The Free Energy of the Two-Dimensional Dilute Bose Gas. I. lower Bound

Abstract

We prove a lower bound for the free energy (per unit volume) of the twodimensional Bose gas in the thermodynamic limit. We show that the free energy at density ρ and inverse temperature β differs from the one of the non-interacting system by the correction term 4πρ 2 | ln a 2 ρ| −1 (2−[1−β c /β] 2 + ). Here a is the scattering length of the interaction potential, [·] + = max{0, ·} and β c is the inverse Berezinskii-Kosterlitz-Thouless critical temperature for superfluidity. The result is valid in the dilute limit a 2 ρ 1 and if βρ 1.2. The lower bound on the o(1) error term given here is uniform in βρ as long as βρ 1. The proof will show that the actual error rate is much better for βρ some distance away from β c ρ (either above or below), see (2.16.13). For very low temperatures, we utilize the proof method of [30]; this way, we recover the ground state energy error rate | ln a 2 ρ| −1/5 for very low temperatures, which was proved for T = 0 in [30].6

Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger

“…• free-energy estimates for dilute gases in the thermodynamic limit [90,286,289,318]. • rigorous bounds on the critical temperature for BEC [292,33].…”

Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons

“…In combination with the matching upper bound that Dyson had proven in 1957 [19], this established its leading order asymptotics. By now, the techniques of Lieb and Yngvason have been significantly extended to prove related results for the ground state energy of the two-dimensional Bose gas [61], the free energy of two-and three-dimensional Bose gases [16,68,76,84], as well as the ground state energy [24,55] and pressure [75] of the Fermi gas. Recently, also the next to leading order correction to the ground state energy of the dilute Bose gas, that is, the Lee-Huang-Yang formula, could be established [31,83].…”

The Excitation Spectrum of Two-Dimensional Bose Gases in the Gross–Pitaevskii Regime

“…The estimate ( 1.5 ) has been extended to general trapping potentials in [ 25 ]. Recently, also the free energy of a two-dimensional dilute Bose gas at positive temperature has been computed to leading order in [ 18 , 29 ] (thermodynamic limit) and in [ 19 ] (Gross–Pitaevskii regime). It is interesting to compare our bound ( 1.3 ) with the second order approximation of the energy per particle in the thermodynamic limit, given by with and where is Euler’s constant.…”

“…The estimate ( 1.5 ) has been extended to general trapping potentials in [ 25 ]. Recently, also the free energy of a two-dimensional dilute Bose gas at positive temperature has been computed to leading order in [ 18 , 29 ] (thermodynamic limit) and in [ 19 ] (Gross–Pitaevskii regime).…”