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

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 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. In this work we re-visit the problem of the absence of classical diamagnetism of a system of charged particles in thermal equilibrium. This vanishing of the classical diamagnetism in equilibrium is generally referred to as the Bohr-van Leeuwen theorem [1–4]. The fact that classically the orbital diamagnetic moment vanishes is quite contrary to our physical expectations inasmuch as a charged particle (of charge −e, position r(t), and velocity v(t) at time t), say, orbiting in a plane perpendicular to the magnetic field B under its Lorentz force should have an orbital magnetic moment M (= −e/2c [ r(t) × v(t) ] ), where c is the speed of light, and with a diamagnetic sign as dictated by the Lenz’s law (see e.g. [5]). Formally, the vanishing of the classical diamagnetic moment follows from the well known fact that the canonical partition function involves the Hamiltonian for the charged particle (coupled minimally to the static magnetic field) and a simple shift of the canonical momentum variable in the integration makes the partition function field-independent, giving zero magnetic moment [3]. (a)email: [email protected] (b)email: [email protected] Physically, the vanishing of the classical diamagnetism is due, however, to a subtle role played by the boundary of the finite sample [1–4]. It turns out that the diamagnetic contribution of the completed cyclotron orbits of the charged particles orbiting around the magnetic field in a plane perpendicular to it, is cancelled by the paramagnetic contribution of the incomplete orbits skipping the boundary in the opposite sense in a cuspidal manner. The cancellation is exact, and that is the surprise. This cancellation was demonstrated explicitly some time back [6] for the case of a harmonic-potential (V (r) = kr/2) confinement, which is equivalent to a soft boundary, and finally letting the spring constant k go to zero. The treatment was based on the classical Langevin equation [7], and the magnetic moment M = −e/2c [