Open Access
March 2011 A variational formula for the free energy of an interacting many-particle system
Stefan Adams, Andrea Collevecchio, Wolfgang König
Ann. Probab. 39(2): 683-728 (March 2011). DOI: 10.1214/10-AOP565


We consider N bosons in a box in ℝd with volume N / ρ under the influence of a mutually repellent pair potential. The particle density ρ ∈ (0, ∞) is kept fixed. Our main result is the identification of the limiting free energy, f(β, ρ), at positive temperature 1 / β, in terms of an explicit variational formula, for any fixed ρ if β is sufficiently small, and for any fixed β if ρ is sufficiently small.

The thermodynamic equilibrium is described by the symmetrized trace of $e^{-\beta {\mathcal{H}}_{N}}$, where ${\mathcal{H}}_{N}$ denotes the corresponding Hamilton operator. The well-known Feynman–Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of our 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.

In our proof of the lower bound for the free energy, we drop all interaction involving “infinitely long” cycles, and their possible presence is signalled by a loss of mass of the “finitely long” cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the “finitely long” cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose–Einstein condensation and intend to analyze it further in future.


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Stefan Adams. Andrea Collevecchio. Wolfgang König. "A variational formula for the free energy of an interacting many-particle system." Ann. Probab. 39 (2) 683 - 728, March 2011.


Published: March 2011
First available in Project Euclid: 25 February 2011

zbMATH: 1225.60147
MathSciNet: MR2789510
Digital Object Identifier: 10.1214/10-AOP565

Primary: 60F10 , 60J65
Secondary: 81S40 , 82B10

Keywords: Bose–Einstein condensation , Brownian bridge , empirical stationary measure , Free energy , interacting many-particle systems , large deviations , symmetrized distribution , variational formula

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 2 • March 2011
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