On the Relation Between Casimir Forces and Bulk Correlations

Marek Napiórkowski,Jarosław Piasecki

doi:10.1007/s10955-014-1050-7

Abstract

Within a microscopic approach we show that in the case of an ideal quantum gas enclosed in a slit the Casimir force can be simply expressed in terms of the bulk one-particle density matrix. The corresponding formula, which holds both for bosons and fermions, allows to relate the range of the Casimir force to the bulk correlation length. The low-temperature behavior of the Casimir forces is derived.

Highlights

The Casimir forces acting on planar walls immersed in a perfect quantum gas have been the object of numerous studies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
We will show that the Casimir forces depend on the boundary conditions imposed by the confining walls, they can be expressed in terms of the one-particle density matrix [1,18] calculated in the thermodynamic limit
In the case of ideal quantum gases the one-particle density matrix is directly related to the correlation function

Summary

Introduction

The Casimir forces acting on planar walls immersed in a perfect quantum gas (the slit geometry) have been the object of numerous studies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We will show that the Casimir forces depend on the boundary conditions imposed by the confining walls, they can be expressed in terms of the one-particle density matrix [1,18] calculated in the thermodynamic limit. In the case of a perfect Bose gas the correlation function χ2B (r ) can be expressed exclusively in terms of ρ1B (r ) [1]. The derived formula forms the basis of the subsequent analysis Most importantly, it permits to establish an explicit relation between the pair correlation function χ(r ) and the Casimir forces. The fact that the Casimir forces can be expressed in terms of the one-particle density matrix implies their direct relation to the correlation function through Eqs. Our analysis permits to recover in a concise way various specific results derived elsewhere by different techniques

Relation Between Casimir Forces and the One-Particle Density Matrix

Range of Casimir Forces and Bulk Correlation Length

F Di r κNFeu λ2 k F 4π 2

D2 h 2k3F m

Concluding Comments