The free energy of the interacting Bose gas: a variational description with loops and interlacements

Abstract

We consider the interacting Bose gas in the thermodynamic limit in a large box in $\R^d$ at positive temperature $1/\beta\in(0,\infty)$ with particle density $\sim\rho\in(0,\infty)$. We follow a path-integral approach and adopt from \cite {ACK10} a description of the free energy in terms of the {\it Brownian loop soup}, a Poisson point process consisting of Brownian bridges, also called loops or cycles. It is the objective of this paper to derive, for any values of $\beta$ and $\rho$, a formula for the limiting free energy with explicit control on the particle numbers in the short and in the long loops. The latter are presumed to play the role of the condensate, according to Feynman's \cite{F53} famous, vague suggestion, and they turn into {\it random interlacements} (bi-infinite, locally finite random processes in $\R^d$) in our formula. In \cite{ACK10} there was no concept that could describe the long loops; only small $\rho$ could be handled successfully. In the present paper we represent the limiting free energy in terms of a variational formula, ranging 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 {\it specific relative entropy density} with respect to the reference process of loops (the Brownian loop soup), together with an independent Markov kernel describing collections of path shreds in large boxes. In $d\geq 3$, the latter can be seen as a projection of the {\em Brownian interlacement Poisson point process with $\beta$-spacing}. Our proof tool box comes from large-deviation theory, both for the derivation of the formula for the free energy and for the proof of the existence of the specific relative entropy.