# 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 m
separated by an initial center-center distance of =3 m) while
`Spheres.trans`

specifies the list of center-center
separation distances 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:

.

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

.