In physics, in kinetic theory the Einstein relation is a previously unexpected connection revealed by Einstein in his 1905 paper on Brownian motion:

$D\; =\; \{\backslash mu\; \backslash ,\; kT\}$

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

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

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

Diffusion of particles

In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient γ. For spherical particles of radius r, Stokes law gives

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

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

$D=\backslash frac\{kT\}\{6\backslash pi\backslash ,\backslash eta\backslash ,r\}$

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

When applied to electrical conduction, it is normal to divide through by the charge q of the charge carriers, defining electron mobility (or hole mobility)

$\backslash mu\; =$

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

$D\; =$

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

$D\; =\; \}\}$

where $\backslash eta$ is the chemical potential and p the particle number Statistical mechanics | Albert Einstein