  We 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, cond-mat/0107494, Physica A 307, 185 (2002)),
is discussed in detail based on the 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, $T_{phys}=1/\beta$. 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. Evaluating the
second variation of free energy, we find that the marginal stability condition
indicated from the specific heat can be exactly recovered from the second
variation of free energy. 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.
