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arXiv:1411.3946

Level density of a Bose gas: beyond the saddle point approximation

A. D. Ribeiro

Published Nov 12, 2014• 0 citations
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Abstract

The present article is concerned with the use of approximations in the calculation of the many-body density of states (MBDS) of a system with total energy E, composed by N bosons. In the mean-field framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we derive a uniform formula for MBDS, which is potentially able to deal with this regime. In the case of the one-dimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method.

DOI
10.48550/arxiv.1411.3946

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The present article is concerned with the use of approximations in the calculation of the many-body density of states (MBDS) of a system with total energy E, composed by N bosons. In the mean-field framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we derive a uniform formula for MBDS, which is potentially able to deal with this regime. In the case of the one-dimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method.
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The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.