Numerical methods for fluctuation-driven interactions between dielectrics

Abstract

We develop a discretized theory of thermal Casimir interactions to numerically calculate the interactions between fluctuating dielectrics. From a constrained partition function we derive a surface free energy, while handling divergences that depend on system size and discretization. We derive analytic results for parallel plate geometry in order to check the convergence of the numerical methods. We use the method to calculate vertical and lateral Casimir forces for a set of grooves.

Worldline approach for numerical computation of electromagnetic Casimir energies: Scalar field coupled to magnetodielectric media

“…To investigate the feasibility of such algorithms, we will discuss the numerical evaluation of the path integrals (27) and (36) for the TE polarization of the electromagnetic field, and compare the numerical solutions to the available analytic solutions in planar geometries. It is important to emphasize that the methods developed here apply in any material geometry, but the solutions correspond to exact electromagnetic solutions in planar layered media.…”

“…For example, for a polarizable atom interacting with a perfectly conducting planar surface (corresponding to r −→ ∞), the conductor imposes Dirichlet boundary conditions on the scalar field, as in previous work on Casimir worldlines with background potentials [18]. In the renormalized form of the path integral (36), the integrand r −3/2 − 1 is averaged over the ensemble. The integrand vanishes for paths that do not touch the surface, but takes the value −1 for those that do.…”

“…To further investigate the worldline path integrals, we will consider their analytic evaluation and show that the dielectric-body path integrals (27) and (36) converge to known solutions in planar geometries. In certain limits, this evaluation is quite straightforward.…”

“…Here we will discuss two such methods in the context of the CasimirPolder path integral, but the same techniques also apply in the general Casimir case. One method comes from rewriting the TE Casmir-Polder path integral (36) in the (unrenormalized) form…”

“…al have considered how to implement regularized Neumann boundary conditions in path integrals, and wrote down a worldline path integral for Neumann boundaries, using a nonlocal (in proper time) representation of the boundary potential [35]. Similar functional-determinant methods have also been used to compute the electrostatic contribution of the Casimir energy in terms of a scalar field [36].…”