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

@article{Deuchert2020GrossPitaevskiiLO,
title={Gross–Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature},
author={Andreas Deuchert and Robert Seiringer},
journal={Archive for Rational Mechanics and Analysis},
year={2020},
volume={236},
pages={1217-1271}
}
• Published 31 January 2019
• Physics, Mathematics
• Archive for Rational Mechanics and Analysis
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…
9 Citations
The free energy of a dilute two-dimensional Bose gas
We study the interacting homogeneous Bose gas in two spatial dimensions in the thermodynamic limit at fixed density. We shall be concerned with some mathematical aspects of this complicated problem
The free energy of the two-dimensional dilute Bose gas. II. Upper bound
• Physics
Journal of Mathematical Physics
• 2020
We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2 \rho \ll 1$ and $\beta \rho \gtrsim 1$ the free energy per unit
On the effect of repulsive pair interactions on Bose–Einstein condensation in the Luttinger–Sy model
• Physics, Mathematics
• 2020
In this paper we investigate the effect of repulsive pair interactions on Bose-Einstein condensation in a well-established random one-dimensional system known as the Luttinger-Sy model at positive
A Path-Integral Analysis of Interacting Bose Gases and Loop Gases
• Mathematics, Physics
Journal of Statistical Physics
• 2020
We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field
Classical field theory limit of many-body quantum Gibbs states in 2D and 3D
• Physics
• 2020
We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the
Optimal rate of condensation for trapped bosons in the Gross--Pitaevskii regime
• Physics, Mathematics
• 2020
We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly
Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger
How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical
Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Physics, Mathematics
• 2020
We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the
THE FREE ENERGY OF THE TWO-DIMENSIONAL DILUTE BOSE GAS. I. LOWER BOUND
• Physics, Mathematics
Forum of Mathematics, Sigma
• 2020
We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\unicode[STIX]{x1D70C}$ and inverse

References

SHOWING 1-10 OF 46 REFERENCES
Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature
• Physics, Mathematics
Communications in Mathematical Physics
• 2018
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
Free Energy of a Dilute Bose Gas: Lower Bound
A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $$\varrho$$ and temperature T. In the dilute regime, i.e., when $$a^3\varrho \ll 1$$ , where a
Free Energies of Dilute Bose Gases: Upper Bound
We derive an upper bound on the free energy of a Bose gas at density ϱ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636,
Ground-state energy of the low-density Fermi gas
• Physics, Mathematics
• 2005
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about
Ground State Energy of the Low Density Bose Gas
• Physics, Mathematics
• 1997
Now that the properties of low temperature Bose gases at low density, $\rho$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago.
Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
• Mathematics, Physics
• 2008
Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let
Quantitative Derivation of the Gross-Pitaevskii Equation
• Physics, Mathematics
• 2012
Starting from first-principle many-body quantum dynamics, we show that the dynamics of Bose-Einstein condensates can be approximated by the time-dependent nonlinear Gross-Pitaevskii equation, giving
Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
• Physics, Mathematics
• 2004
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V (N(xi − xj)), where x = (x1, . . ., xN) denotes the positions of the particles. Let HN
Ground states of large bosonic systems: The gross-pitaevskii limit revisited
• Mathematics, Physics
• 2015
We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schr\"odinger functional whose quartic term is
Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases
• Physics, Mathematics
• 2006
We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body