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Title: Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities

Abstract

In one of the very few exact quantum mechanical calculations of fugacity coefficients, [L.R. Dodd, A.M. Gibbs. J. Math. Phys. 15 (1974) 41] obtained b{sub 2} and b{sub 3} for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For b{sub 2}, we have shown [A. Amaya-Tapia, S.Y. Larsen, M. Lassaut. Mol. Phys. 103 (2005) 1301-1306. < (arXiv:physics/0405150)>] that Dodd and Gibbs' result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms, usually contributing only in one dimension. Now, we develop an exact expression for b{sub 3}-b{sub 3}{sup 0} (where b{sub 3}{sup 0} is the free particle fugacity coefficient) in terms of sums and differences of three-body eigenphase shifts. Further, we show that if we obtain these eigenphase shifts in a Distorted-Born approximation, then, to first order, we reproduce the leading low temperature behaviour, obtained from an expansion of the twofold integral of Dodd and Gibbs. The contributions of the oscillating terms cancel. The formalism that we propose is not limited to one dimension, but seeks to provide a general method to obtain virial coefficients, fugacity coefficients, in terms of asymptotic quantities. The exactmore » one dimensional results allow us to confirm the validity of our approach in this domain.« less

Authors:
 [1];  [2];  [3]
  1. Instituto de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico AP 48-3, Cuernavaca, Mor. 62251 (Mexico)
  2. Department of Physics, Temple University, Philadelphia, PA 19122 (United States)
  3. Institut de Physique Nucleaire, IN2P3-CNRS, Universite Paris-Sud 11, F-91406 Orsay Cedex (France)
Publication Date:
OSTI Identifier:
21579846
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 326; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2010.10.004; PII: S0003-4916(10)00181-8; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; BORN APPROXIMATION; BOSE-EINSTEIN GAS; BOSONS; DELTA FUNCTION; INTEGRALS; INTERACTIONS; ONE-DIMENSIONAL CALCULATIONS; PHASE SHIFT; QUANTUM MECHANICS; TEMPERATURE RANGE 0065-0273 K; THREE-BODY PROBLEM; WAVE FUNCTIONS; APPROXIMATIONS; CALCULATION METHODS; FUNCTIONS; MANY-BODY PROBLEM; MATHEMATICAL SOLUTIONS; MECHANICS; TEMPERATURE RANGE

Citation Formats

Amaya-Tapia, A, Larsen, S Y, and Lassaut, M. Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities. United States: N. p., 2011. Web. doi:10.1016/j.aop.2010.10.004.
Amaya-Tapia, A, Larsen, S Y, & Lassaut, M. Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities. United States. https://doi.org/10.1016/j.aop.2010.10.004
Amaya-Tapia, A, Larsen, S Y, and Lassaut, M. 2011. "Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities". United States. https://doi.org/10.1016/j.aop.2010.10.004.
@article{osti_21579846,
title = {Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities},
author = {Amaya-Tapia, A and Larsen, S Y and Lassaut, M},
abstractNote = {In one of the very few exact quantum mechanical calculations of fugacity coefficients, [L.R. Dodd, A.M. Gibbs. J. Math. Phys. 15 (1974) 41] obtained b{sub 2} and b{sub 3} for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For b{sub 2}, we have shown [A. Amaya-Tapia, S.Y. Larsen, M. Lassaut. Mol. Phys. 103 (2005) 1301-1306. < (arXiv:physics/0405150)>] that Dodd and Gibbs' result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms, usually contributing only in one dimension. Now, we develop an exact expression for b{sub 3}-b{sub 3}{sup 0} (where b{sub 3}{sup 0} is the free particle fugacity coefficient) in terms of sums and differences of three-body eigenphase shifts. Further, we show that if we obtain these eigenphase shifts in a Distorted-Born approximation, then, to first order, we reproduce the leading low temperature behaviour, obtained from an expansion of the twofold integral of Dodd and Gibbs. The contributions of the oscillating terms cancel. The formalism that we propose is not limited to one dimension, but seeks to provide a general method to obtain virial coefficients, fugacity coefficients, in terms of asymptotic quantities. The exact one dimensional results allow us to confirm the validity of our approach in this domain.},
doi = {10.1016/j.aop.2010.10.004},
url = {https://www.osti.gov/biblio/21579846}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 2,
volume = 326,
place = {United States},
year = {Tue Feb 15 00:00:00 EST 2011},
month = {Tue Feb 15 00:00:00 EST 2011}
}