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' law giveswhere 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 . 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" protein density 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 arbitrary density 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,
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Last updated on Sunday October 05, 2008 at 18:13:06 PDT (GMT -0700)
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