Einstein relation (kinetic theory)

In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works on Brownian motion. The more general form of the equation in the classical case is

D=\mu \,k_{\text{B}}T,

where

$D$ is the diffusion coefficient;
$μ$ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, $μ = v d / F$ ;
$k B$ is the Boltzmann constant;
$T$ is the absolute temperature.

This equation is an early example of a fluctuation-dissipation relation. Note that the equation above describes the classical case and should be modified when quantum effects are relevant.

Two frequently used important special forms of the relation are:

Einstein–Smoluchowski equation, for diffusion of charged particles: $D={\frac {\mu _{q}\,k_{\text{B}}T}{q}}$
Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number: $D={\frac {k_{\text{B}}T}{6\pi \,\eta \,r}}$

Here

$q$ is the electrical charge of a particle;
$μ q$ is the electrical mobility of the charged particle;
$η$ is the dynamic viscosity;
$r$ is the radius of the spherical particle.

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