Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-01-31
J. Stat. Mech. (2007) P02002
Physics
Condensed Matter
Statistical Mechanics
19 pages, to appear in J. Stat. Mech
Scientific paper
10.1088/1742-5468/2007/02/P02002
For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions, the Wigner-Kirkwood formalism provides a semiclassical expansion of the Boltzmann density in configuration space in even powers of the thermal de Broglie wavelength $\lambda$. This result permits one to generate an analogous $\lambda$-expansion for the bulk free energy and many-body densities. The present paper brings a technically nontrivial generalization of the Wigner-Kirkwood technique to semi-infinite quantum fluids, constrained by a plane hard wall impenetrable to particles. In contrast to the bulk case, the resulting Boltzmann density involves also position-dependent terms of type $\exp(-2x^2/\lambda^2)$ ($x$ denotes the distance from the wall boundary) which are non-analytic in $\lambda$. Under some condition, the analyticity in $\lambda$ is restored by integrating the Boltzmann density over configuration space; however, in contrast to the bulk free energy, the semiclassical expansion of the surface part of the free energy (surface tension) contains odd powers of $\lambda$, too. Explicit expressions for the leading quantum corrections in the presence of the boundary are given for the one-body and two-body densities. As model systems for explicit calculations, we use Coulomb fluids, in particular the one-component plasma defined in the $\nu$-dimensional (integer $\nu\ge 2$) space.
Jancovici Bernard
Samaj Ladislav
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