Einstein relation (kinetic theory)

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

$${\displaystyle D={\frac {\mu _{q}\,k_{\text{B}}T}{q}},}$$

where

• ${\displaystyle D}$ is the diffusion coefficient (${\displaystyle \mathrm {m^{2}s^{-1}} }$).
• ${\displaystyle \mu _{q}}$ is the electrical mobility (${\displaystyle \mathrm {m^{2}V^{-1}s^{-1}} }$).
• ${\displaystyle q}$ is the electric charge of particle (C, coulombs)
• ${\displaystyle T}$ is the electron temperature or ion temperature in plasma (K).[9]

If the temperature is given in Volt, which is more common for plasma:

$${\displaystyle D={\frac {\mu _{q}\,T}{Z}},}$$

where

• ${\displaystyle Z}$ is the Charge number of particle (unitless)
• ${\displaystyle T}$ 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:

$${\displaystyle D={\frac {\mu _{q}\,E_{F}}{q}},}$$

where ${\displaystyle E_{F}}$ is Fermi energy.

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

$${\displaystyle \zeta =6\pi \,\eta \,r,}$$

where ${\displaystyle \eta }$ is the viscosity of the medium. Thus the Einstein–Smoluchowski relation results into the Stokes–Einstein relation

$${\displaystyle D={\frac {k_{\text{B}}T}{6\pi \,\eta \,r}}.}$$

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.[10]

In the case of rotational diffusion, the friction is ${\displaystyle \zeta _{\text{r}}=8\pi \eta r^{3}}$, and the rotational diffusion constant ${\displaystyle D_{\text{r}}}$ is

$${\displaystyle D_{\text{r}}={\frac {k_{\text{B}}T}{8\pi \,\eta \,r^{3}}}.}$$

This is sometimes referred to as the Stokes–Einstein–Debye relation.

In a semiconductor with an arbitrary density of states, i.e. a relation of the form ${\displaystyle p=p(\varphi )}$ between the density of holes or electrons ${\displaystyle p}$ and the corresponding quasi Fermi level (or electrochemical potential) ${\displaystyle \varphi }$, the Einstein relation is[11][12]

$${\displaystyle D={\frac {\mu _{q}p}{q{\frac {dp}{d\varphi }}}},}$$

where ${\displaystyle \mu _{q}}$ is the electrical mobility (see § Proof of the general case 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):

$${\displaystyle p(\varphi )=N_{0}e^{\frac {q\varphi }{k_{\text{B}}T}},}$$

where ${\displaystyle N_{0}}$ is the total density of available energy states, which gives the simplified relation:

$${\displaystyle D=\mu _{q}{\frac {k_{\text{B}}T}{q}}.}$$

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:

$${\displaystyle \Lambda _{e}={\frac {z_{i}^{2}F^{2}}{RT}}(D_{+}+D_{-}).}$$