Einstein relation (kinetic theory)

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

[ D = ]

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

[ \gamma = 6 \pi \, \eta \, r, ]

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

[ D=\frac ]

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 = ]

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 [ \eta ] is the chemical potential and p the particle number

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