Statistical equilibrium of nonrelativistic multiply periodic classical systems and random classical electromagnetic radiation

The statistical equilibrium of charged nonrelativistic multiply periodic classical mechanical systems and random classical electromagnetic radiation is considered in the narrow-linewidth approximation using calculations by van Vleck based on action-angle variables. It is found that the nonrelativistic classical mechanical systems and the classical radiation are both in equilibrium when the distribution of the mechanical systems in phase space is given by the Boltzman distribution and the radiation spectrum is given by the Rayleigh-Jeans law. This result was reported by van Vleck in 1924 but the calculations were never published. It is also shown that nonrelativistic periodic systems with no harmonics are in equilibrium in classical zero-point radiation provided the phase-space distribution is given by $e^{-\frac{2J}{h}}$, where $J$ is the action variable for the system and $h$ is Planck's constant. In the case of a nonrelativistic nonlinear dipole oscillator the action-angle description is connected with earlier work, also leading to the Rayleigh-Jeans law, which did not use the narrow-linewidth approximation. It is pointed out that the use of relativity for the mechanical systems will lead to expressions different from those used in this work, and it is conjectured that the appearance of the Rayleigh-Jeans law as an equilibrium spectrum may reflect the nonrelativistic character of the scattering systems.