Diffusion: A Relativistic Appraisal

The classic equation describing diffusion, $${\mathrm{\delta n/\delta t\hspace{0.17em}=\hspace{0.17em}D\nabla }}^{2}n$$ , 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, $$v^{\text{−1}}\left({\delta }^2{\mathrm{n/\delta t}}^2\right)\mathrm{\hspace{0.17em}+\hspace{0.17em}}\left(\mathrm{\delta n/\delta t}\right){\mathrm{\hspace{0.17em}=\hspace{0.17em}D\nabla }}^{2}n$$ , 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.