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# Einstein relation (kinetic theory)

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

$D = {\mu_p \, k_B T}$

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

The mobility μp is the ratio of the particle's terminal drift velocity to an applied force, μp = vd / 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

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

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

$D=\frac{k_B T}{6\pi\,\eta\,r}$

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

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)

$\mu = {{v_d}\over{E}}$

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

$D = {{\mu \, k_B T}\over{q}}$

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

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

where η is the chemical potential and p the particle number