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Classical Orbital Magnetic Moment in a Dissipative Stochastic System

We present an analytical treatment of the dissipative-stochastic dynamics of a charged classical particle confined bi-harmonically in a plane with a uniform static magnetic field directed perpendicular to the plane. The stochastic dynamics gives a steady state in the long-time limit. We have examined the orbital magnetic effect of introducing a parametrized deviation ($η$ -1) from the second fluctuation-dissipation (II-FD) relation that connects the driving noise and the frictional memory kernel in the standard Langevin dynamics. The main result obtained here is that the moving charged particle generates a finite orbital magnetic moment in the steady state, and that the moment shows a crossover from para-to dia-magnetic sign as the parameter $η$ is varied. It is zero for $η= 1$ that makes the steady state correspond to equilibrium, as it should. The magnitude of the orbital magnetic moment turns out to be a non-monotonic function of the applied magnetic field, tending to zero in the limit of an infinitely large as well as an infinitesimally small magnetic field. These results are discussed in the context of the classic Bohr-van Leeuwen theorem on the absence of classical orbital diamagnetism. Possible realization is also briefly discussed.