Generalized Temperature Of Nonextensive Thermodynamics And Its Application To A Self-gravitating System

The start point of nonextensive thermodynamics is nonextensive entropy introduced firstly by C. Tsallis, which inherits the most characters of BG entropy except its nonextensivity. By the optimization of entropy under the confines of normalization condition and internal energy condition, we can obtain a distribution obviously different from the classical BG statistics. This formalism has been proved by a series of experiments.The thermodynamic theorems are proved in the nonextensive framework, but we find that there are some disaccords between the zero and the first theorem of thermodynamics, that is, the concepts of temperature appearing the two theorems are different from each other. The Legendre relation also appears the same problem.By the comparison of BG stationary state and Tsallis stationary state, and comparison of Tsalis stationary and equilibium state, we define the concept of generalized equilibrium state; and verify that the temperature defined in the generalized zero theorem of thermodynamics, which is called generalized temperature, is just the state function to describe the generalized equilibrium state. Then we deform the Legendre relation, and import the generalized temperature into the fist law of thermodynamics, to resolve the conflict between it and the zero law of thermodynamics.The fact, that there is generalized equilibrium state while there is not equilibrium state in one self-gravitating system, reminds us that the self-gravitating system just is the objective described by the nonextensive thermodynamics. Through the thermodynamic method we discussed the physical basement of the Jeans instability in one self-gravitating system.