  We revisit the issues on the thermodynamic property of stellar
self-gravitating system arising from Tsallis' non-extensive entropy. Previous
papers (Taruya & Sakagami, Physica A 307 (2002) 185 (cond-mat/0107494); ibid.
(2002) in press (cond-mat/0204315)) have revealed that the extremum-state of
Tsallis entropy characterized by the so-called stellar polytrope has consistent
thermodynamic structure, which predicts the thermodynamic instability due to
the negative specific heat. However, their analyses heavily relies on the old
Tsallis formalism using standard linear mean values. In this paper, extending
our previous study, we focus on the equilibrium structure based on the standard
framework by means of the normalized q-expectation values. It then turns out
that the new extremum-state of Tsallis entropy essentially remains unchanged
from the previous result, i.e., the stellar quasi-equilibrium distribution can
be described by the stellar polytrope. While the thermodynamic stability for a
system confined in an adiabatic wall completely agrees with the previous study
and thereby the stability/instability criterion remains unchanged, the
stability analysis reveals a new equilibrium property for the system surrounded
by a thermal bath. In any case, the stability/instability criteria are
consistently explained from the presence of negative specific heat and within
the formalism, the stellar polytrope is characterized as a plausible
non-extensive meta-equilibrium state.
