- Einstein relation (kinetic theory)
physics(namely, in kinetic theory) the Einstein relation (also known as Einstein–Smoluchowski relation) is a previously unexpected connection revealed independently by Albert Einsteinin 1905 and by Marian Smoluchowski(1906) in their papers on Brownian motion:
linking "D", the diffusion constant, and "μp", the mobility of the particles; where "" 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. It is frequently used in the electrodiffusion phenomena.
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' lawgives
where "η" is the
viscosityof the medium. Thus the Einstein relation becomes
This equation is also known as the Stokes–Einstein Relation or Stokes–Einstein–Sutherland equation [http://www.physics.emory.edu/~weeks/lab/papers/sendai2007.pdf] . It can be used to estimate the
Diffusion coefficientof a globular proteinin aqueous solution:For a 100 kDalton protein, we obtain "D" ~10-10 m² s-1, assuming a "standard" proteindensity of ~1.2 103 kg m-3.
When applied to
electrical conduction, it is normal to define an electrical mobility by multiplying the mechanical mobility by the charge of the particle "q" of the charge carriers:
or alternatively formulated:
where "E" is the applied electric field; so the Einstein relation becomes
semiconductorwith an arbitrary density of statesthe Einstein relation is
where is the
chemical potentialand p the particle number.
*"Fluctuation-Dissipation: Response Theory in Statistical Physics" by Umberto Marini Bettolo Marconi, Andrea Puglisi, Lamberto Rondoni, Angelo Vulpiani, [http://arxiv.org/abs/0803.0719]
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