Gravothermal Catastrophe and Tsallis’ Generalized Entropy of Self-Gravitating Systems II. Thermodynamic properties of stellar polytrope

In this paper, we continue to investigate the thermodynamic properties of stellar self-gravitating system arising from the Tsallis generalized entropy. In particular, physical interpretation of the thermodynamic instability, as has been revealed by previous paper(Taruya & Sakagami, Physica A 307 (2002) 185), is discussed in detail based on the framework of non-extensive thermostatistics. Examining the Clausius relation in a quasi-static experiment, we obtain the standard result of thermodynamic relation that the physical temperature of the equilibrium non-extensive system is identified with the inverse of the Lagrange multiplier, Tphys = 1/β. Using this relation, the specific heat of total system is computed, and confirm the common feature of self-gravitating system that the presence of negative specific heat leads to the thermodynamic instability. In addition to the gravothermal instability discovered previously, the specific heat shows the curious divergent behavior at the polytrope index n > 3, suggesting another type of thermodynamic instability in the case of the system surrounded by the thermal bath. Evaluating the second variation of free energy, we check the condition for onset of this instability and find that the zero-eigenvalue problem of the second variation of free energy exactly recovers the marginal stability condition indicated from the specific heat. Thus, the stellar polytropic system is consistently characterized by the non-extensive thermostatistics as a plausible thermal equilibrium state. We also clarify the non-trivial scaling behavior appeared in specific heat and address the origin of non-extensive nature in stellar polytrope.