Classical Langevin dynamics of a charged particle moving on a sphere and diamagnetism: A surprise

TL;DR

This study demonstrates that a classical charged particle constrained to a spherical surface exhibits a non-zero diamagnetic response under a magnetic field, challenging the traditional understanding of classical diamagnetism as null.

Contribution

The paper introduces a real space-time Langevin simulation showing classical diamagnetism on a sphere, revealing a non-zero magnetic moment contrary to the Bohr-van Leeuwen theorem.

Findings

Classical diamagnetism can be non-zero on a spherical surface.

Boundary effects are crucial in classical magnetic response.

First demonstration of finite classical diamagnetism due to boundary avoidance.

Abstract

It is generally known that the orbital diamagnetism of a classical system of charged particles in thermal equilibrium is identically zero -- the Bohr-van Leeuwen theorem. Physically, this null result derives from the exact cancellation of the orbital diamagnetic moment associated with the complete cyclotron orbits of the charged particles by the paramagnetic moment subtended by the incomplete orbits skipping the boundary in the opposite sense. Motivated by this crucial, but subtle role of the boundary, we have simulated here the case of a finite but \emph{unbounded} system, namely that of a charged particle moving on the surface of a sphere in the presence of an externally applied uniform magnetic field. Following a real space-time approach based on the classical Langevin equation, we have computed the orbital magnetic moment which now indeed turns out to be non-zero, and has the diamagnetic sign. To the best of our knowledge, this is the first report of the possibility of finite classical diamagnetism in principle, and it is due to the avoided cancellation.