- Einstein relation (kinetic theory)
In
physics (namely, inkinetic theory ) the Einstein relation (also known as Einstein–Smoluchowski relation) is a previously unexpected connection revealed independently byAlbert Einstein in 1905 and byMarian Smoluchowski (1906) in their papers onBrownian motion ::
linking "D", the diffusion constant, and "μp", the mobility of the particles; where "" is
Boltzmann's constant , and "T" is theabsolute 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' law gives:
where "η" is the
viscosity of 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 coefficient of aglobular protein in 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.Electrical conduction
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
:
In a
semiconductor with an arbitrarydensity of states the Einstein relation is:
where is the
chemical potential and p the particle number.References
*"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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