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]
[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: Here 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.
where 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:
In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient
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
This has been applied for many years to estimating the self-diffusion coefficient in liquids, and a version consistent with isomorph theory has been confirmed by computer simulations of the Lennard-Jones system.
In a semiconductor with an arbitrary density of states, i.e. a relation of the form
and the corresponding quasi Fermi level (or electrochemical potential)
An example assuming a parabolic dispersion relation for the density of states and the Maxwell–Boltzmann statistics, which is often used to describe inorganic semiconductor materials, one can compute (see density of states):
is the total density of available energy states, which gives the simplified relation:
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:
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
(for example, an electric force) on a particle located at a given position
We assume that the particle would respond by moving with velocity
Now assume that there are a large number of such particles, with local concentration
After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy
, called the drift current, perfectly balances the tendency of particles to spread out due to diffusion, called the diffusion current (see drift-diffusion equation).
The net flux of particles due to the drift current is
The flow of particles due to the diffusion current is, by Fick's law,
where the minus sign means that particles flow from higher to lower concentration.
Second, for non-interacting point particles, the equilibrium density
is solely a function of the local potential energy
, it implies the general form of the Einstein relation:
for classical particles can be modeled through Maxwell-Boltzmann statistics
Under this assumption, plugging this equation into the general Einstein relation gives: