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Wikipedia / 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

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–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number

Here

Special cases

Electrical mobility equation (classical case)

For a particle with electrical charge , its electrical mobility is related to its generalized mobility by the equation . The parameter is the ratio of the particle's terminal drift velocity to an applied electric field

where

If the temperature is given in volts where

  • 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 is Fermi energy.

Stokes–Einstein–Sutherland equation

In the limit of low Reynolds number, the mobility μ 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 is the viscosity 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 the case of rotational diffusion This is sometimes referred to as the Stokes–Einstein–Debye relation.

Semiconductor

In a semiconductor with an arbitrary density of states and the corresponding quasi Fermi level (or electrochemical potential, the Einstein relation is is the electrical mobility (see for a proof of this relation). 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

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 conductivitywere R is the gas constant.

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. The following derives it from the steady state of the convection-diffusion equation with a velocity proportional to a conservative force.

Suppose some fixed, external potential energy generates a conservative force (see Drag (physics), but still will be spread out to some extent because of diffusion, called the drift current, perfectly balances the tendency of particles to spread out due to diffusion, called the diffusion current.

i.e., the number of particles flowing past a given position equals the particle concentration times the average velocity.

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.

(e.g. see Maxwell-Boltzmann statistics as discussed below.) That means, applying the chain rule

for classical particles can be modeled through Maxwell-Boltzmann statistics

which corresponds to the classical Einstein relation.

See also

References