Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box $Λ$ of side length $L$. We take the limit $N, L \to \infty$ while keeping the density $ρ= N / L^3$ fixed and small. We prove a new lower bound for its ground state energy per particle $$\frac{E(N, Λ)}{N} \geq 4 πa ρ[ 1 - O(ρ^{1/3} |\log ρ|^3) ],$$ as $ρ\to 0$, where $a$ is the scattering length of $V$.
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.
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 $e(ρ) \geq 4πa ρ^2 (1- C \sqrt{ρa^3} \,)$.
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.
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…
The leading term of the ground state energy/particle of a dilute gas of bosons with mass $m$ in the thermodynamic limit is $2π\hbar^2 a ρ/m$ when the density of the gas is $ρ$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a>0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.
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.
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.
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 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.
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 ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be $E_0/N = (2π\hbar^2ρ/m){|\ln (ρa^2)|^{-1}}$, to leading order, with a relative error at most ${\rm O} (|\ln (ρa^2)|^{-1/5})$. Here $N$ is the number of particles, $ρ=N/V$ is the particle density and $a$ is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, $E_0$ is not simply $N(N-1)/2$ tim…
A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $ρ$ and temperature $T$. In the dilute regime, i.e., when $a^3ρ\ll 1$, 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 $4πa (2ρ^2 - [ρ-ρ_c]_+^2)$. Here, $ρ_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critic…
According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density $ρ$ is $2π\hbar^2ρa/m$ to leading order in $ρa^3\ll 1$, where $a$ is the scattering length of the interaction potential and $m$ the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than $1/r^3$ …
We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a^2 ρis small and the temperature T satisfies T > 4 πρ/ \ln |\ln(a^2ρ). Here, a is the scattering length of the repulsive interaction potential and ρis the density. To leading order in a^2 ρ, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when ΔT_c / T_c^0 > 5 \sqrt{a ρ^{1/3}}, where T_c^0 is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it g…
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 $ρ$ and inverse temperature $β$ differs from the one of the non-interacting system by the correction term $4 πρ^2 |\ln a^2 ρ|^{-1} (2 - [1 - β_{\mathrm{c}}/β]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $β_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit $a^2ρ\ll 1$ and if $βρ\gtr…
A dilute two-dimensional (2D) Bose gas at zero temperature is studied by the method developed earlier by the authors. Low density expansions are derived for the chemical potential, ground state energy, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+\ln u=-\ln(na^2π)-2γ, where na^2 and γare the gas parameter and the Euler constant, respectively. It is shown that the ground state energy is mostly kinetic in the low density limit; this result does not depend on a specific form of the pairwise inter…
