$D\; =\; \{mu\_p\; ,\; k\_B\; T\}$

linking D, the Diffusion constant, and μ_p, the mobility of the particles; where $k\_B$ is Boltzmann's constant, and T is the absolute temperature.

The mobility μ_p is the ratio of the particle's terminal drift velocity to an applied force, μ_p = v_d / F.

This equation is an early example of a fluctuation-dissipation relation.

Diffusion of particles

$gamma\; =\; 6\; pi\; ,\; eta\; ,\; r,$

where η is the viscosity of the medium. Thus the Einstein relation becomes

$D=frac\{k\_B\; T\}\{6pi,eta,r\}$

This equation is also known as the Stokes-Einstein Relation. We can use this to estimate the Diffusion coefficient of a globular protein in aqueous solution: For a 100 kDalton protein, we obtain D ~10^-10 m² s^-1, assuming a "standard" protein density of ~1.2 10³ kg m^-3.

Electrical conduction

$mu\_q\; =\; q*\{mu\_p\}$

or alternatively formulated:

$mu\_q\; =\; \{\{v\_d\}over\{E\}\}$

where E is the applied electric field; so the Einstein relation becomes

$D\; =\; \{\{mu\_q\; ,\; k\_B\; T\}over\{q\}\}$

In a semiconductor with an arbitrary density of states the Einstein relation is

$D\; =\; \{\{mu\_q\; ,\; p\}over\{q\; \{\{d\; ,\; p\}over\{d\; eta\}\}\}\}$

where $eta$ is the chemical potential and p the particle number