Dynamical approach to the Casimir effect

TL;DR

This paper develops a general dynamical formalism to compute Casimir forces in various media, encompassing equilibrium and nonequilibrium conditions, and applies it to systems with different boundary conditions and noise correlations.

Contribution

It introduces a novel dynamical approach to derive Casimir forces directly from fluctuation equations without relying on Hamiltonian formalism.

Findings

Nonequilibrium dynamics can produce Casimir forces.

Finite correlations in noise lead to nonequilibrium Casimir effects.

Anisotropic stress tensors can result in Casimir forces in equilibrium.

Abstract

Casimir forces can appear between intrusions placed in different media driven by several fluctuation mechanisms, either in equilibrium or out of it. Herein, we develop a general formalism to obtain such forces from the dynamical equations of the fluctuating medium, the statistical properties of the driving noise, and the boundary conditions of the intrusions (which simulate the interaction between the intrusions and the medium). As a result, an explicit formula for the Casimir force over the intrusions is derived. This formalism contains the thermal Casimir effect as a particular limit and generalizes the study of the Casimir effect to such systems through their dynamical equations, with no appeal to their Hamiltonian, if any exists. In particular, we study the Casimir force between two infinite parallel plates with Dirichlet or Neumann boundary conditions, immersed in several media with finite correlation lengths (reaction--diffusion system, liquid crystals, and two coupled fields with non-Hermitian evolution equations). The driving Gaussian noises have vanishing or finite spatial or temporal correlation lengths; in the first case, equilibrium is reobtained and finite correlations produce nonequilibrium dynamics. The results obtained show that, generally, nonequilibrium dynamics leads to Casimir forces, whereas Casimir forces are obtained in equilibrium dynamics if the stress tensor is anisotropic.