Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2003-04-17
Physics
Condensed Matter
Statistical Mechanics
33 pages, 6 figures, submitted to Phys.Rev. E
Scientific paper
10.1103/PhysRevE.68.016113
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 $V_{\gamma}(\br)=\gamma^{3}V(\gamma r)$ 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.
Martin Philippe A.
Piasecki Jaroslaw
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