# Non-equilibrium fluctuation-induced interactions between spheres: heat radiation and non-equilibrium Casimir forces

This test validates the scuff-neq application module of the scuff-em code suite by using it to compute (a) the temperature-dependent rate of heat radiation from individual spheres, (b) radiative heat-transfer rates and non-equilibrium Casimir forces between spheres at various separation distances.

## Analytical solution

Analytical formulas for single-sphere heat radiation and sphere-sphere heat-transfer rates and non-equilibrium Casimir forces were obtained by M. Krueger and are discussed in this paper:

The formulas are a bit complicated to reproduce here, but here is a simple julia code that implements them:

This code may be used to

## scuff-em solution

Values of the flux quantity \$\phi^{\text{\small heat radiation and between two dielectric spheres may be computed using scuff-cas3d as follows:

 % scuff-neq --geometry SiO2Sphere_501.scuffgeo --OmegaFile OmegaFile

 % scuff-neq --geometry SiO2Spheres_501.scuffgeo --OmegaFile OmegaFile


Here the two .scuffgeo files (PECSpheres_501.scuffgeo and E10Spheres_501.scuffgeo] describe the two geometric configurations (two PEC spheres and two dielectric spheres of radius $R=1\, \mu$m separated by an initial center-center distance of $d$=3 $\mu$m) while Spheres.trans specifies the list of center-center separation distances $d$ at which we compute the energy and force. (Both geometries refer to the same surface mesh file for the sphere, Sphere_327.msh.

The above calculations produce output files named PECSpheres_327.out and E10Spheres_327.out. Plotting against the theoretical predictions of Emig et. al (referenced above) yields good agreement:

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Here's the gnuplot script I used to produce this plot: Plotter.gp.