Flux Quantization and Dimensionality

A system of charged particles in a ring of radius $R$ is considered, assuming separability of the tangential component of the total momentum. The dependence of the free energy upon the magnetic flux through the ring is shown to be strongly affected by the dimensionality. Irrespective of the specific dynamical properties of the system, one finds in the one-dimensional case that a realistic choice of $R$ leads to the exclusion of thermodynamically stable flux trapping. The entirely different criterion for the three-dimensional case is separately discussed and is seen to be closely related to the mean-square fluctuation of the total momentum.