TY - JOUR

T1 - ANALYSIS OF A SIMPLE EQUATION FOR THE GROUND STATE OF THE BOSE GAS II

T2 - MONOTONICITY, CONVEXITY, AND CONDENSATE FRACTION

AU - Carlen, Eric A.

AU - Jauslin, Ian

AU - Lieb, Elliott H.

N1 - Funding Information:
\ast Received by the editors October 30, 2020; accepted for publication (in revised form) April 19, 2021; published electronically September 23, 2021. https://doi.org/10.1137/20M1376820 Funding: The work of the first author was supported by the National Science Foundation grant DMS-1764254. The work of the second author was supported by the National Science Foundation grant DMS-1802170. \dagger Depatment of Mathematics, Rutgers University, Piscataway, NJ 08854-8019 USA (carlen@math. rutgers.edu). \ddagger Department of Physics, Princeton University, Princeton, NJ 08544 USA (ijauslin@princeton. edu). \S Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544 USA (lieb@ math.princeton.edu).
Publisher Copyright:
© 2021 Eric A. Carlen, Ian Jauslin, and Elliott H. Lieb

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Bose gas

KW - Bose-Einstein condensation

KW - partial differential equations

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U2 - 10.1137/20M1376820

DO - 10.1137/20M1376820

M3 - Article

AN - SCOPUS:85125034437

SN - 0036-1410

VL - 53

SP - 5322

EP - 5360

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 5

ER -