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

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

Nonextensive and extensive thermostatistic properties of Fermi systems trapped in different external potentials

Research Foundation of Ministry of Education, China [20050384005]; Science Research Fund, Huaqiao University, China [07BS105]The thermostatistic properties of a q-generalized Fermi system trapped in a generic power-law potential are studied, based on the generalized statistic distribution derived from the Tsallis entropy. The total number of particles, the total energy, and the heat capacity at constant volume of the system are derived. The thermostatistic characteristics of the system are discussed in detail. It is found that the thermostatistic properties of such a system depend closely on parameter q, dimensional number of the space, kinetic characteristics of particles and shapes of the external potential, and the external potential has a great influence on the thermostatistic properties of the system. Moreover, it is shown that the results obtained here are very general and can be used to unify the description of the nonextensive and extensive thermostatistic properties of a class of Fermi systems trapped in different external potentials so that the important conclusions of many typical Fermi systems in the literature may be directly derived from the present paper

Finite-size effects in a D-dimensional ideal Fermi gas

National Natural Science Foundation of China [10875100]By using the Euler-MacLaurin formula, this paper studies the thermodynamic properties of an ideal Fermi gas confined in a D-dimensional rectangular container. The general expressions of the thermodynamic quantities with the finite-size corrections are given explicitly and the effects of the size and shape of the container on the properties of the system are discussed. It is shown that the corrections of the thermodynamic quantities due to the finite-size effects are significant to be considered for the case of strong degeneracy but negligible for the case of weak degeneracy or non-degeneracy. It is important to find that some familiar conclusions under the thermodynamic limit are no longer valid for the finite-size systems and there are some novel characteristics resulting from the finite-size effects, such as the nonextensivity of the system, the anisotropy of the pressure, and so on