Einstein relation (kinetic theory)

In physics (namely, in kinetic theory) the Einstein relation (also known as Einstein–Smoluchowski relation) is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski (1906) in their papers on Brownian motion:

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

linking "D", the diffusion constant, and "μ_p", the mobility of the particles; where "$k\_B$" is Boltzmann's constant, and "T" is the absolute 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 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" 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 $mu\_p$ 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 arbitrary density 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]