We consider the interacting Bose gas in the thermodynamic limit at positive temperature with fixed particle density. We follow a path-integral approach and use a description in terms of the Brownian loop soup, a Poisson point process consisting of Brownian loops. We derive a variational formula for the limiting free energy with explicit control on the particle densities in the short and in the long loops. The latter are presumed to play the role of the condensate, according to Feynman's famous, vague suggestion from 1953, and they turn into random interlacements (bi-infinite, locally finite random processes in ᵈ) in our formula. Our formula ranges over the set of all stationary point processes with loops and with interlacements, having each a given particle density, and minimizing the sum of the interaction energy and a characteristic entropy term. The latter is a new kind of a specific relative entropy density with respect to the reference process of loops, together with an independent Markov kernel describing collections of path shreds in large boxes. Our proof tool box is based on large-deviation theory and random point-process theory.
Wolfgang König (Thu,) studied this question.