Third Bose Fugacity Coefficient in One Dimension, as a Function of Asymptotic Quantities

TL;DR

This paper derives an exact expression for the third Bose fugacity coefficient in one dimension using eigenphase shifts, confirming the approach's validity and proposing a general method for calculating virial coefficients from asymptotic data.

Contribution

It develops a formalism expressing third fugacity coefficients in terms of eigenphase shifts, extending previous results and providing a general method beyond one dimension.

Findings

Exact expression for $b_3 - b_3^0$ in terms of 3-body eigenphase shifts.

First-order reproduction of low-temperature behavior using distorted-Born approximation.

Oscillating term contributions cancel in the formalism.

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.