Fluctuation-induced forces in confined ideal and imperfect Bose gases

摘要

Fluctuation-induced ("Casimir") forces caused by thermal and quantum fluctuations are investigated for ideal and imperfect Bose gases confined to d-dimensional films of size ∞d~(-1) × D under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions γ_d~(BC) (x_λ = D/λ_(th), x_ξ = D/ξ) of the residual reduced grand potential per area Φ_(res,d)~(BC) (T,μ,D) = D~(-(d-1))γ_d~(BC) (x_λ,x_ξ ) are determined for the ideal gas casewith these BCs, where λ_(th) and ξ are the thermal deBroglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functions BCd (xξ ) ≡γ_d~(BC) (∞,x_ξ ) describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive freeO(2) theory for BC = P,A,DD,DN,NN. For d = 3, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions γ_d~(BC) (x_λ, xξ ,c_1D,c_2D) and R_d (x_ξ ,c_1D,c_2D) under the RBCs (?_z - c_1)Φ|z=0 = (?_z + c_2)Φ|_z=D = 0 with c1 ≥ 0 and c_2≥ 0 are also determined. The corresponding scaling functions γ_(∞,d)~P (x_λ,x_ξ) and P_(∞,d) (x_ξ ) for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with n internal degrees of freedom in the limit n→∞. Hence, for d = 3,P∞,d (x_ξ ) is known exactly in closed analytic form. To account for the breakdown of translation invariance in the direction perpendicular to the boundary planes implied by free BCs such as DDBCs, a modified imperfect Bose gas model is introduced that corresponds to the limit n→∞ of this interacting Bose gas. Numerically and analytically exact results for the scaling function DD∞,3 (x_ξ ) therefore follow from those of the O(2n) Φ~4 model for n→∞.