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arXiv:1910.07313

Determination of the second virial coefficient of interacting Bosons using the method of Wigner distribution function

Anirban Bose

Published Sep 27, 2019• 0 citations
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Abstract

In a previous article \cite{kn:anirban1} a method has been introduced to derive the all order Bose-Einstein distribution of the non interacting Bosons as the solution of the Wigner equation. The process was a perturbative one where the Bose-Einstein distribution was taken as the unperturbed solution. In this article it is shown that the same formalism is also applicable in the case of interacting Bosons. The formalism has been applied to calculate the quantum second virial coefficient of the Bosons interacting pairwise via Lenard-Jones potential and compared with the previous result.

DOI
10.48550/arxiv.1910.07313

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In a previous article \cite{kn:anirban1} a method has been introduced to derive the all order Bose-Einstein distribution of the non interacting Bosons as the solution of the Wigner equation. The process was a perturbative one where the Bose-Einstein distribution was taken as the unperturbed solution. In this article it is shown that the same formalism is also applicable in the case of interacting Bosons. The formalism has been applied to calculate the quantum second virial coefficient of the Bosons interacting pairwise via Lenard-Jones potential and compared with the previous result.
Abstract A method has been introduced to derive the all order Bose–Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose–Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons at finite temperature. This process can also be applied to obtain the all order Fermi distribution.
A simple method has been introduced to derive the all order quantum corrected Bose-Einstein distribution as the solution of the Wigner equation. The process is a perturbative one where the Bose-Einstein distribution has been taken as the unperturbed solution. This solution has been applied to calculate the number density of the bosons at finite temperature. The study may be important to investigate the properties of bosons and bose condensates at finite temperature. This process can also be applied to obtain the quantum corrected Fermi distribution.
We present a field theoretic method for the calculation of the second and third virial coefficients b2{b}_{2}b2​ and b3{b}_{3}b3​ of two-species fermions interacting via a contact interaction. The method is mostly analytic. We find a closed expression for b3{b}_{3}b3​ in terms of the two- and three-body TTT matrices. We recover numerically, at unitarity, and also in the whole Bose-Einstein-condensate--BCS crossover, previous numerical results for the third virial coefficient b3{b}_{3}b3​.
We consider the statistical mechanics of an interacting anyon gas with a repulsive pair potential of the form g/r2{\mathit{r}}^{2}r2. The second virial coefficient is calculated and shown to be a smooth function of the statistical parameter \ensuremath{\alpha} for g>0, without the cusps previously found at the Bose point in a noninteracting anyon gas. As g\ensuremath{\rightarrow}0 the cusps remain absent but a different type of singularity develops at the Bose point, showing that the order of limits is crucial.
Abstract We apply the formalism of the free Wigner transform to the derivation of a kinetic equation valid for dilute quantum gases; this formalism is compatible with a precise treatment of two‐body correlations and allows the inclusion of second virial corrections. We generalize the results obtained in previous articles where the calculation of the collision integral was limited to zero (“local”) and first order gradient terms; here we use a more general (“non‐local”) form for the interaction term which is an integral of higher dimension (18). The conservation laws are studied in this formalism, which implies non‐linear expressions for the local hydrodynamic quantities in terms of the function appearing in the kinetic equation. The pressure at thermal equilibrium is then evaluated and found to be equal to the value predicted by the Beth‐Uhlenbeck formula, without adjusting any free parameter of the theory.
The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.