rotate. There will therefore be an average magnetic moment of zero. Since the distribution of motions does not depend on the magnetic field, the moment in thermal equilibrium remains zero in any magnetic field. A more formal proof So as to lower the complexity of the proof, a system with electrons will be used. This is appropriate, since most of the magnetism in a solid is carried by electrons, and the proof is easily generalized to more than one type of charged particle. Each electron has a negative charge and mass . If its position is and velocity is , it produces a current and a magnetic moment The above equation shows that the magnetic moment is a linear function of the velocity coordinates, so the total magnetic moment in a given direction must be a linear function of the form where the dot represents a time derivative and are vector coefficients depending on the position coordinates . Maxwell–Boltzmann statistics gives the probability that the nth particle has momentum and coordinate as where is the Hamiltonian, the total energy of the system. The thermal average of any function of these generalized coordinates is then In the presence of a magnetic field, where is the magnetic vector potential and is the electric scalar potential. For each particle the components of the momentum and position are related by the equations of Hamiltonian mechanics: Therefore, so the moment is a linear function of the momenta . The thermally averaged moment, is the sum of terms proportional to integrals of the form where represents one of the momentum coordinates. The integrand is an odd function of , so it vanishes. Therefore, . Applications The Bohr–Van Leeuwen theorem is useful in several applications including plasma physics: "All these references base their discussion of the Bohr–Van Leeuwen theorem on Niels Bohr's physical model, in which perfectly reflecting walls are necessary to provide the currents that cancel the net contribution from the interior of an element of plasma, and result in zero net diamagnetism for the plasma element." Diamagnetism of a purely classical nature occurs in plasmas but is a consequence of thermal disequilibrium, such as a gradient in plasma density. Electromechanics and electrical engineering also see practical benefit from the Bohr–Van Leeuwen theorem. See also List of plasma (physics) articles References External links The early 20th century: Relativity and quantum mechanics bring understanding at last Classical