Variance of the Casimir force in an ideal Bose gas

TL;DR

This paper calculates the variance of the thermal Casimir force in an ideal Bose gas confined in a slab, revealing its decay behavior and dependence on non-universal scaling variables, with implications for Bose-Einstein condensates.

Contribution

It provides a detailed analysis of the variance of the Casimir force in an ideal Bose gas, including its scaling behavior and non-universal amplitude in different thermodynamic states.

Findings

Variance decays as 1/D for large D

Amplitude depends on scaling variables λ/ξ and D/ξ

Ratio of standard deviation to force scales with (D/L)^{(d-1)/2} (D/λ)^{d/2}

Abstract

We consider an ideal Bose gas enclosed in a $d$-dimensional slab of thickness $D$. Using the grand canonical ensemble we calculate the variance of the thermal Casimir force acting on the slab's walls. The variance evaluated per unit wall area is shown to decay like $ \Delta_{var}/D$ for large $D$. The amplitude $ \Delta_{var}$ is a non-universal function of two scaling variables $\lambda/ \xi$ and $D/ \xi$, where $\lambda$ is the thermal de Broglie wavelength and $\xi$ is the bulk correlation length. It can be expressed via the bulk pressure, the Casimir force per unit wall area, and its derivative with respect to chemical potential. For thermodynamic states corresponding to the presence of the Bose-Einstein condensate the amplitude $ \Delta_{var}$ retains its non-universal character while the ratio of the mean standard deviation and the Casimir force takes the scaling form $(D/L)^{\frac{d-1}{2}}\, (D/\lambda)^{d/2}$, where $L$ is the linear size of the wall.