Equilibrium distributions for relativistic free particles in thermal radiation within classical electrodynamics

The equilibrium velocity distribution of classical free particles interacting with random classical radiation is investigated using the model of Einstein and Hopf, and extended to allow relativistic particle velocities. The model considers a massive free particle which has an electric dipole oscillator mounted inside; the oscillator provides the interaction between the particle and the random radiation. In this paper we give the calculations leading to a Fokker-Planck equation for the equilibrium distribution of particle momenta, and evaluate the equation for a number of radiation spectra. We find the following results: (i) If the random classical radiation is the Rayleigh-Jeans law for thermal radiation, then the equilibrium particle distribution follows the Boltzmann distribution only at low temperatures where nonrelativistic particle velocities are involved. The Rayleigh-Jeans law does not lead to the Boltzmann distribution for relativistic free particles. (ii) If the random classical radiation is the zero-point radiation spectrum, then the equilibrium particle distribution is the Lorentz-invariant distribution. (iii) If the random classical radiation is the Planck spectrum with zero-point radiation, then the equilibrium particle distribution goes over asymptotically to the Lorentz-invariant distribution at velocities near the speed of light. Thus no finite number of free particles can form an equilibrium velocity distribution if zero-point radiation is present. The particles will diffuse to velocities ever closer to the speed of light. We conclude that equilibrium for a finite number of particles must involve some explicit mechanism for confinement if classical zero-point radiation is present.