arXiv:1901.11363 · Archive for Rational Mechanics and Analysis

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

Andreas Deuchert, Robert Seiringer

Published Mar 9, 2020• 22 citations

Abstract

Abstract We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering lengthais 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 $$4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right) $$ 4πa2ϱ2-ϱ02 . Here $$\varrho $$ ϱ denotes the density of the system and $$\varrho _0$$ ϱ0 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.

DOI: 10.1007/s00205-020-01489-4

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arXiv:1901.11363 · Archive for Rational Mechanics and Analysis

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

Andreas Deuchert, Robert Seiringer

Published Mar 9, 2020• 22 citations

arXiv HTML PDF

Abstract

4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right)

4πa2ϱ2-ϱ02 . Here

\varrho

ϱ denotes the density of the system and

\varrho _0

ϱ0 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.

DOI: 10.1007/s00205-020-01489-4

Related papers

Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

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We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ${\omega}$ , the temperature T, and the particle number N are related by ${N \sim (T / \omega)^{3} \to\infty}$ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency $${\omega}$$ , the temperature T, and the particle number N are related by $${N \sim (T / \omega)^{3} \to\infty}$$ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.

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A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density and temperature T. In the dilute regime, i.e., when a 3 ≪ 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(2 2 −[−c] 2 +). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ ·]+ = max { · , 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ∼ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one. 1 Introduction and Main Result The advance of experimental techniques for studying ultra-cold atomic gasesA lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density and temperature T. In the dilute regime, i.e., when a 3 ≪ 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(2 2 −[−c] 2 +). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ ·]+ = max { · , 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ∼ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one. 1 Introduction and Main Result The advance of experimental techniques for studying ultra-cold atomic gases

Jan 1, 200615 citations

4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right)

4πa2ϱ2-ϱ02 . Here

\varrho

ϱ denotes the density of the system and

\varrho _0

{\omega}

, the temperature T, and the particle number N are related by

{N \sim (T / \omega)^{3} \to\infty}

while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.