Einstein relation (kinetic theory)

In physics (specifically, in kinetic theory) the Einstein relation (also known as Einstein–Smoluchowski relation) is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their papers on Brownian motion. The more general form of the equation is

where

This equation is an early example of a fluctuation-dissipation relation.

Two frequently used important special forms of the relation are:

where

For a particle with electrical charge q, its electrical mobility μ_q is related to its generalized mobility μ by the equation μ=μ_q/q. The parameter μ_q is the ratio of the particle's terminal drift velocity to an applied electric field. Hence, the equation in the case of a charged particle is given as

In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient ${\displaystyle \zeta }$. A damping constant ${\displaystyle \gamma =\zeta /m}$ is frequently used for the inverse momentum relaxation time (time needed for the inertia momentum to become negligible compared to the random momenta) of the diffusive object. For spherical particles of radius r, Stokes' law gives

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