# Fluctuation-induced forces in confined ideal and imperfect Bose gases

## TL;DR

This paper derives exact scaling functions for fluctuation-induced forces in confined ideal and imperfect Bose gases under various boundary conditions, revealing their critical behavior and connections to interacting Bose gases and field theories.

## Contribution

It provides comprehensive analytical results for fluctuation-induced forces in Bose gases with different boundary conditions, including novel exact solutions for the imperfect Bose gas and its relation to field theory models.

## Key findings

- Scaling functions are derived for ideal Bose gases with various boundary conditions.
- Exact analytical forms are obtained for the three-dimensional case.
- Connections established between imperfect Bose gases and interacting Bose gases in the large internal degrees of freedom limit.

## 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 $\Upsilon^{\text{BC}}_d(x_\lambda=D/\lambda_{th},{x_\xi=D/\xi})$ of the residual reduced grand potential per area, $\varphi_{\text{res},d}^{\text{BC}}(T,\mu,D)=D^{-(d-1)}\Upsilon_d^{\text{BC}}(x_\lambda,x_\xi)$, are determined for the ideal gas case with these BCs, where $\lambda_{th}$ and $\xi$ are the thermal de-Broglie wavelength and the bulk correlation length, respectively. The scaling functions $\Theta^{\text{BC}}_d(x_\xi)\equiv \Upsilon_d^{\text{BC}}(\infty,x_\xi)$ describing the critical behavior at the bulk condensation transition are shown to agree with those previously determined from a massive free $O(2)$ 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 $\Upsilon_d^{\text{BC}}(x_\lambda,x_\xi,c_1D,c_2D)$ and $\Theta^{\text{R}}_d(x_\xi,c_1D,c_2D)$ under the RBCs $(\partial_z-c_1)\phi|_{z=0}=(\partial_z+c_2)\phi|_{z=D}=0$ with $c_1\ge 0$ and $c_2\ge 0$ are also determined. The functions $\Upsilon_{\infty,d}^{\text{P}}(x_\lambda,x_\xi)$ and $\Phi_{\infty,d}^{\text{P}}(x_\xi)$ 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 ${d=3}$, $\Phi_{\infty,d}^{\text{P}}(x_\xi)$ 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 $\Theta_{\infty,3}^{\mathbb{DD}}(x_\xi)$ therefore follow from those of the $O(2n)$ $\phi^4$ model for $n\to\infty$.