Fluctuation-induced forces in confined ideal and imperfect Bose gases

Abstract

Fluctuation-induced forces are investigated for ideal and imperfect Bose gases confined to $d$-dimensional films of size $\infty^{d-1}\times D$ under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions $Υ^{\text{BC}}_d(x_λ=D/λ_{th},{x_ξ=D/ξ})$ of the residual reduced grand potential per area, $φ_{\text{res},d}^{\text{BC}}(T,μ,D)=D^{-(d-1)}Υ_d^{\text{BC}}(x_λ,x_ξ)$, are determined for the ideal gas case with these BCs, where $λ_{th}$ and $ξ$ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The scaling functions $Θ^{\text{BC}}_d(x_ξ)\equiv Υ_d^{\text{BC}}(\infty,x_ξ)$ describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free $O(2)$ theory for $\text{BC}=\text{P},\text{A},\text{DD},\text{DN},\text{NN}$. For $d=3$, they are expressed in closed analytical form. The analogous functions $Υ_d^{\text{BC}}(x_λ,x_ξ,c_1D,c_2D)$ and $Θ^{\text{R}}_d(x_ξ,c_1D,c_2D)$ under the RBCs $(\partial_z-c_1)ϕ|_{z=0}=(\partial_z+c_2)ϕ|_{z=D}=0$ with $c_1\ge 0$ and $c_2\ge 0$ are also determined. The functions $Υ_{\infty,d}^{\text{P}}(x_λ,x_ξ)$ and $Φ_{\infty,d}^{\text{P}}(x_ξ)$ for the imperfect Bose gas are shown to agree with those of the interacting Bose gas with $n\to\infty$ internal degrees of freedom. Hence for ${d=3}$, $Φ_{\infty,d}^{\text{P}}(x_ξ)$ is known exactly in closed analytic form. A modified imperfect Bose-gas model with free BC is introduced that corresponds to the limit $n\to\infty$ of this interacting Bose gas. Exact results for the function $Θ_{\infty,3}^{\mathbb{DD}}(x_ξ)$ therefore follow from those of the $O(2n)$ $ϕ^4$ model for $n\to\infty$.