A micro-macro variational formula for the free energy of a many-body system with unbounded marks

The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose–Einstein condensation phase transition is sought for. A qualitative understanding of the free energy would be helpful, but this is currently far out of reach. In this paper, we demonstrate a path towards gaining such an understanding for a simplified version of the model with deterministic boxes instead of Brownian cycles. This model is a marked Poisson point process with unbounded marks containing particles and bounded-reach interactions between the particles. Even though it is not a quantum model, it is close to that in spirit. We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step meso-macro large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks; an application of well-known level-three principles á la Georgii/Zessin is not possible because of the appearance of macro marks. The characteristic variational formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition.