Inphysics (specifically, thekinetic theory of gases), theEinstein relation is a previously unexpected^{[clarification needed]} connection revealed independently byWilliam Sutherland in 1904,^[1][2][3]Albert Einstein in 1905,^[4] and byMarian Smoluchowski in 1906^[5] in their works onBrownian motion. The more general form of the equation in the classical case is^[6]

$${\displaystyle D=\mu k_{\text{B}}T,}$$where

• D is thediffusion coefficient;
• μ is the "mobility", or the ratio of the particle'sterminaldrift velocity to an appliedforce,μ =v_d/F;
• k_B is theBoltzmann constant;
• T is theabsolute temperature.

This equation is an early example of afluctuation-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 ofcharged particles:^[8]$${\displaystyle D=\frac{{\mu }_q{k}_{\text{B}}T}{q}}$$
• Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with lowReynolds number:$${\displaystyle D=\frac{k_{\text{B}}T}{6\pi \eta r}}$$

Here

• q is theelectrical charge of a particle;
• μ_q is theelectrical mobility of the charged particle;
• η is the dynamicviscosity;
• r is theStokes radius of the spherical particle.

For a particle withelectrical chargeq, itselectrical mobilityμ_q is related to its generalized mobilityμ by the equationμ =μ_q/q. The parameterμ_q is the ratio of the particle's terminaldrift velocity to an appliedelectric 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 (${\displaystyle {\mathrm{m}}^2{\mathrm{s}}^{-1}}$).
• ${\displaystyle {\mu }_q}$ is theelectrical mobility (${\displaystyle {\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}}$).
• ${\displaystyle q}$ is theelectric charge of particle (C, coulombs)
• ${\displaystyle T}$ is the electron temperature or ion temperature in plasma (K).^[9]

If the temperature is given involts, which is more common for plasma:$${\displaystyle D=\frac{{\mu }_{q}T}{Z},}$$where

• ${\displaystyle Z}$ is thecharge number of particle (unitless)
• ${\displaystyle T}$ is electron temperature or ion temperature in plasma (V).

For the case ofFermi gas or aFermi liquid, relevant for the electron mobility in normal metals like in thefree electron model, Einstein relation should be modified:$${\displaystyle D=\frac{{\mu }_q{E}_{\mathrm{F}}}{q},}$$where${\displaystyle E_{\mathrm{F}}}$ isFermi energy.

In the limit of lowReynolds number, the mobilityμ 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 radiusr,Stokes' law gives$${\displaystyle \zeta =6\pi \eta r,}$$where${\displaystyle \eta }$ is theviscosity of the medium. Thus the Einstein–Smoluchowski relation results into the Stokes–Einstein–Sutherland 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 theLennard-Jones system.^[10]

In the case ofrotational 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 asemiconductor with an arbitrarydensity of states, i.e. a relation of the form${\displaystyle p=p(\phi )}$ between the density of holes or electrons${\displaystyle p}$ and the correspondingquasi Fermi level (orelectrochemical potential)${\displaystyle \phi }$, the Einstein relation is^[11][12]$${\displaystyle D=\frac{{\mu }_{q}p}{q\frac{dp}{d\phi }},}$$where${\displaystyle {\mu }_q}$ is theelectrical mobility (see§ Proof of the general case for a proof of this relation). An example assuming aparabolic dispersion relation for the density of states and theMaxwell–Boltzmann statistics, which is often used to describeinorganicsemiconductor materials, one can compute (seedensity of states):$${\displaystyle p(\phi )=N_0{e}^{\frac{q\phi }{k_{\text{B}}T}},}$$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 theequivalent conductivity of an electrolyte the Nernst–Einstein equation is derived:$${\displaystyle {\mathrm{\Lambda }}_e=\frac{z_i^2{F}^2}{RT}(D_{+}+D_{-}).}$$wereR is thegas constant.

The proof of the Einstein relation can be found in many references, for example see the work ofRyogo Kubo.^[13] The following derives it from the steady state of theconvection-diffusion equation with a velocity proportional to a conservative force.

Suppose some fixed, externalpotential energy${\displaystyle U}$ generates aconservative force${\displaystyle F(\mathbf{x})=-\mathrm{\nabla }U(\mathbf{x})}$ (for example, an electric force) on a particle located at a given position${\displaystyle \mathbf{x}}$. We assume that the particle would respond by moving with velocity${\displaystyle v(\mathbf{x})=\mu (\mathbf{x})F(\mathbf{x})}$ (seeDrag (physics)). Now assume that there are a large number of such particles, with local concentration${\displaystyle \rho (\mathbf{x})}$ as a function of the position. After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy${\displaystyle U}$, but still will be spread out to some extent because ofdiffusion. At equilibrium, there is no net flow of particles: the tendency of particles to get pulled towards lower${\displaystyle U}$, called thedrift current, perfectly balances the tendency of particles to spread out due to diffusion, called thediffusion current.

The net flux of particles due to the drift current is$${\displaystyle {\mathbf{J}}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}(\mathbf{x})=\mu (\mathbf{x})F(\mathbf{x})\rho (\mathbf{x})=-\rho (\mathbf{x})\mu (\mathbf{x})\mathrm{\nabla }U(\mathbf{x}),}$$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, byFick's law,$${\displaystyle {\mathbf{J}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}(\mathbf{x})=-D(\mathbf{x})\mathrm{\nabla }\rho (\mathbf{x}),}$$where the minus sign means that particles flow from higher to lower concentration.

Now consider the equilibrium condition. First, there is no net flow, i.e.${\displaystyle {\mathbf{J}}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}+{\mathbf{J}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}=0}$. Second, for non-interacting point particles, the equilibrium density${\displaystyle \rho }$ is solely a function of the local potential energy${\displaystyle U}$, i.e. if two locations have the same${\displaystyle U}$ then they will also have the same${\displaystyle \rho }$ (e.g. seeMaxwell-Boltzmann statistics as discussed below.) That means, applying thechain rule,$${\displaystyle \mathrm{\nabla }\rho =\frac{\mathrm{d}\rho }{\mathrm{d}U}\mathrm{\nabla }U.}$$

Therefore, at equilibrium:$${\displaystyle 0={\mathbf{J}}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}+{\mathbf{J}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}=-\mu \rho \mathrm{\nabla }U-D\mathrm{\nabla }\rho =\left(-\mu \rho -D\frac{\mathrm{d}\rho }{\mathrm{d}U}\right)\mathrm{\nabla }U.}$$

As this expression holds at every position${\displaystyle \mathbf{x}}$, it implies the general form of the Einstein relation:$${\displaystyle D=-\mu \frac{\rho }{\frac{\mathrm{d}\rho }{\mathrm{d}U}}.}$$

The relation between${\displaystyle \rho }$ and${\displaystyle U}$ forclassical particles can be modeled throughMaxwell-Boltzmann statistics$${\displaystyle \rho (\mathbf{x})=A{e}^{-\frac{U(\mathbf{x})}{k_{\text{B}}T}},}$$where${\displaystyle A}$ is a constant related to the total number of particles. Therefore$${\displaystyle \frac{\mathrm{d}\rho }{\mathrm{d}U}=-\frac{1}{k_{\text{B}}T}\rho .}$$

Under this assumption, plugging this equation into the general Einstein relation gives:$${\displaystyle D=-\mu \frac{\rho }{\frac{\mathrm{d}\rho }{\mathrm{d}U}}=\mu k_{\text{B}}T,}$$which corresponds to the classical Einstein relation.

• Smoluchowski factor
• Conductivity (electrolytic)
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