Einstein relation

The Einstein relation connects the diffusion coefficient of a particle to the drag it feels when pushed: D = k_BT/γ, where γ is the drag coefficient. For a spherical particle of radius r in a fluid of viscosity η, Stokes's law gives γ = 6πηr, yielding the Einstein–Stokes form D = k_BT/(6πηr). It states that the same molecular collisions that randomly kick a particle also resist its steady motion.

The relation is the original fluctuation-dissipation theorem: the strength of thermal fluctuations (diffusion) and the strength of dissipation (drag) are not independent but are locked together by the temperature. One cannot have random thermal jiggling without the accompanying friction, because both arise from the very same impacts of surrounding molecules.

Practically, the Einstein relation turned Brownian motion into a measuring instrument. By observing D and knowing η, r and T, Perrin extracted k_B and hence Avogadro's number. The same relation governs the mobility of ions in solution and charge carriers in semiconductors, where it links diffusion to electrical mobility.

Derived by Albert Einstein in his 1905 paper on Brownian motion, combining the diffusion equation with Stokes's law of viscous drag; generalised by Nyquist, Onsager and Kubo into the modern fluctuation-dissipation theorem.