  For realistic crystals, the free energy strictly formulated in ensemble
theory can hardly be obtained because of the difficulty in solving the
high-dimension integral of the partition function, the dilemma of which makes
it even a doubt if the rigorous ensemble theory is applicable to phase
transitions of condensed matters. In the present work, the partition function
of crystal vanadium under compression up to $320$ GPa at room temperature is
solved by an approach developed very recently, and the derived equation of
state is in a good agreement with all the experimental measurements, especially
the latest one covering the widest pressure range up to $300$ GPa. Furthermore,
the derived Gibbs free energy proves the very argument to understand most of
the experiments reported in the past decade on the pressure-induced phase
transition, and, especially, a novel phase transition sequence concerning three
different phases observed very recently and the measured angles of two phases
agree with our theoretical results excellently.
