Stability Analysis Of Self-gravitating System In Nonextensive Statistical Mechanics

In this paper, we state the contents of the traditional Boltzmann-Gibbs statistics and the difficulties of them. In order to solve these problems, Tsallis, a physicist brings forward the so called non-extensive statistical mechanics. Nonextensive statistical mechanics is based on the nonextensive entropy, which inherits all characters of BG entropy except of its nonextensivity. Based on the frame work of nonextensive statistical mechanics, we introduce the fundamental properties of self-gravitating system, and we apply generalized entropy to seek the equilibrium criterion, then the system can be reduced to the stellar polytrope system and we find the relation about the nonextensive parameter with the polytrope index. After this, we obtain the physical temperature of the equilibrium nonextensive system is identified with the inverse of the Lagrange multiplier, using this relation, the specific heat of total system is computed, which shows two types of thermodynamic instability.For the instability of adiabatic self-gravitating system, according to the principle of maximum entropy, the stable state can be attained only when the second variation around the extremum solution becomes negativeδ2 Sq< 0, conversely, the solution becomes unstable if one obtainδ2 Sq> 0.The conditionδ2 Sq= 0 corresponds to the critical case in which the equilibrium becomes neither stable nor unstable. The stability/instability criterion can be extracted fromδ2 Sq= 0. Evaluating the second variation of Tsallis entropy, we obtain the stability/instability criterion, which exactly matches with the results derived from the standard turning-point analysis. The other type of thermodynamic instability is the system surrounded by the thermal bath. Evaluating the second variation of free energy, we obtain the stability/instability criterion which exactly recovers the marginal stability condition indicated from the specific heat. The results gives an important conclusion that Tsallis generalized entropy is indeed applicable and viable to the long-range nature of the self-gravitating system.