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1、 - 1 - The nonextensive and extensive thermostatistic properties of Fermi systems trapped in different external potentials1 Huang Zhifu1，Ou Congjie1,2，Chen Jincan1,* 1Department of Physics and Institute of Theoretical Physics and Astrophysics，Xiamen University，Xiamen （361005） 2College of Information
2、 Science and Engineering，Huaqiao University，Quanzhou （362021） *E-mail：jcchen Abstract The thermostatistic propertites of a q-generalized Fermi system trapped in a generic power-law potential are studied, based on the generalized statistic distribution derived from Tsallis entropy. The total number o
3、f particles, total energy, and 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 the parameter q, dimensional number of the space, k
4、inetic 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 describe the nonextensive and extensive
5、 thermostatistic properties of a class of Fermi systems trapped in different external potentials in a unified way so that the important conclusions of many typical Fermi systems in literature may be directly derived from the present paper. Keywords：Generalized Fermi system；External potential；Thermos
6、tatistic property PACS numbers：05.30.-d Quantum statistical mechanics；05.20.-y Classical statistical mechanics； 05.70.-a Thermodynamics 1. Introduction In recent years, it has been considered that the systems with spatial and/or temporal long-range interactions are nonextensive and the conventional
7、Boltzmann-Gibbs (BG) statistical mechanics may need to be generalized for the statistical description of the features of the systems. The nonextensive generalization of BG statistical mechanics was firstly done by constructing a new form of entropy with a nonextensive parameter q different from unit
8、y 1. It has been developed by many researchers and widely used to analyze the thermostatistic properties of many such nonextensive systems 28. A representative set of examples are the dynamic linear response theory 7, the Lvy distributions 9, the ground-state geometry of silicon cluster 10, the noni
9、onized hydrogen atom system 11 with q+ = q, the chemical potential of a nonextensive Fermi system is always less than the Fermi energy and there is a cut-off of the chemical potential. The straight lines in Fig.1 stand for the critical condition of the cut-off. The slope of the straight line increas
10、es with the decrease of q and the cut-off temperature decreases with the increase of the slope of the straight line. It implies the fact that the curves of the chemical potential of a nonextensive Fermi system varying with temperature are only allowed to be situated in the right side of the straight
11、 line which has a slope of ) 1/(qkT. When q tends to 1, the slope of the straight line becomes infinite, the cut-off temperature tends to zero, and the chemical potential at the cut-off temperature is equal to the Fermi energy. When 11 is larger than that of the system with q=1 and increases monoton
12、ically with temperature. When 1q, the heat capacity at constant volume of a Fermi system will tend to zero when temperature approaches to absolute zero no matter what the q value is, and consequently, the third law of thermodynamics still holds. At any temperature, the heat capacities of the systems
13、 with q1 - 8 - are always less than that of the system with q=1. When temperature is very low, the difference between the heat capacities of the systems with q1 and q=1 will not be obvious. The heat capacity of the system with q=1 is a monotonically increasing function of temperature, while the heat
14、 capacities of the systems with q1 are not monotonic functions of temperature. However, unlike the heat capacity of the nonextensive Bose system 23, 27, 28 which has a phase transition point and the heat capacity at the critical temperature of BEC may be discontinuous, the heat capacity of the q-gen
15、eralized Fermi system continuously varies with temperature. It first increases and then decreases as temperature is increased so that there is a maximum heat capacity at a certain value of temperature. The main cause is that the Fermions must be constrained by the Pauli principle. In addition, it is
16、 significant to note that at high temperatures, the heat capacities of the generalized Fermi systems do not tend to a constant. This is very abnormal but coincides with the case of the generalized Bose system 23, 27, 28. It implies the fact that at high temperatures, the quantum effects of the generalized Bose and Fermi systems are negligible and consequently their thermostatistic properties tend to unanimity. 4. Discussion It is interesting to note that the result
