Gross–Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature

Abstract

We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross-Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4πa 2̺ 2 − ̺ 2 0 . Here ̺ denotes the density of the system and ̺ 0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose-Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.

Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

“…Although a universal behavior can still be expected at the phase transition, the phase diagram depends on w at leading order and a mathematical treatment seems out of reach with the present techniques. A simpler behavior is however expected in the dilute regime ρ → 0 with ρ ∼ T d/2 (Gross-Pitaevskii regime [48]). In dimension d = 3 and at our macroscopic scale, the Gross-Pitaevskii limit corresponds to replacing λw by λw λ with w λ (x) = λ −3 w(x/λ) in our many-particle Hamiltonian.…”

“…In dimension d = 3 and at our macroscopic scale, the Gross-Pitaevskii limit corresponds to replacing λw by λw λ with w λ (x) = λ −3 w(x/λ) in our many-particle Hamiltonian. In this case one would expect the phase transition to be described by the (appropriately renormalized) nonlinear Gibbs measure µ over the torus T 3 , with w replaced by the Dirac delta 8πaδ 0 where a is the scattering length of w [115,114,48]. Proving such a result seems a formidable task.…”

“…Other rigorous mathematical works on the Bose gas taking temperature into account include [17,144,145,147,167,49,48]. In particular, the rigorous derivation of the Bose-Einstein phase transition in interacting Bose gases still seems way out of reach, except for special lattice models [114,Chapter 11] and for the trapped case in the Gross-Pitaevskii limit which was recently solved by Deuchert-Seiringer-Yngvason [49,48]. To our knowledge, the only mathematical works devoted to the study of the behavior close to the Bose-Einstein phase transition are [103,107,60,154,62] for equilibrium states and [61] in the one-dimensional dynamical case.…”

“…is actually a difference of two quantities which are individually infinite µ 0almost surely. Estimates on relative one-particle density matrices related to (3.9) are recently obtained in [49,48], but in a different setting, without divergences.…”

Ground state energy of a dilute two-dimensional Bose gas from the Bogoliubov free energy functional

“…There exist only very limited rigorous results about the free energy of a bosonic system starting from a many-body Schrödinger Hamiltonian. In fact, in the dilute regime, the homogeneous gas in three dimensions has been treated by Seiringer [35] and Yin [38] (see also [8,9] for recent developments for the trapped and Gross-Pitaevskii cases). The lower [35] and upper bound [38] prove the free energy asymptotics F (T, ρ) = F 0 (T, ρ) + 4πa 2ρ 2 − [ρ − ρ fc ] 2 + + o(aρ 2 ) as ρa 3 → 0.…”