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Third Bose Fugacity Coefficient in One Dimension, as a Function of Asymptotic Quantities
A. Amaya-Tapia, S. Y. Larsen, M. Lassaut1

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.
1 :  IPNO - Institut de Physique Nucléaire d'Orsay
Physique/Matière Condensée/Gaz Quantiques

Physique/Matière Condensée/Mécanique statistique

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