Equation in Brownian motion
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]
D
=
μ
k
B
T
,
{\displaystyle D=\mu \,k_{\text{B}}T,}
where
This equation is an early example of a fluctuation-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 of charged particles:[ 8]
D
=
μ
q
k
B
T
q
{\displaystyle D={\frac {\mu _{q}\,k_{\text{B}}T}{q}}}
Stokes–Einstein–Sutherland equation , for diffusion of spherical particles through a liquid with low Reynolds number :
D
=
k
B
T
6
π
η
r
{\displaystyle D={\frac {k_{\text{B}}T}{6\pi \,\eta \,r}}}
Here
^ World Year of Physics – William Sutherland at the University of Melbourne . Essay by Prof. R Home (with contributions from Prof B. McKellar and A./Prof D. Jamieson) dated 2005. Accessed 2017-04-28.
^ Sutherland William (1905). "LXXV. A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin" . Philosophical Magazine . Series 6. 9 (54): 781–785. doi :10.1080/14786440509463331 .
^ P. Hänggi, "Stokes–Einstein–Sutherland equation" .
^ Einstein, A. (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" . Annalen der Physik (in German). 322 (8): 549–560. Bibcode :1905AnP...322..549E . doi :10.1002/andp.19053220806 .
^ von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen" . Annalen der Physik (in German). 326 (14): 756–780. Bibcode :1906AnP...326..756V . doi :10.1002/andp.19063261405 .
^ Dill, Ken A.; Bromberg, Sarina (2003). Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology . Garland Science. p. 327. ISBN 9780815320517 .
^ Umberto Marini Bettolo Marconi, Andrea Puglisi, Lamberto Rondoni, Angelo Vulpiani, "Fluctuation-Dissipation: Response Theory in Statistical Physics" .
^ Van Zeghbroeck, "Principles of Semiconductor Devices", Chapter 2.7 Archived 2021-05-06 at the Wayback Machine .