Einstein relation (kinetic theory)
Equation in Brownian motion / From Wikipedia, the free encyclopedia
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In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected[clarification needed] connection revealed independently by William Sutherland in 1904,[1][2][3] Albert Einstein in 1905,[4] and by Marian Smoluchowski in 1906[5] in their works on Brownian motion. The more general form of the equation in the classical case is[6]
where
- D is the diffusion coefficient;
- μ is the "mobility", or the ratio of the particle's terminal drift velocity to an applied force, μ = vd/F;
- kB is the Boltzmann constant;
- T is the absolute temperature.
This equation is an early example of a fluctuation-dissipation relation.[7] Note that the equation above describes the classical case and should be modified when quantum effects are relevant.
Two frequently used important special forms of the relation are:
- Einstein–Smoluchowski equation, for diffusion of charged particles:[8]
- Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number:
Here
- q is the electrical charge of a particle;
- μq is the electrical mobility of the charged particle;
- η is the dynamic viscosity;
- r is the radius of the spherical particle.