- 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 ::$D\; =\; \{mu\_p\; ,\; k\_B\; T\}$

linking "D", the diffusion constant, and "μ

_{p}", the mobility of the particles; where "$k\_B$" isBoltzmann's constant , and "T" is theabsolute temperature .The mobility "μ

_{p}" is the ratio of the particle's terminal drift velocity to an applied force, "μ_{p}= v_{d}/ 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:$gamma\; =\; 6\; pi\; ,\; eta\; ,\; r,$

where "η" is the

viscosity of the medium. Thus the Einstein relation becomes:$D=frac\{k\_B\; T\}\{6pi,eta,r\}$

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 theDiffusion 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 10^{3}kg m^{-3}.**Electrical conduction**When applied to

electrical conduction , it is normal to define an electrical mobility by multiplying the mechanical mobility $mu\_p$ by the charge of the particle "q" of the charge carriers::$mu\_q\; =\; q*\{mu\_p\}$

or alternatively formulated:

:$mu\_q\; =$v_d}over{E

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

:$D\; =$mu_q , k_B T}over{q

In a

semiconductor with an arbitrarydensity of states the Einstein relation is:$D\; =$mu_q , p}over{q d , p}over{d eta

where $eta$ 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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