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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]
- 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.
Special cases
Electrical mobility equation (classical case)
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).[9]
If the temperature is given in volts, which is more common for plasma:
- is the charge number of particle (unitless)
- is electron temperature or ion temperature in plasma (V).
Electrical mobility equation (quantum case)
For the case of Fermi gas or a Fermi liquid, relevant for the electron mobility in normal metals like in the free electron model, Einstein relation should be modified:
Stokes–Einstein–Sutherland equation
In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient . A damping constant 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
In the case of rotational diffusion, the friction is , and the rotational diffusion constant is
Semiconductor
In a semiconductor with an arbitrary density of states, i.e. a relation of the form between the density of holes or electrons and the corresponding quasi Fermi level (or electrochemical potential) , the Einstein relation is[11][12]
Nernst–Einstein equation
By replacing the diffusivities in the expressions of electric ionic mobilities of the cations and anions from the expressions of the equivalent conductivity of an electrolyte the Nernst–Einstein equation is derived:
Proof of the general case
The proof of the Einstein relation can be found in many references, for example see the work of Ryogo Kubo.[13]
Suppose some fixed, external potential energy generates a conservative force (for example, an electric force) on a particle located at a given position . We assume that the particle would respond by moving with velocity
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