Study On Some Basic Problems In Nonextensive Statistics

Nonextensive statistics (Tsallis statistics) is a new statistical approach which develops on the basis of Tsallis entropy. Nowadays, it it believed that Tsallis statistics is a useful generalization of the traditional Boltzmann statistics. In this dissertation, some basic problems in Tsallis statistics are investigated, the main contents are organized as follows:First, we discuss the applications of Tsallis statistics. Based on the fact that Tsallis statistics is suitable for the description of long-rang interaction systems, we investigate the dispersion relation and Landau damping of ion acoustic waves in the collisionless and magnetic-free plasma if the palasma is described by the generalized q-velocity distribution in Tsallis statistics. The results show that the Landau damping of the ion acoustic waves in the framework of Tsallis statistics is related to the nonextensive parameter, and the strengthened or weakned models of damping can be reasonably explained by the increased number of superthermal or low velocity particles contained in the system. Due to the nonextensive parameter is related to the inhomogeneity of the temperature and potential energy, thus, the properties of ion acoustic waves derived in the framework of Tsallis statistics are actually the ones of plasmas in nonequilibrium steady state.In the following section, we present a stability analysis of the generalized classical ideal gas in the framework of Tsallis satistics and ues this therory to interpret the phenomenon of negative specific heat appeared in the nonextensive gas. The satability analysis is made on the basis of the second variation of Tsallis entropy. It shows that the appearence of negative specific heat can be reasonably explained from the viewpoint of the system thermodynamic instability. In other words, when the system is in the minimum entropy state and exhibits thermodynamic instability, the negtive specific heat will appear correspondingly.Fianlly, we investigate the energy fluctuations for the canonical enseble in Tsallis statistics and further to check the ensemble equivalence in this new statistics. By taking the generalized ideal gas and harmonic oscillators in Tsallis statistics as exmples, we see that when the particle number N is large enough, the relative energy fluctuation of the caonical ensemble is proportional to 1/N. Thus, in the large particle number limit, the energy fluctuations is so small that it can be negligible. The system described by canonical ensemble approach is the same as that in microcanonical ensemble. Therefore, the equivalence between the microcanonical ensemble and canonical ensemble still holds in Tsallis statistics.