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A variational formula for the free energy of an interacting many-particle system. (English) Zbl 1225.60147

The paper considers the Bose-Einstein condensation. The main purpose is to provide a formula for the limiting free energy for fixed positive particle density \(\rho\) and temperature \(1/\beta\). An explicit variational formula is given for any fixed \(\rho\) if \(\beta\) is sufficiently small and for any fixed \(\beta\) if \(\rho\) is sufficiently small.
The mathematical description of the \(N\) bosons is given in terms of the symmetrized trace of the negative exponential of the Hamiltonian times \(\beta\). The Feynman-Kac formula reformulates this trace in terms of \(N\) interactive Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of the approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60J65 Brownian motion
82B10 Quantum equilibrium statistical mechanics (general)
81S40 Path integrals in quantum mechanics
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