We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.
For a dilute system of non-relativistic bosons interacting through a positive, radial potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1- C \sqrt{ρa^3} \,)$.
We establish the Lee-Huang-Yang formula for the ground state energy of a dilute Bose gas for a broad class of repulsive pair-interactions in 3D as a lower bound. Our result is valid in an appropriate parameter regime of soft potentials and confirms that the Bogolubov approximation captures the right second order correction to the ground state energy.
Non-relativistic interacting bosons at zero temperature, in two and three dimensions, are expected to exhibit a fascinating critical phase, famously known as condensate phase. Even though a proof of Bose-Einstein condensation in the thermodynamic limit is still beyond reach of the current available methods, in the past decades the mathematical physics community has gained an enhanced comprehension of other aspects of the macroscopic behavior of dilute Bose gases at zero temperature. In these notes we review part of these advances, by focusing on the strict relation among the occurrence of Bose…
We extend the analysis of the Bogoliubov free energy functional to two dimensions at very low temperatures. For sufficiently weak interactions, we prove two term asymptotics for the ground state energy.
Lee-Huang-Yang Energy for Dilute Bose Gases: an Upper Bound for General Potentials
Department of Mathematics “Guido Castelnuovo”, La Sapienza（罗马第一大学数学系）；Institute of Mathematics, University of Zurich（苏黎世大学数学研究所）；Gran Sasso Science Institute（格兰萨索科学研究所）；Dipartimento di Matematica, Politecnico di Milano（米兰理工大学数学系）
It is well known that the ground state energy of a three dimensional dilute Bose gas in the thermodynamic limit is $E=4πa ρN$ when the particles interact via a non-negative, finite range, symmetric, two-body potential. Here, $N$ is the number of particles, $ρ$ is the density of the gas, and $a$ is the scattering length of the potential. In this paper, we prove the same result without the non-negativity condition on the potential, provided the negative part is small.
The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.
For a dilute system of non-relativistic bosons interacting through a positive $L^1$ potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving the Lee-Huang-Yang formula for the energy density.
For a dilute system of non-relativistic bosons interacting through a positive potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving a lower bound consistent with the Lee-Huang-Yang formula for the energy density. The proof allows for potentials with large $L^1$-norm, in particular, the case of hard core interactions is included. Thereby, we solve a problem in mathematical physics that had been a major challenge since the 1960's.
We construct a kinetic equation simulating the behavior of degenerate quantum Bose gases with the collision rate proportional to the molecule velocity. We obtain an analytic solution of the half--space boundary--value Smoluchowski problem of the temperature jump at the interface between the degenerate Bose gas and the condensed phase.
We derive a upper bound on the free energy of a Bose gas system at density $ρ$ and temperature $T$. In combination with the lower bound derived previously by Seiringer \cite{RS1}, our result proves that in the low density limit, i.e., when $a^3ρ\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $Δf$ the free energy difference per volume between interacting and ideal Bose gases is equal to $4πa (2ρ^2-[ρ-\rhoc]^2_+)$. Here, $\rhoc(T)$ denotes the critical density for Bose-Einstein condensation (for the ideal gas), and $[\cdot ]_+$ $=$ $\max\{\c…
Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.
A commonly used theoretical definition of superfluidity in the ground state of a Bose gas is based on the response of the system to an imposed velocity field or, equivalently, to twisted boundary conditions in a box. We are able to carry out this program in the case of a dilute interacting Bose gas in a trap. We prove that a gas with repulsive interactions is 100% superfluid in the dilute limit in which the Gross-Pitaevskii equation is exact. This is the first example in an experimentally realistic continuum model in which superfluidity is rigorously verified.
We compute the critical temperature of Bose-Einstein condensation in dilute three-dimensional homogeneous Bose gases. Our method involves the models of spatial permutations and it should be exact to lowest order in the scattering length of the particle interactions. We find that the change in the critical temperature is proportional to a rho^{1/3} with constant c = -2.33; this contradicts the current consensus among physicists.
The ground state of bosonic atoms in a trap has been shown experimentally to display Bose-Einstein condensation (BEC). We prove this fact theoretically for bosons with two-body repulsive interaction potentials in the dilute limit, starting from the basic Schroedinger equation; the condensation is 100% into the state that minimizes the Gross-Pitaevskii energy functional. This is the first rigorous proof of BEC in a physically realistic, continuum model.
We derive linear fluctuating hydrodynamics as the low density limit of a deterministic system of particles at equilibrium. The proof builds upon previous results of the authors where the asymptotics of the covariance of the fluctuation field is obtained, and on the proof of the Wick rule for the fluctuation field.
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 limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose-Einstein condensation, polymers and $\vert \boldsymbolϕ \vert^{4}$-theory.
In these notes of six lectures on selected topics in the theory of cold, dilute Bose gases, presented at the 5th Warsaw School of Statistical Physics in June 2013, the following topics are discussed: 1) The concept of BEC, 2) the ground state energy of a dilute Bose gas with short range interactions, 3) Gross-Pitaevskii theory and BEC in trapped gases, 4) Bose gases in rotating traps and quantized vortices, and 5) strongly correlated phases in the lowest Landau level generated by rapid rotation.
We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second order term, as predicted by the Lee-Huang-Yang formula. This result has been first established in J. Stat. Phys. 136(3) by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported $V \in L^3 (\mathbb{R}^3)$, gives better rates and, in our opinion, is substantially simpler.
The evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford showed that the Boltzmann equation emerges as the law of large numbers in the low density limit, at least for very short times. The goal of this survey is t…
