  In an attempt to derive thermodynamics from classical mechanics, an
approximate expression for the equilibrium temperature of a finite system has
been derived [M. Bianucci, R. Mannella, B. J. West, and P. Grigolini, Phys.
Rev. E 51, 3002 (1995)] which differs from the one that follows from the
Boltzmann principle S = k log (Omega(E)) via the thermodynamic relation 1/T=
dS/dE by additional terms of "dynamical" character, which are argued to correct
and generalize the Boltzmann principle for small systems (here Omega(E) is the
area of the constant-energy surface). In the present work, the underlying
definition of temperature in the Fokker-Planck formalism of Bianucci et al. is
investigated and shown to coincide with an approximate form of the
equipartition temperature. Its exact form, however, is strictly related to the
"volume" entropy S = k log (Phi(E)) via the thermodynamic relation above for
systems of any number of degrees of freedom (Phi(E) is the phase space volume
enclosed by the constant-energy surface). This observation explains and
clarifies the numerical results of Bianucci et al. and shows that a dynamical
correction for either the temperature or the entropy is unnecessary, at least
within the class of systems considered by those authors. Explicit analytical
and numerical results for a particle coupled to a small chain (N~10) of quartic
oscillators are also provided to further illustrate these facts.
