  This study shows that the generalized Boltzmann distribution is the only
distribution mathematically consistent with thermodynamics when the system is
described by an ensemble of a certain mathematical form. This mathematical form
is very general, such that the canonical, grand-canonical, or
isothermal-isobaric ensemble theories are all special cases of this form.
Compared with the standard textbook formalism of the statistical mechanics
(SM), this approach does not require a prior distribution, does not assume the
functional form or maximization of entropy, and employs fewer assumptions.
Therefore, this new insight challenges the belief on the requirement of a prior
distribution in SM and provides a new way to derive the Boltzmann distribution.
This study also reveals the logical and mathematical constraints of SM's
fundamental components; therefore, it could potentially benefit researchers on
non-Boltzmann-Gibbs SM and philosophers studying the foundations of SM.
