Diffusion: A Relativistic Appraisal

Abstract

The classic equation describing diffusion, δn/δt = D∇2n, is not consistent with the principles of special relativity. Specifically, its solutions permit the diffusive flow of particles to proceed at speeds in excess of the speed of light. A careful derivation, proceeding from the covariant form of the Boltzmann equation, results in a diffusion equation which exhibits the correct causal behavior. This equation assumes a particularly simple form, v−1(δ2n/δt2) + (δn/δt) = D∇2n, when the collision frequency for momentum transfer, v, is independent of particle speed. The general solution for diffusion in one dimension is presented, revealing the causal character of the diffusive flow.