2018/09/30 by Joshua Eby, Madelyn Leembruggen, Lauren Street +3 · 1 citation
Mathematics · Physics and Astronomy · #Ansatz #Cold Atom Physics and Bose-Einstein Condensates #Cosmology and Gravitation Theories #Dimensionless quantity #Double exponential function #Exponential function #Function (biology) #Gaussian #Mathematical analysis #Mathematics #Physics #Pulsars and Gravitational Waves Research #Quantum mechanics #Space (punctuation) #Statistical physics #Wave function #astro-ph.CO #hep-ph
paper · pdf · doi:10.1103/physrevd.98.123013
published as Phys. Rev. D 98, 123013 (2018) · 20 pages, 2 appendices, 4 figures. v2: Citations added, typos corrected
arxiv created 2018/10/06 · openalex publication_date 2018/12/18 · arxiv updated 2018/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We analyze the accuracy of the variational method in computing physical quantities relevant for gravitationally bound Bose-Einstein condensates. Using a variety of spherically symmetric variational ans"atze found in existing literature, we determine physical quantities and compare them to numerical solutions. We conclude that a ``linear+exponential'' wave function proportional to (1+\ensuremathξ)exp(\ensuremath-\ensuremathξ) (where \ensuremathξ is a dimensionless radial variable) is the best fit for attractive self-interactions along the stable branch of solutions, while for small particle number N it is also the best fit for repulsive self-interactions. For attractive self-interactions along the unstable branch, a single exponential is the best fit for small N, while a sech wave function fits better for large N. The Gaussian wave function ansatz, which is used often in the literature, is exceedingly poor across most of the parameter space, with the exception of repulsive interactions for large N. We investigate a ``double exponential'' ansatz with a free constant parameter, which is computationally efficient and can be optimized to fit the numerical solutions in different limits. We show that the double exponential can be tuned to fit the sech ansatz, which is computationally slow. We also show how to generalize the addition of free parameters in order to create more computationally efficient ans"atze using the double exponential. Determining the best ansatz, according to several comparison parameters, will be important for analytic descriptions of dynamical systems. Finally, we examine the underlying relativistic theory, and critically analyze the Thomas-Fermi approximation often used in the literature.