In this chapter, we will review the results obtained thus far by Eric A. Carlen, Elliott H. Lieb, and me on a Simplified Approach to the Bose gas. The Simplified Approach yields a family of effective one-particle equations, which capture some non-trivial physical properties of the Bose gas at both low and high densities, and even some of the behavior at intermediate densities. In particular, the Simplified Approach reproduces Bogolyubov’s estimates for the ground state energy and condensate fraction at low density, as well as the mean-field estimate for the energy at high densities.We will also discuss a phase that appears at intermediate densities with liquid-like properties. The simplest of the effective equations in the Simplified Approach can be studied analytically, and we will review several results about it; the others are so far only amenable to numerical analysis, and we will discuss several numerical results. We will start by reviewing some results and conjectures on the Bose gas and then introduce the Simplified Approach and its derivation from the Bose gas.We will then discuss the predictions of the Simplified Approach and compare these to results and conjectures about the Bose gas. Finally, we will discuss a few open problems about the Simplified Approach.In this chapter, we will review the results obtained thus far by Eric A. Carlen, Elliott H. Lieb, and me on a Simplified Approach to the Bose gas. The Simplified Approach yields a family of effective one-particle equations, which capture some non-trivial physical properties of the Bose gas at both low and high densities, and even some of the behavior at intermediate densities. In particular, the Simplified Approach reproduces Bogolyubov’s estimates for the ground state energy and condensate fraction at low density, as well as the mean-field estimate for the energy at high densities.We will also discuss a phase that appears at intermediate densities with liquid-like properties. The simplest of the effective equations in the Simplified Approach can be studied analytically, and we will review several results about it; the others are so far only amenable to numerical analysis, and we will discuss several numerical results. We will start by reviewing some results and conjectures on the Bose gas and then introduce the Simplified Approach and its derivation from the Bose gas.We will then discuss the predictions of the Simplified Approach and compare these to results and conjectures about the Bose gas. Finally, we will discuss a few open problems about the Simplified Approach.

arXiv:2202.07637EMS Press eBooksJun 29, 20224 citations

The Ideal Bose Gas

W. Cassing

Springer eBooksJan 1, 20250 citations

Analysis of a Simple Equation for the Ground State of the Bose Gas II: Monotonicity, Convexity, and Condensate Fraction

Eric A. Carlen, Ian Jauslin, Élliott H. Lieb

In a recent paper we studied an equation (called the “simple equation'') introduced by one of us in 1963 for an approximate correlation function associated with the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $\rho$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $\rho(e)$ for the density as a function of the energy $e$ . We had conjectured that $\rho(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(\rho)$ . We had also conjectured that $\rho e(\rho)$ is convex as a function of $\rho$ . We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities $\rho$ . Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.In a recent paper we studied an equation (called the “simple equation'') introduced by one of us in 1963 for an approximate correlation function associated with the ground state of an interacting Bose gas. Solving the equation yields a relation between the density $\rho$ of the gas and the energy per particle. Our construction of solutions gave a well-defined function $\rho(e)$ for the density as a function of the energy $e$. We had conjectured that $\rho(e)$ is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function $e(\rho)$. We had also conjectured that $\rho e(\rho)$ is convex as a function of $\rho$. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities $\rho$. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.

arXiv:2010.13882SIAM Journal on Mathematical AnalysisJan 1, 202111 citations

\rho

of the gas and the energy per particle. Our construction of solutions gave a well-defined function

\rho(e)

for the density as a function of the energy

e

. We had conjectured that

\rho(e)

is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function

e(\rho)

. We had also conjectured that

\rho e(\rho)

is convex as a function of

\rho

. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities

\rho

. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities.