Symmetry Results for Finite-Temperature, Relativistic Thomas-Fermi Equations

TL;DR

This paper proves that in the semi-classical limit, relativistic electron-ion beams described by finite-temperature Thomas-Fermi equations must be cylindrically symmetric, using isoperimetric inequalities and the virial theorem.

Contribution

It establishes symmetry results for finite-temperature, relativistic Thomas-Fermi equations in the semi-classical limit, extending previous work on Bennett equations.

Findings

All particle density functions are cylindrically symmetric.

Non-symmetric densities violate the virial theorem.

Symmetry holds under the semi-classical limit.

Abstract

In the semi-classical limit, the quantum mechanics of a stationary beam of counter-streaming relativistic electrons and ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature / low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.