Einstein relation (kinetic theory)

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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:

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.

[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

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 ^[1]. 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 define an electrical mobility by multiplying the mechanical mobility μ_p by the charge of the particle q of the charge carriers:

μ_q = q * μ_p

or alternatively formulated:

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

In a semiconductor with an arbitrary density of states the Einstein relation is

Failed to parse (Cannot write to or create math output directory): D = {{\mu_q \, p}\over{q {{d \, p}\over{d \eta}}}}

where η is the chemical potential and p the particle number.

[edit] References