Self-consistent equation for an interacting Bose gas

Abstract

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that $0<\int d\br V(r) = a<\infty$. Expressing the partition function by the Feynman-Kac functional integral yields a classical-like polymer representation of the quantum gas. With Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\rho (\mu )=F(\mu -a\rho (\mu ))$ between the density $\rho$ and the chemical potential $\mu$, valid in the range of convergence of Mayer series. The function $F$ is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling $V_γ(\br)=γ^{3}V(γr)$ we prove that in the mean-field limit $\gamma \to 0$ only tree diagrams contribute and function $F$ reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of Mayer series (vicinity of Bose-Einstein condensation) and study dominant corrections to mean field. At lowest order, the form of function $F$ is shown to depend on single polymer partition function for which we derive lower and upper bounds and on the resummation of ring diagrams which can be analytically performed.

33 pages, 6 figures, submitted to Phys.Rev. E

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• Quantum Coulomb systems : screening, recombination and van der Waals forces
• A possible mechanism of concurring diagonal and off-diagonal long-range order for soft interactions