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:
- <math> D = {\mu \, kT} </math>
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 = vd / F.
This equation is an early example of a fluctuation-dissipation relation.
[edit] 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
- <math> \gamma = 6 \pi \, \eta \, r, </math>
where η is the viscosity of the medium. Thus the Einstein relation becomes
- <math> D=\frac{kT}{6\pi\,\eta\,r} </math>
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.
[edit] 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)
- <math>\mu = {{v_d}\over{E}}</math>
where E is the applied electric field; so the Einstein relation becomes
- <math> D = {{\mu \, kT}\over{q}}</math>
In a semiconductor with an arbitrary density of states the Einstein relation is
- <math> D = {{\mu \, p}\over{q {{d \, p}\over{d \eta}}}} </math>
where <math> \eta </math> is the chemical potential and p the particle number
