Einstein relation for diffusion

Remarks

The Fokker–Planck equation of the process is $\partial_t p = \partial_x\big(\mu V' p + D\, \partial_x p\big)$. A stationary density with zero probability current solves $\mu V' p + D p' = 0$, so $p \propto e^{-\mu V/D}$. Comparing the exponent with $-V/(k_B T)$ for a non-constant $V$ forces $\mu/D = 1/(k_B T)$. For a sphere of radius $r$ in a fluid of viscosity $\eta$, Stokes drag gives $\mu = 1/(6\pi\eta r)$, and the relation becomes the Stokes–Einstein formula $D = k_B T/(6\pi\eta r)$. For a charged particle of charge $q$, the electrical mobility $q\mu$ gives the form $D = \mu_q k_B T / q$ with $\mu_q = q\mu$.

Going deeper

For a student

The relation is the simplest instance of the fluctuation–dissipation principle: the friction that damps motion and the noise that drives it come from the same collisions. In the Langevin model it is exactly the condition under which the equilibrium state is the Boltzmann distribution. Einstein used it in 1905 to connect Brownian motion with the size of molecules, and Perrin's measurements of $D$ then gave a value of Avogadro's number. The Green–Kubo formula generalises the diffusion coefficient side, and linear response theory generalises the mobility side.