A variational formula for the free energy of an interacting many-particle system

Abstract

We consider N bosons in a box in R d 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 −βH N , where HN 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.

Randomization and the Gross–Pitaevskii Hierarchy

“…An alternative probabilistic approach was taken in [3]. Here, the authors consider a many-body system of mutually repellent bosons and they derive an explicit variational expression for the corresponding limiting free energy.…”

“…As a result, we can control the factorial number of Duhamel terms which we obtain in the expansion. Consequently, we can prove Theorem 5.2 with a quantity T > 0, which is independent of n. A possible approach in the context of the hierarchy (6) is to argue directly and prove a good spacetime estimate for higher-order Duhamel expansions without directly using (3). This reduces to a purely combinatorial problem of possible pairings of the frequencies.…”

“…The constant C 0 is the same as in (3). In particular, we see that by (4), the range of regularity exponents is extended to α > 3 4 , if one is willing to take the L 2 norm in the probability space .…”

“…This is in contrast to the deterministic setting in which it is known that one has to restrict to α > 1 [92]. Even though the bound (4) is a direct extension of the known bound in the deterministic setting, in our further analysis, we have to use the stronger bound given by (3). The bound in (3) is the content of Theorem 3.1 below.…”

“…The randomization given in (23) and (24) below is the one which leads to the randomized spacetime bound given by (3).…”