A variational formula for the free energy of a many-Boson system

We consider $N$ Bosons in a box with volume $N/\rho$ under the presence of a mutually repellent pair potential. Denote by $H_N$ the corresponding Hamilton operator with either zero or periodic boundary condition. The symmetrised trace of $e^{-\beta H_N}$ describes the Bosons at positive temperature $1/\beta$. Our main result is a variational formula for the limiting free energy, for any fixed values of the particle density $\rho$ and the inverse temperature $\beta$, in any dimension. The main tools are a description in terms of a marked Poisson point process and a large-deviation analysis of the stationary empirical field. The resulting variational formula in particular describes the asymptotic cycle structure that is induced by the symmetrisation in the Feynman-Kac formula. We close with a short discussion of the relation to Bose-Einstein condensation.

(joint work in progress with S. Adams and A. Collevecchio)