# Einstein relation (kinetic theory)

The Einstein relation is a statement of kinetic theory and equilibrium statistical mechanics connecting the diffusion coefficient \( D \) of a particle or charge carrier to its mobility \( \mu \), the drift velocity produced per unit applied force: \( D = \mu k_{\mathrm{B}}T \), where \( k_{\mathrm{B}} \) is Boltzmann's constant and \( T \) the absolute temperature<sup>[1](https://www.feynmanlectures.caltech.edu/I_43.html)</sup>. It links a fluctuation property of the system, how far random thermal motion spreads a particle, to a dissipation property, how strongly the system resists being driven. It is regarded as the first form of the fluctuation-dissipation theorem based on a microscopic understanding of thermal motion<sup>[2](https://terpconnect.umd.edu/~cjarzyns/CHEM-CHPH-PHYS_703_Spr_20/resources/Rep%20Prog%20Phys%202012%20Seifert.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Relation | \( D = \mu k_{\mathrm{B}}T \); the mobility \( \mu \) is the reciprocal of the friction coefficient \( \xi \), defined by terminal drift velocity \( v_d = F_{\mathrm{ext}}/\xi \)<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup> |
| Stokes–Einstein–Sutherland form | For a sphere of radius \( a \) in a fluid of viscosity \( \eta \): \( D = k_{\mathrm{B}}T/(6\pi\eta a) \) with no slip at the surface, \( D = k_{\mathrm{B}}T/(4\pi\eta a) \) with perfect slip<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup> |
| Origin | Derived by Albert Einstein in his 1905 Brownian motion paper (received 11 May 1905), independently by William Sutherland (June 1905), and by Marian von Smoluchowski (1906)<sup>[4](https://link.springer.com/article/10.1007/s00407-024-00337-1)</sup> |
| Experimental role | Jean Baptiste Perrin used the relation to determine Avogadro's number, \( N = 6.022 \times 10^{23}\ \mathrm{mol^{-1}} \), within 6.3% error, settling the dispute over the atomic nature of matter<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup> |
| Validity | Holds for systems in thermal equilibrium with a perturbation applied in the linear response regime; it is violated in nonequilibrium steady states<sup>[5](https://pmc.ncbi.nlm.nih.gov/articles/PMC9857877/)</sup><sup> • </sup><sup>[6](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.71.046143)</sup> |
| Semiconductor form | For electrons in a semiconductor, \( \mu_n = qD_n/(kT) \), widely used in device analysis and design<sup>[7](https://www.ee.ucla.edu/wp-content/uploads/ee/Volume-2_Basic-Semiconductor-Devices-for-Electrical-Engineers.pdf)</sup> |

## Derivation

The standard derivation balances two opposing particle currents in a dilute suspension. Suppose a force field acts on suspended particles, giving each particle a drift velocity \( v_{\mathrm{drift}} = \mu F \), where \( \mu \) is the mobility. The drift current is \( J_{\mathrm{drift}} = n_a \mu F \), with \( n_a \) the number density. At the same time the particles diffuse, giving a diffusion current \( J_x = -D\, dn_a/dx \). The force can be adjusted so that the drift current exactly cancels the diffusion current and there is no net flow<sup>[1](https://www.feynmanlectures.caltech.edu/I_43.html)</sup>.

Because this adjusted state is an equilibrium state, the density must follow the [Boltzmann distribution](https://www.edgepedia.org/boltzmann-distribution) of statistical mechanics, \( n_a = n_0 e^{-U/kT} \), where \( U \) is the potential energy. Differentiating gives \( dn_a/dx = n_a F/(kT) \) for a force in the x-direction. Comparing this with the balance condition \( D\, dn_a/dx = n_a \mu F \) yields \( D = \mu kT \) exactly, with no correction factors<sup>[1](https://www.feynmanlectures.caltech.edu/I_43.html)</sup>. The argument uses only the equilibrium distribution and the definitions of \( D \) and \( \mu \), so the result holds in complicated situations, such as a suspension in a liquid, where a kinetic-theory calculation of the mean free path would not apply<sup>[1](https://www.feynmanlectures.caltech.edu/I_43.html)</sup>.

Einstein's own 1905 derivation proceeded differently. He computed the diffusion coefficient from osmotic pressure considerations, \( D = RT/(N \cdot 6\pi k P) \) in his notation, for spherical particles of radius \( P \) in a liquid of viscosity \( k \), and combined it with the mean-square displacement \( \langle \Delta x^2 \rangle = 2Dt \) that follows from the diffusion equation<sup>[8](https://sites.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/Einstein_1905_docs/Einstein_Brownian_English.pdf)</sup>.

## The Stokes–Einstein–Sutherland formula

Combining the Einstein relation with [Stokes' law](https://www.edgepedia.org/stokes-law) for the mobility of a sphere in an incompressible fluid gives the Stokes–Einstein–[Sutherland](https://www.edgepedia.org/sutherland) formula. With the no-slip boundary condition adopted by George Gabriel Stokes in 1851, the mobility is \( \mu = 1/(6\pi\eta a) \), so

\[ D = \frac{k_{\mathrm{B}}T}{6\pi\eta a}. \]

For a perfect-slip interface the denominator becomes \( 4\pi\eta a \), and a slip length \( b = \eta/\alpha \) interpolates between the two cases, where \( \alpha \) is the surface friction coefficient<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>.

This formula connects the mass transport of the particle with the momentum transport of the fluid, so one unknown quantity can be obtained from the others. Measuring \( D \) from the mean-square displacement, with known \( k_{\mathrm{B}}T \) and \( \eta \), determines the particle radius \( a \); conversely, a known radius allows determination of Avogadro's number through \( k_{\mathrm{B}} = R_g/N_A \)<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>. In dynamic light scattering, the Stokes–Einstein relationship is the standard route from a measured diffusion coefficient to a particle size<sup>[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC5425802/)</sup>.

## Place in fluctuation-dissipation theory

The Einstein relation is the simplest instance of the fluctuation-dissipation theorem (FDT), the general statement that the amplitude of spontaneous thermal fluctuations is fixed by the strength of the system's dissipative response. In Langevin's 1908-style model of a Brownian particle with inertia, the random force must satisfy \( \langle \tilde{F}(t)\tilde{F}(t') \rangle = \Gamma \delta(t-t') \) with a strength tied to the friction \( \xi \), so that temperature is well defined; the friction and random forces together act as a thermostat<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>.

The time-integral form makes the connection explicit. For [Langevin dynamics](https://www.edgepedia.org/langevin-dynamics) the velocity autocorrelation function decays exponentially, and

\[ \int_0^{\infty} \langle v(0)v(t) \rangle\, dt = \frac{k_{\mathrm{B}}T}{\xi} = D, \]

which is the simplest example of the Green–Kubo relations, connecting macroscopic transport coefficients to correlation functions of microscopic fluctuations<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>. In the overdamped limit \( m/\xi \to 0 \), Langevin's dynamics reduces to Einstein's pure-diffusion model<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>. Other instances of the same theorem include [Johnson–Nyquist noise](https://www.edgepedia.org/johnson-nyquist-noise), the finite current fluctuations of a shorted resistor, \( S_I^{(\omega)} = 2k_{\mathrm{B}}T/(R\pi) \), where the inverse resistance plays the role of the linear response coefficient<sup>[10](https://cfaed.tu-dresden.de/files/user/bfriedrich/teaching/Stochastic%20processes/Lectures_Notes_Stochastic_Processes.pdf)</sup>. Later work generalized the theorem: Callen and Welton gave a quantum formulation, and [Ryogo Kubo](https://www.edgepedia.org/ryogo-kubo) extended it to generalized Langevin processes, culminating in linear response theory<sup>[11](https://onlinelibrary.wiley.com/doi/full/10.1002/andp.202500062)</sup><sup> • </sup><sup>[12](https://www.sciencedirect.com/science/article/abs/pii/S0370157308000768)</sup>.

## History and priority

Einstein's paper "On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat" was received on 11 May 1905<sup>[8](https://sites.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/Einstein_1905_docs/Einstein_Brownian_English.pdf)</sup>. The Scottish-born Australian physicist and physical chemist William Sutherland published an essentially equivalent diffusion-viscosity relation in June 1905 in the *Philosophical Magazine*, derived by the same argument of balancing an osmotic force against Stokes drag, but motivated by determining the masses of large molecules such as albumin rather than by [Brownian motion](https://www.edgepedia.org/brownian-motion); some historians argue the relation should be called the Sutherland–Einstein relation<sup>[4](https://link.springer.com/article/10.1007/s00407-024-00337-1)</sup>. Marian von Smoluchowski, who had worked on the problem since 1900, published his own kinetic derivation in 1906; his expression for the root-mean-square displacement carried an extra factor \( \sqrt{64/27} \approx 1.54 \) from his approximations, and he confirmed Einstein's version in 1912<sup>[4](https://link.springer.com/article/10.1007/s00407-024-00337-1)</sup>.

The experimental validation of the theory by [Jean Baptiste Perrin](https://www.edgepedia.org/jean-baptiste-perrin), using measurements of Brownian motion to determine Avogadro's number, unambiguously verified the atomic nature of matter and was recognized with the 1926 [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics)<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)</sup>.

## Beyond equilibrium

The relation is an equilibrium statement, and it fails when the system is driven. In a one-dimensional driven lattice gas studied by numerical experiment, the equilibrium form \( D\chi = \sigma T \), connecting the diffusion constant \( D \), the fluctuation intensity \( \chi \), the conductivity \( \sigma \), and the environment temperature \( T \), does not hold in the nonequilibrium steady state; replacing \( T \) with the real part of a complex effective temperature \( \Theta - i\Phi \) restores a relation \( D\chi = \sigma\Theta \)<sup>[6](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.71.046143)</sup>. For a tracer in an active bath of self-propelling particles, the relation \( D = \mu k_{\mathrm{B}}T \) cannot be expected to hold a priori because the medium is out of equilibrium<sup>[13](https://arxiv.org/abs/2203.05459)</sup>. For interacting Brownian particles under shear, the Einstein relation holds only for the short-time in-cage motion and is violated at long times<sup>[14](https://arxiv.org/abs/0911.1632)</sup>.

Generalized versions extend the relation's reach. A generalized Einstein relation holds for a dissipative quantum particle in a washboard potential, including sub-Ohmic dissipation where the motion is subdiffusive<sup>[15](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.46.10743)</sup>, and analogous relations have been derived for the mutual diffusion coefficient of binary fluid mixtures and for nonlinear nonequilibrium thermodynamic states<sup>[16](https://arxiv.org/abs/1707.00423)</sup><sup> • </sup><sup>[17](https://www.sciencedirect.com/science/article/abs/pii/0378437195001646)</sup>.

## Applications

In semiconductor physics the relation takes the form \( \mu_n = qD_n/(kT) \) for electrons; since \( kT/q \) has the dimension of a volt, the rule of thumb is that mobility times \( kT/q \) gives the diffusion constant. The relation is frequently used in semiconductor device analysis and design<sup>[7](https://www.ee.ucla.edu/wp-content/uploads/ee/Volume-2_Basic-Semiconductor-Devices-for-Electrical-Engineers.pdf)</sup><sup> • </sup><sup>[18](https://www.sciencedirect.com/science/article/pii/003811017390110X/pdf?md5=472095d3fa1609524ef1893b83df14d5&pid=1-s2.0-003811017390110X-main.pdf)</sup>. In soft matter, the Brownian motion of colloidal particles and polymer molecules, quantified through the diffusion coefficient, determines the dynamical response of the material, and the fluctuation-dissipation theorem built on the Einstein relation underlies the Onsager principle used to derive macroscopic time-evolution equations<sup>[19](https://academic.oup.com/book/32680/chapter-abstract/270665255)</sup>.

## References

1. [The Feynman Lectures on Physics, Vol. I, Ch. 43: Diffusion](https://www.feynmanlectures.caltech.edu/I_43.html)
2. [U. Seifert (2012). Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on Progress in Physics.](https://terpconnect.umd.edu/~cjarzyns/CHEM-CHPH-PHYS_703_Spr_20/resources/Rep%20Prog%20Phys%202012%20Seifert.pdf)
3. [Li et al. (2016). 111 years of Brownian motion. PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC5476231/)
4. [Einstein–Perrin dilemma on the Brownian motion (Avogadro's number) resolved? Archive for History of Exact Sciences.](https://link.springer.com/article/10.1007/s00407-024-00337-1)
5. [Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond. PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC9857877/)
6. [Extended Einstein relations with a complex effective temperature in a one-dimensional driven lattice gas. Phys. Rev. E 71, 046143 (2005).](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.71.046143)
7. [Basic Semiconductor Devices for Electrical Engineers, UCLA, Section 2.2.7 Einstein Relation.](https://www.ee.ucla.edu/wp-content/uploads/ee/Volume-2_Basic-Semiconductor-Devices-for-Electrical-Engineers.pdf)
8. [A. Einstein (1905, trans.). Investigations on the Theory of Brownian Movement, English translation.](https://sites.pitt.edu/~jdnorton/lectures/Rotman_Summer_School_2013/Einstein_1905_docs/Einstein_Brownian_English.pdf)
9. [Dynamic light scattering: a practical guide and applications in biomedical sciences. PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC5425802/)
10. [Lecture notes on Stochastic Processes, Ch. 7: Fluctuation-Dissipation Theorem, TU Dresden.](https://cfaed.tu-dresden.de/files/user/bfriedrich/teaching/Stochastic%20processes/Lectures_Notes_Stochastic_Processes.pdf)
11. [Giona et al. (2025). Stochastic Hydrodynamic Velocity Field and the Representation of Langevin Equations. Annalen der Physik.](https://onlinelibrary.wiley.com/doi/full/10.1002/andp.202500062)
12. [Crisanti, Ritort (2008). Fluctuation–dissipation: Response theory in statistical physics. Physics Reports 461, 111–195.](https://www.sciencedirect.com/science/article/abs/pii/S0370157308000768)
13. [On the Einstein relation between mobility and diffusion coefficient in an active bath. arXiv:2203.05459.](https://arxiv.org/abs/2203.05459)
14. [Non-Equilibrium relation between mobility and diffusivity of interacting Brownian particles under shear. arXiv:0911.1632.](https://arxiv.org/abs/0911.1632)
15. [Quantum particle in a washboard potential. I. Linear mobility and the Einstein relation. Phys. Rev. B 46, 10743 (1992).](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.46.10743)
16. [Generalized Einstein relation for the mutual diffusion coefficient of a binary fluid mixture. arXiv:1707.00423.](https://arxiv.org/abs/1707.00423)
17. [Diffusion and mobility and generalized Einstein relation. Physica A 221, 495–510 (1995).](https://www.sciencedirect.com/science/article/abs/pii/0378437195001646)
18. [A generalized Einstein relation for semiconductors. Solid-State Electronics.](https://www.sciencedirect.com/science/article/pii/003811017390110X/pdf?md5=472095d3fa1609524ef1893b83df14d5&pid=1-s2.0-003811017390110X-main.pdf)
19. [Brownian motion and thermal fluctuations. Soft Matter Physics, Oxford Academic.](https://academic.oup.com/book/32680/chapter-abstract/270665255)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Statistical mechanics and kinetic theory › Fluctuations, Brownian motion, and noise*

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