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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, Dodd and Gibbs (\textit{J. Math.Phys}.,\textbf{15}, 41 (1974)) obtained $b_{2}$ and $b_{3}$ for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For $b_{2}$, we have shown (\textit{Mol. Phys}.,\textbf{103}, 1301 (2005)) 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 1 dimension. Now, we develop an exact expression for $b_{3}-b_{3}^{0}$ (where $b_{3}^{0}$ is the free particle fugacity coefficient) in terms of sums and differences of 3-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 two-fold 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.
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Contributor : Sophie Heurteau <>
Submitted on : Wednesday, February 16, 2011 - 10:46:31 AM
Last modification on : Wednesday, September 16, 2020 - 4:07:55 PM

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A. Amaya-Tapia, S.Y. Larsen, M. Lassaut. Third Bose fugacity coefficient in one dimension, as a function of asymptotic quantities. Annals of Physics, 2011, 326 (2), pp.406-425. ⟨10.1016/j.aop.2010.10.004⟩. ⟨in2p3-00566419⟩

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