论物理学号：B20032400

欧聪杰

Xiamen University Institutional Repository

    学校编码：10384                                     分类号      密级        学号：B200324001                                               UDC             博  士  学  位  论  文                                            非广延统计物理中的四个基本问题与广义量子气体的热力学性质 Four basic problems in nonextensive statistical physics and the thermodynamic properties of generalized quantum gases   欧 聪 杰  指导教师姓名：陈 金 灿  教授 专  业 名 称：理  论  物  理 论文提交日期：2006 年 07 月 论文答辩时间：2006 年 08 月 学位授予日期：2006 年   答辩委员会主席：           评    阅    人：           2006 年 07 月 厦门大学博硕士论文摘要库   厦门大学博硕士论文摘要库    厦门大学学位论文原创性声明  兹呈交的学位论文，是本人在导师指导下独立完成的研究成果。本人在论文写作中参考的其他个人或集体的研究成果，均在文中以明确方式标明。本人依法享有和承担由此论文产生的权利和责任。                            声明人（签名）： 年   月   日                      厦门大学博硕士论文摘要库   厦门大学博硕士论文摘要库    厦门大学学位论文著作权使用声明  本人完全了解厦门大学有关保留、使用学位论文的规定。厦门大学有权保留并向国家主管部门或其指定机构送交论文的纸质版和电子版，有权将学位论文用于非赢利目的的少量复制并允许论文进入学校图书馆被查阅，有权将学位论文的内容编入有关数据库进行检索，有权将学位论文的标题和摘要汇编出版。保密的学位论文在解密后适用本规定。 本学位论文属于   1、保密（ ），在   年解密后适用本授权书。   2、不保密（√） （请在以上相应括号内打“√”）   作者签名：      日期：  年 月 日 导师签名：      日期：  年 月 日                 厦门大学博硕士论文摘要库   厦门大学博硕士论文摘要库    摘  要  非广延统计物理是传统的广延统计物理的一个延伸和拓展，主要用于处理包含长程相互作用和/或有记忆效应的复杂系统。它是一个全新的理论，能够解释广延统计物理无法解释的一些奇异实验现象，是当前国际基础物理研究的一大热点。由于对具有复杂内部作用的系统有不同的处理方法，非广延统计物理已经发展出许多分支，其中有Tsallis 统计、非完整统计、q形变统计等等。本学位论文主要研究非广延统计物理中存在的四个基本问题和广义量子气体的热力学性质，主要内容安排如下： 第一章简要介绍了非广延统计诞生的背景。首先引用了关于自引力系统的一个例子，具体分析了长程相互作用的判断依据，阐明处理此类系统时需要非广延统计理论。然后通过一个纯电子等离体的二维涡流试验，再次说明非广延统计可以很好地解释一些无法用传统理论来解释的奇异试验现象。 第二章简要介绍了 Tsallis 熵的构造过程以及由它为基础的非广延统计的一些内在性质。主要研究了 Tsallis 统计中存在的两个基本问题，一是在非广延系统中能量是否为广延的，二是是否有必要引入所谓的广义热力学第零定律。通过严格的数学推导得到的结论清楚地表明，正如 Tsallis 熵一样，在 Tsallis 统计中能量也是非广延的，热力学第零定律自从 1988 年首次提出 Tsallis 统计时就一直被隐含地应用在该统计理论中。进而证明多年来一直被大量研究者们引用的标准能量可相加律对 Tsallis 统计并不适合，因为它不仅违背了能量守恒定理而且它的推论与热力学第零定律相互矛盾。 在第三章中针对非完整统计中出现的一个基本问题做了详细分析，提出了一种关于配分函数对内能的偏微分的严格数学推导，导出了物理温度的定义以及几个重要的热力学基本关系，纠正了相关文献中的错误，有利于非完整统计的进一步发展与完善。 在第四章中，主要研究了包括广延与非广延在内的几种统计方法与热力学理论之间的一个等价性问题。利用概率分布函数的泛函，可以把各种不同的统计用一种高度概括的方式统一起来。当系统处在热力学平衡状态下，它的熵函数达到极大值，通过变分法的分析，发现求 概然几率分布与著名的热力学关系 TdSdU = 在本质上是等价的。采用不同形式的泛函可导出各种广延统计和非广延统计的已有结论。 在第五章中，在由 Tsallis 熵推导出的统计分布的基础上，首先研究了 n 维谐振势阱厦门大学博硕士论文摘要库   中广义玻色系统的热力学性质，推导出了广义玻色系统的态密度、总粒子数、玻色-爱因斯坦凝聚（BEC）发生的临界温度、内能、定容热容量的表达式，具体讨论了系统发生 BEC 时的热力学性质。由获得的结果发现这种系统的热力学性质和非广延参数的取值、空间的维数以及谐振势阱的振动频率密切相关；而且外势对系统的热力学性质有明显的影响，获得的结论还可以为实验上获得较高的 BEC 临界温度提供理论上的依据。在此基础上，把外势的形状进一步拓展成广义指数的形状，对广义玻色系统的一系列参量进行分析，获得的结论可用于统一描述一类玻色系统的热统计性质，由此可直接推出文献和教科书中许多重要的结论。 在第六章中，基于广义费米统计分布式，研究了q -广义费米系统的热统计属性。在1q ≤ 和 1q ≥ 两种不同的情况下分别导出系统的总粒子数、内能、定容热容量等相关物理量，其中q是评估系统非广延性的一个重要参数。发现当 1q ≤ 时非广延系统的费米能和参数q无关且等于广延系统的费米能，但其他的热统计属性则和q密切相关，譬如在1q < 时，系统的化学势在特定的温度区间可以超过费米能；当 1q > 时，系统的化学势、热容量等热力学参数在低温区间存在截止。  关键词：非广延统计，广义量子气体，非广延参数 厦门大学博硕士论文摘要库    Abstract  Nonextensive statistical physics is the extension and deploitation of the conventional (extensive) statistical physics. It’s a new theory to deal with some complex systems which present long-range interactions and/or long-duration memory. It can explain some abnormal experimental phenomena that cannot be explained by the theory of the extensive statistical physics. It is a hot problem in the basic physical research fields at present international circumstance. Since there are various different methods in dealing with some complex systems, the nonextensive statistical physics has developed several branches such as Tsallis’ statistics, incomplete statistics, q-deformed statistics, etc. Four basic problems in nonextensive statistical physics and the thermodynamic properties of generalized quantum gases are investigated in this dissertation. The main contents are organized as follows. The background and history of the naissance of the nonextensive statistical physics are simply introduced in chapter 1. A research work on self-gravitating systems is cited to analyze the criteria of the so-called “long-range interactions”, and consequently, it is expounded that nonextensive statistical physics is good at dealing with such systems. Then, an abnormal phenomenon of 2D turbulence in pure electron plasma is introduced to illuminate that nonextensive statistical physics is the most suitable theory for complex systems which present long-range interactions and/or long-duration memory. The mathematical conformation of Tsallis’ entropy and some inherent properties of nonextensive Tsallis’ statistics is briefly introduced in chapter 2. On the basis of Tsallis’ entropy and the joint probability factorization condition, two controversial problems existing in Tsallis’ statistics are investigated, where one is whether energy is extensive or not and the other is whether it is necessary to introduce the so-called generalized zeroth law of thermodynamics or not. The results obtained by strictly mathematical derivation show clearly that like entropy, energy is also nonextensive in Tsallis’ statistics, and that the zeroth law of thermodynamics has been implicitly used in Tsallis’ statistics since 1988. Moreover, it is expounded that the standard energy additivity rule adopted by a great number of researchers 厦门大学博硕士论文摘要库   is not suitable in Tsallis’ statistics, because it violates the energy conservation principle as well as its corollary is in contradiction with the zeroth law of thermodynamics. In chapter 3, a basic problem appearing in Incomplete Statistics is analyzed. Through a precise mathematical calculation related to the partial differential of the partition function with respect to the internal energy, the definition of the physical temperature and several important fundamental thermodynamic relations are obtained. The errors existing in some IS references are corrected. It is helpful for the further development and improvement of Incomplete Statistics. In chapter 4, an equivalent problem of the relations between the maximum entropy principle and the thermodynamic law is investigated. By introducing the functional form of the probability distribution function, the different physical statistics may be generalized in a unified method. When the entropy of the system in thermodynamic equilibrium reaches it’s maximum, it is found by using the variation of the functional form of the probability distribution function that the maximum entropy principle is in equivalence with the well-known thermodynamic relation dU TdS= . It is pointed out that many conclusions in extensive and nonextensive statistics may be directly derived by adopting the different functional forms of the probability distribution function.  In chapter 5, the thermostatistic properties of a q-generalized boson system trapped in an n-dimensional harmonic oscillator potential are firstly studied, based on the generalized statistic distribution derived from Tsallis’ entropy. The density of states, total number of particles, critical temperature that Bose-Einstein condensation occurs, internal energy, and heat capacity at constant volume are derived. The characteristics of Bose-Einstein condensation of the system are discussed in detail. It is found from the results obtained here that the thermostatistic properties of such a system depend closely on the parameter q, dimensional number of the space, and frequency of the harmonic oscillator; and the external potential has a great influence on the thermostatistic properties of the system. The results obtained here will be helpful to one searching some q-boson systems which have a higher critical temperature of BEC. On the basis of the above analysis, a generic power-law potential 厦门大学博硕士论文摘要库    is introduced to q-boson systems. General expressions for some important parameters of the system are derived. It is expounded that the results obtained here can describe the thermostatistic properties of a class of Bose systems in a unified way so that many important conclusions in literatures and textbooks are included in this chapter. Based on the generalized Fermi distribution function, the thermostatistic properties of a q-generalized Fermi system are studied in chapter 6. The total number of particles, internal energy, and heat capacity at constant volume are calculated for two different cases of 1≤q  and 1≥q , respectively, where q  is an important parameter to evaluate the nonextensivity of the system. It is found from the results obtained here that the Fermi energy of such a system is independent of the parameter q  and is equal to that of an original ideal Fermi system when 1q ≤ , while other thermostatistic properties of the system depend closely on the parameter q . For example, when 1q < , the chemical potential of the system in certain region of temperature may be larger than the Fermi energy; when 1q > , some thermodynamic parameters of the system at low temperatures, such as the chemical potential, heat capacity, and so on, must be cut off.  Key words: nonextensive statistics, generalized quantum gas, nonextensive parameter  厦门大学博硕士论文摘要库 厦门大学博硕士论文摘要库   目  录 第 1 章 引言 .................................................. 1 §1.1 非广延统计热力学的诞生..........................................1 §1.2 非广延统计物理的发展和应用......................................2 §1.2.1 自引力系统 ................................................3 §1.2.2 纯电子等离体中的二维涡流 ..................................4 §1.3 本论文的研究方向和内容提要......................................7 参考文献 ..................................................... 9 第 2 章 Tsallis 统计中的两个基本热力学问题 .................... 21 §2.1 Tsallis 熵及其统计简介 .........................................21 §2.1.1 从玻尔兹曼-吉布斯(BG)熵到非广延熵 ........................21 §2.1.2 Tsallis 熵的构造思路 ......................................24 §2.1.3 Tsallis 熵的数学特性 ......................................26 §2.2 Tsallis 统计中的两个基本问题 ...................................28 §2.2.1 基本问题的提出 ...........................................28 §2.2.2 Tsallis 统计中出现的三种内能约束..........................29 §2.2.3 内能的表达 ...............................................30 §2.2.4 内能的非广延性和热力学第零定律 ...........................32 §2.3 Tsallis 统计与热力学间的联系 ...................................32 §2.3.1 Tsallis 统计中的热力学第零定律............................32 §2.3.2 自相矛盾的广义第零定律 ...................................33 §2.4 结论 ...........................................................34 参考文献 .................................................... 35 第 3 章 非完整统计中的基本热力学问题.......................... 38 §3.1 非完整统计的提出...............................................38 §3.2 非完整统计中存在的问题 .........................................39 厦门大学博硕士论文摘要库    §3.3 拉格朗日参数的意义.............................................40 §3.4 非完整统计中的热力学关系.......................................41 参考文献 .................................................... 42 第 4 章 非广延统计与宏观热力学间内在联系中的一个基本问题.... 43 §4.1 微观统计和宏观热力学...........................................43 §4.2 一个等价性原理的数学表式.......................................43 §4.3 举例讨论.......................................................47 §4.3.1 0δ = 的情况...............................................47 §4.3.2 1δ = 的情况 ...............................................49 §4.4 结论 ...........................................................49 参考文献 .................................................... 50 第 5 章 广义玻色气体的热力学性质.............................. 52 §5.1 n 维谐振势阱中广义玻色系统的热力学性质 .........................52 §5.1.1 总粒子数与临界温度 ........................................53 §5.1.2 内能和热容量 .............................................58 §5.1.3 一个特例 .................................................62 §5.1.4 小结 .....................................................63 §5.2 一类玻色系统热统计属性的统一描述...............................64 §5.2.1 广义玻色分布函数的适用范围 ...............................64 §5.2.2 广义指数外势 .............................................65 §5.2.3 系统热力学参量的表达 .....................................66 §5.2.4 系统的热力学性质 ..........................................68 §5.2.5 小结 .....................................................72 参考文献 .................................................... 73 第 6 章 广义费米气体的热力学性质.............................. 79 §6.1 广义费米-狄拉克统计............................................79 厦门大学博硕士论文摘要库   §6.2 系统的总粒子数和内能...........................................80 §6.3 系统的一般性质.................................................82 §6.3.1 1q ≤ 的情况 ...............................................83 §6.3.2 1q ≥ 的情况 ...............................................87 §6.4 讨论 ...........................................................90 §6.4.1 系统在高温时的热力学性质 .................................90 §6.4.2 广延系统的热力学性质 .....................................91 §6.5 结论 ...........................................................92 参考文献 .................................................... 93 第 7 章 结束语 ............................................... 95 §7.1 本论文所获得的一些结论.........................................95 §7.2 本论文的特色与创新之处.........................................96 §7.3 今后的研究方向.................................................96 附录 A: ...................................................... 98 附录 B：.................................................... 100 附录 C：.................................................... 102 附录 D: ..................................................... 104 附录 E: ..................................................... 105 附录 F: ..................................................... 109 厦门大学博硕士论文摘要库 厦门大学博硕士论文摘要库Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”. Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details.厦门大学博硕士论文摘要库

Four basic problems in nonextensive statistical physics and the thermodynamic properties of generalized quantum gases

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Four basic problems in nonextensive statistical physics and the thermodynamic properties of generalized quantum gases

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