Microscopic theory of a phase transition in a critical region: Bose–Einstein condensation in an interacting gas

•A microscopic theory of a phase transition in a critical region is found.•Critical-region extension of Gross–Pitaevskii and Beliaev–Popov equations is found.•Exact Hamiltonian for Bose–Einstein condensation in a mesoscopic system is found.•Exact recurrence equations for the basis contraction superoperators are found.•A failure of previous phase-transition theories in a critical region is analyzed.

We present a microscopic theory of the second-order phase transition in an interacting Bose gas that allows one to describe formation of an ordered condensate phase from a disordered phase across an entire critical region continuously. We derive the exact fundamental equations for a condensate wave function and the Green's functions, which are valid both inside and outside the critical region. They are reduced to the usual Gross–Pitaevskii and Beliaev–Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions, in particular, in the physics of condensed matter and quantum fields.