  In this paper, we present an exactly solvable phase transition model in which
the phase transition is purely statistically derived. The phase transition in
this model is a generalized Bose-Einstein condensation. The exact expression of
the thermodynamic quantity which can simultaneously describe both gas phase and
condensed phase is solved with the help of the homogeneous Riemann-Hilbert
problem, so one can judge whether there exists a phase transition and determine
the phase transition point mathematically rigorously. A generalized statistics
in which the maximum occupation numbers of different quantum states can take on
different values is introduced, as a generalization of Bose-Einstein and
Fermi-Dirac statistics.
