2020/10/26 by Eric A. Carlen, Carlen, Eric A., Ian Jauslin +3
Computer Science · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Gases (cond-mat.quant-gas) #Quantum Information and Cryptography #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2010.13882
openalex publication_date 2020/10/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In a recent paper we studied an equation (called the "simple equation")\nintroduced by one of us in 1963 for an approximate correlation function\nassociated to the ground state of an interacting Bose gas. Solving the equation\nyields a relation between the density \ρ of the gas and the energy per\nparticle. Our construction of solutions gave a well-defined function \ρ(e)\nfor the density as a function of the energy e. We had conjectured that\n\ρ(e) is a strictly monotone increasing function, so that it can be\ninverted to yield the strictly monotone increasing function e(\ρ). We had\nalso conjectured that \ρ e(\ρ) is convex as a function of \ρ. We\nprove both conjectures here for small densities, the context in which they have\nthe most physical relevance, and the monotonicity also for large densities.\nBoth conjectures are grounded in the underlying physics, and their proof\nprovides further mathematical evidence for the validity of the assumptions\nunderlying the derivation of the simple equation, at least for low or high\ndensities, if not intermediate densities, although the equation gives\nsurprisingly good predictions for all densities \ρ. Another problem left\nopen in our previous paper was whether the simple equation could be used to\ncompute accurate predictions of observables other than the energy. Here, we\nprovide a recipe for computing predictions for any one- or two-particle\nobservables for the ground state of the Bose gas. We focus on the condensate\nfraction and the momentum distribution, and show that they have the same low\ndensity asymptotic behavior as that predicted for the Bose gas. Along with the\ncomputation of the low density energy of the simple equation in our previous\npaper, this shows that the simple equation reproduces the known and conjectured\nproperties of the Bose gas at low densities.\n