Einstein relation (kinetic theory)

In physics (specifically, the kinetic theory of gases), the Einstein 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 works on Brownian motion. The more general form of the equation in the classical case is 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. 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: Stokes–Einstein 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. For a particle with electrical charge q, its electrical mobility μq is related to its generalized mobility μ by the equation μ = μq/q. The parameter μq 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 where is the diffusion coefficient (). is the electrical mobility (). is the electric charge of particle (C, coulombs) is the electron temperature or ion temperature in plasma (K). If the temperature is given in Volt, which is more common for plasma: where is the Charge number of particle (unitless) is electron temperature or ion temperature in plasma (V). For the case of Fermi gas (Fermi liquid), relevant for the electron mobility in normal metals, Einstein relation should be modified: where is Fermi energy. In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient .