A note on the possibility of classical orbital diamagnetism for an unbounded system --the Bohr-van Leeuwen Theorem

TL;DR

This paper challenges the classical understanding of diamagnetism by showing that a finite orbital diamagnetic moment can exist in classical systems, contradicting the Bohr-van Leeuwen theorem, through analysis of the Fokker-Planck equation and exact solutions for simpler geometries.

Contribution

It demonstrates that the classical orbital diamagnetism can be non-zero in unbounded systems by analyzing the time-dependent Fokker-Planck equation, providing exact solutions for simpler geometries.

Findings

Finite orbital diamagnetic moment observed in simulations.

Long-time limit of expectation value is non-zero.

Steady-state Fokker-Planck density satisfies canonical distribution.

Abstract

Recently [EPL, 86, (2009) 17001], we had simulated the classical Langevin dynamics of a charged particle on the surface of a sphere in the presence of an externally applied magnetic field, and found a finite value for the orbital diamagnetic moment in the long-time limit. This result is surprising in that it seems to violate the classic Bohr-van Leeuwen Theorem on the absence of classical diamagnetism. It was indeed questioned by some workers [EPL, 89, (2010) 37001] who verified that the Fokker-Planck (FP) equation derived from our Langevin equation, was satisfied by the classical canonical density in the steady state, obtained by setting d/dt=0 in the FP equation. Inasmuch as the canonical density does not contain the magnetic field, they concluded that the diamagnetic moment must be zero. The purpose of this note is to show that this argument and the conclusion are invalid -- instead of setting d/dt=0 one must first obtain the fundamental time-dependent solution for the FP equation, and then calculate the expectation value of the diamagnetic moment, and finally consider its long-time limit (i.e., $t \to \infty$). This would indeed correspond to our numerical simulation of the dynamics. That this is indeed so is shown by considering the simpler analytically solvable problem, namely that for an unbounded plane for which the above procedure can be carried out exactly. We then find that the limiting value for the expectation of the diamagnetic moment is indeed non-zero, and yet the steady-state FP equation obtained by setting d/dt=0 is satisfied by the canonical density. Admittedly, the exact analytical solution for the sphere is not available. But, the exact solution obtained for the case of the unbounded 2D-plane illustrates our point all right. We also present some further new results for other finite but unbounded surfaces such the ellipsoids of revolution.