Constraints on stable equilibria with fluctuation-induced forces

TL;DR

This paper investigates whether fluctuation-induced forces like Casimir and van der Waals can produce stable levitation, concluding that such forces generally do not lead to stable equilibria due to inherent instabilities.

Contribution

The study extends Earnshaw's theorem to include thermal and quantum fluctuation-induced forces, demonstrating their inability to stabilize levitation in typical material configurations.

Findings

Classical objects with charges are unstable to small perturbations.

Quantum fluctuation forces do not stabilize equilibrium positions.

Stability is not achieved when objects' permittivities differ from the medium.

Abstract

We examine whether fluctuation-induced forces can lead to stable levitation. First, we analyze a collection of classical objects at finite temperature that contain fixed and mobile charges, and show that any arrangement in space is unstable to small perturbations in position. This extends Earnshaw's theorem for electrostatics by including thermal fluctuations of internal charges. Quantum fluctuations of the electromagnetic field are responsible for Casimir/van der Waals interactions. Neglecting permeabilities, we find that any equilibrium position of items subject to such forces is also unstable if the permittivities of all objects are higher or lower than that of the enveloping medium; the former being the generic case for ordinary materials in vacuum.