Fluctuation-induced forces in confined ideal and imperfect Bose gases

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 ${\infty }^{d-1}\times D$ under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions ${\mathrm{{\rm Y}}}_d^{\text{BC}}(x_{\lambda }=D/{\lambda }_{\text{th}},x_{\xi }=D/\xi )$ of the residual reduced grand potential per area ${\phi }_{\text{res},d}^{\text{BC}}(T,\mu ,D)=D^{-(d-1)}{\mathrm{{\rm Y}}}_d^{\text{BC}}(x_{\lambda },x_{\xi })$ are determined for the ideal gas case with these BCs, where ${\lambda }_{\text{th}}$ and $\xi$ are the thermal de Broglie wavelength and the bulk correlation length, respectively. The associated limiting scaling functions ${\mathrm{\Theta }}_d^{\text{BC}}\left(x_{\xi }\right)\equiv {\mathrm{{\rm Y}}}_d^{\text{BC}}(\infty ,x_{\xi })$ describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free $O\left(2\right)$ theory for $\text{BC}=\text{P},\text{A},\text{DD},\text{DN},\text{NN}$. For $d=3$, they are expressed in closed analytical form in terms of polylogarithms. The analogous scaling functions ${\mathrm{{\rm Y}}}_d^{\text{BC}}(x_{\lambda },x_{\xi },c_{1}D,c_{2}D)$ and ${\mathrm{\Theta }}_d^{\text{R}}(x_{\xi },c_{1}D,c_{2}D)$ under the RBCs $({\partial }_z-c_1)\varphi {{|}_{z=0}=({\partial }_z+c_2)\varphi |}_{z=D}=0$ with $c_1\ge 0$ and $c_2\ge 0$ are also determined. The corresponding scaling functions ${\mathrm{{\rm Y}}}_{\infty ,d}^{\text{P}}(x_{\lambda },x_{\xi })$ and ${\mathrm{\Theta }}_{\infty ,d}^{\text{P}}\left(x_{\xi }\right)$ 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 $n\to \infty$. Hence, for $d=3, {\mathrm{\Theta }}_{\infty ,d}^{\text{P}}\left(x_{\xi }\right)$ 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 $n\to \infty$ of this interacting Bose gas. Numerically and analytically exact results for the scaling function ${\mathrm{\Theta }}_{\infty ,3}^{\mathbb{DD}}\left(x_{\xi }\right)$ therefore follow from those of the $O\left(2n\right){\varphi }^4$ model for $n\to \infty$.

Figure 1

Classical scaling functions ${\mathrm{\Theta }}_3^{\text{BC}}\left(x_{\xi }\right)$ for $n=1$ and $\text{BC}=\text{P},\text{A},\text{DD},\text{NN}$, and $\text{DN}$. Their values at $x_{\xi }$ are the corresponding Casimir amplitudes ${\mathrm{\Delta }}_d^{\text{BC}}$.

Figure 2

Classical scaling functions ${\vartheta }_3^{\text{BC}}\left(x_{\xi }\right)$ for $n=1$ and $\text{BC}=\text{P},\text{A},\text{DD},\text{NN}$, and $\text{DN}$.

Figure 3

Deformation of the contour ${\mathcal{C}}_1$ into the union of contours ${\gamma }_j,j\in \mathbb{Z}$ plus a circle around the isolated pole at $E=-ic$.

Figure 4

The scaling functions ${\mathrm{\Theta }}_{3,\infty }^{\text{P}}\left(x\right)$ and ${\vartheta }_{3,\infty }^{\text{P}}\left(x\right)$.