View source for Brownian motion

{{short description|Random motion of particles suspended in a fluid}}
{{About|Brownian motion as a natural phenomenon|the stochastic process|Wiener process|Temperature|Thermodynamic temperature|internal energy|Equipartition theorem|the mobility model|Random walk}}
{{Use dmy dates|date=July 2012}}
[[File:2d random walk ag adatom ag111.gif|thumb|2 dimensional random walk of a silver adatom on a Ag(111) surface <ref>Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments, Jan Philipp Meyburg and Detlef Diesing, Journal of Chemical Education, '''(2017)''', 94, 9, 1225–1231, [https://pubs.acs.org/doi/abs/10.1021/acs.jchemed.6b01008 J.Chem.Edu]</ref> ]]
[[File:Brownianmotion5particles150frame.gif|thumb|This is a simulation of the Brownian motion of 5 particles (yellow) that collide with a large set of 800 particles. The yellow particles leave 5 blue trails of random motion and one of them has a red velocity vector.]]
[[File:Brownian motion large.gif|thumb|This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions.]]
'''Brownian motion''' or '''pedesis''' (from {{lang-grc|πήδησις}} {{IPA|/pɛ̌ːdɛːsis/}} "leaping"<!--not to be confused with πέδησις "braking"-->) is the random motion of [[particle]]s suspended in a [[fluid]] (a [[liquid]] or a [[gas]]) resulting from their collision with the fast-moving [[molecule]]s in the fluid.<ref name=Feynman-41>
{{cite book
 |last=Feynman |first=R.
 |year=1964
 |chapter=The Brownian Movement
 |chapter-url=http://www.feynmanlectures.caltech.edu/I_41.html
 |title=The Feynman Lectures of Physics, Volume I
 |pages=41{{hyphen}}1
}}</ref>

This pattern of motion typically alternates random fluctuations in a particle's position inside a fluid sub-domain with a relocation to another sub-domain.  Each relocation is followed by more fluctuations within the new closed volume.  This pattern describes a fluid at [[thermal equilibrium]], defined by a given [[temperature]]. Within such a fluid, there exists no preferential direction of flow as in [[transport phenomena]].  More specifically, the fluid's overall [[Linear momentum|linear]] and [[Angular momentum|angular]] momenta remain null over time.  The [[Kinetic energy|kinetic energies]] of the molecular Brownian motions, together with those of molecular rotations and vibrations, [[Equipartition theorem|sum up to the caloric component]] of a fluid's [[internal energy]].

This motion is named after the [[Botany|botanist]] [[Robert Brown (botanist, born 1773)|Robert Brown]], who first described the phenomenon in [[1827]], while looking through a [[microscope]] at [[pollen]] of the [[plant]] ''[[Clarkia pulchella]]'' immersed in water. In 1905, almost eighty years later,  [[List of theoretical physicists|theoretical physicist]] [[Albert Einstein]] published [[Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|a paper]] where he modeled the motion of the pollen as being moved by individual water molecules, making one of his first major scientific contributions.<ref name="Einstein1905">{{Cite journal|last=Einstein|first=Albert|title=Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen |trans-title=On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat |language=de |journal=Annalen der Physik |volume=322 |issue=8 |pages=549–560 |doi=10.1002/andp.19053220806 |date=1905 |url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_549-560.pdf |bibcode=1905AnP...322..549E }}</ref> This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by [[Jean Baptiste Perrin|Jean Perrin]] in 1908. Perrin was awarded the [[Nobel Prize in Physics]] in 1926 "for his work on the discontinuous structure of matter".<ref>{{Cite web|url=https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/|title=The Nobel Prize in Physics 1926|website=NobelPrize.org|language=en-US|access-date=2019-05-29}}</ref> The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion.

The [[Many-body problem|many-body interactions]] that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence, only probabilistic models applied to [[Statistical ensemble|molecular populations]] can be employed to describe it.  Two such models of the [[statistical mechanics]], due to Einstein and Smoluchowski are presented below.  Another, pure probabilistic class of models is the class of the [[stochastic process]] models.  There exist both simpler and more complicated stochastic processes which in extreme ("taken to the [[limit (mathematics)|limit]]") {{Citation needed|date=December 2018}} may describe the Brownian Motion (see [[random walk]] and [[Donsker's theorem]]).

==History==

The Roman philosopher [[Lucretius]]' scientific poem "[[On the Nature of Things]]" (c. 60 BC) has a remarkable description of Brownian motion of [[dust]] particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms:
[[File:PerrinPlot2.svg|thumb|Reproduced from the book of [[Jean Baptiste Perrin]], ''Les Atomes'', three tracings of the motion of colloidal particles of radius 0.53&nbsp;µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2&nbsp;µm).<ref>{{cite book |last=Perrin |first=Jean |year=1914 |title=Atoms |url=https://archive.org/stream/atomsper00perruoft#page/115/mode/1up |page=115|publisher=London : Constable }}</ref>]]

"Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible."

Although the mingling motion of dust particles is caused largely by air currents, the glittering, tumbling motion of small dust particles is, indeed, caused chiefly by true Brownian dynamics.

While [[Jan Ingenhousz]] described the irregular motion of [[coal]] [[dust]] particles on the surface of [[ethanol|alcohol]] in 1785, the discovery of this phenomenon is often credited to the botanist [[Robert Brown (botanist, born 1773)|Robert Brown]] in 1827. Brown was studying [[pollen]] grains of the plant ''[[Clarkia pulchella]]'' suspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained.

The first person to describe the mathematics behind Brownian motion was [[Thorvald N. Thiele]] in a paper on the method of [[least squares]] published in 1880. This was followed independently by [[Louis Bachelier]] in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the [[stock market]] is often cited, but [[Benoit Mandelbrot]] rejected its applicability to stock price movements in part because these are discontinuous.<ref>
{{cite book
 |last1=Mandelbrot |first1=B.
 |last2=Hudson |first2=R.
 |year=2004
 |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward
 |url=https://archive.org/details/misbehaviorofmar00beno |url-access=registration |publisher=[[Basic Books]]
 |isbn=978-0-465-04355-2
}}</ref>

[[Albert Einstein]] (in one of his [[Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|1905 papers]]) and [[Marian Smoluchowski]] (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of [[Jean Baptiste Perrin]] in 1908.

==Statistical mechanics theories==

=== Einstein's theory ===
There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the [[mean squared displacement]] of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.<ref name=Eistein1956>
{{cite book
 |last=Einstein |first=Albert
 |year=1956 |origyear=1926
 |title=Investigations on the Theory of the Brownian Movement
 |url=http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf
 |publisher=Dover Publications
 |accessdate=2013-12-25
}}</ref> In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas.<ref>
{{cite book
 |editor-last=Stachel |editor-first=J.
 |year=1989
 |chapter=Einstein's Dissertation on the Determination of Molecular Dimensions
 |chapter-url=http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf
 |title=The Collected Papers of Albert Einstein, Volume 2
 |publisher=Princeton University Press
}}</ref> In accordance to [[Avogadro's law]] this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the [[Avogadro number]], and the determination of this number is tantamount to the knowledge of the mass of an atom since the latter is obtained by dividing the mass of a mole of the gas by the [[Avogadro constant]].

[[File:Diffusion of Brownian particles.svg|thumb|300px|The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a [[Dirac delta function]], indicating that all the particles are located at the origin at time ''t'' = 0, and for increasing times they become flatter and flatter until the distribution becomes uniform in the asymptotic time limit.]]

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.<ref name="Einstein1905"/> Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 10<sup>14</sup> collisions per second.<ref name=Feynman-41/> Thus Einstein was led to consider the collective motion of Brownian particles.{{citation needed|date=October 2012}}

He regarded the increment of particle positions in time <math>\tau</math> in a one dimensional (''x'') space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable (<math>\Delta</math>) with some probability density function <math>\varphi(\Delta)</math>. Further, assuming conservation of particle number, he expanded the density (number of particles per unit volume) at time <math>t + \tau</math> in a Taylor series,

:<math>
\begin{align}
\rho(x,t) + \tau \frac{\partial\rho(x)}{\partial t} + \cdots = \rho(x, t+\tau) ={}& \int_{-\infty}^{+\infty} \rho(x - \Delta, t) \cdot \varphi(\Delta) \, \mathrm{d} \Delta = \mathbb{E}_\Delta[\rho(x, t + \tau)]
\\
={}& \rho(x, t) \cdot \int_{-\infty}^{+\infty} \varphi(\Delta) \, d \Delta -
 \frac{\partial\rho}{\partial x} \cdot \int_{-\infty}^{+\infty} \Delta \cdot \varphi(\Delta) \, \mathrm{d} \Delta
\\
&{}+
 \frac{\partial^2 \rho}{\partial x^2} \cdot \int_{-\infty}^{+\infty} \frac{\Delta^2}{2} \cdot \varphi(\Delta) \, \mathrm{d} \Delta + \cdots
\\
={}& \rho(x, t) \cdot 1 + 0 +
 \frac{\partial^2 \rho}{\partial x^2} \cdot \int_{-\infty}^{+\infty} \frac{\Delta^2}{2} \cdot \varphi(\Delta) \, \mathrm{d} \Delta + \cdots
\end{align}
</math>

where the second equality in the first line is by definition of <math>\varphi</math>. The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation:

:<math>
\frac{\partial\rho}{\partial t} = \frac{\partial^2 \rho}{\partial x^2} \cdot \int_{-\infty}^{+\infty} \frac{\Delta^2}{2\, \tau} \cdot \varphi(\Delta) \, \mathrm{d} \Delta + \text{higher-order even moments.}
</math>

Where the coefficient after the Laplacian, the second moment of probability of displacement <math>\Delta</math>, is interpreted as [[mass diffusivity]] ''D'':

:<math> D = \int_{-\infty}^{+\infty} \frac{\Delta^2}{2\, \tau} \cdot \varphi(\Delta) \, \mathrm{d} \Delta. </math>

Then the density of Brownian particles ''ρ'' at point ''x'' at time ''t'' satisfies the [[diffusion equation]]:

:<math>\frac{\partial\rho}{\partial t}=D\cdot \frac{\partial^2\rho}{\partial x^2},</math>

Assuming that ''N'' particles start from the origin at the initial time ''t'' = 0, the diffusion equation has the solution

:<math>\rho(x,t)=\frac{N}{\sqrt{4\pi Dt}}e^{-\frac{x^2}{4Dt}}.</math>

This expression (which is a [[normal distribution]] with the mean <math> \mu=0</math> and variance <math> \sigma^2=2Dt</math> usually called Brownian motion <math> B_t</math>) allowed Einstein to calculate the [[Moment (mathematics)|moments]] directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by

:<math>\overline{x^2}=2\,D\,t.</math>

This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.<ref name=Eistein1956/> His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.<ref>
{{cite book
 |last=Lavenda |first=Bernard H.
 |year=1985
 |title=Nonequilibrium Statistical Thermodynamics
 |page=20
 |publisher=John Wiley & Sons
 |isbn=978-0-471-90670-4
}}</ref>

The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of Avogadro's number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium.

In his original treatment, Einstein considered an [[osmotic pressure]] experiment, but the same conclusion can be reached in other ways.

Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of ''v'' = ''μmg'', where ''m'' is the mass of the particle, ''g'' is the acceleration due to gravity, and ''μ'' is the particle's [[Einstein relation (kinetic theory)|mobility]] in the fluid. [[Sir George Stokes, 1st Baronet|George Stokes]] had shown that the mobility for a spherical particle with radius ''r'' is <math>\mu=\tfrac{1}{6\pi\eta r}</math>, where ''η'' is the [[dynamic viscosity]] of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the [[Barometric formula|barometric distribution]]

:<math>\rho=\rho_0e^{-\frac{mgh}{k_{\rm B}T}},</math>

where ''ρ'' − ''ρ''<sub>0</sub> is the difference in density of particles separated by a height difference of ''h'', ''k''<sub>B</sub> is the [[Boltzmann constant]] (the ratio of the [[universal gas constant]], ''R'', to the Avogadro constant, ''N''{{sub|A}}), and ''T'' is the [[Thermodynamic temperature|absolute temperature]].

[[File:Brownian motion gamboge.jpg|thumb|The equilibrium distribution for particles of [[gamboge]] shows the tendency for granules to move to regions of lower concentration when affected by gravity.]]
[[Dynamic equilibrium]] is established because the more that particles are pulled down by [[gravity]], the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by [[Fick's laws of diffusion|Fick's law]],

:<math>J=-D\frac{d\rho}{dh},</math>

where ''J'' = ''ρv''. Introducing the formula for ''ρ'', we find that

:<math>v=\frac{Dmg}{k_{\rm B}T}.</math>

In a state of dynamical equilibrium, this speed must also be equal to ''v'' = ''μmg''. Both expressions for ''v'' are proportional to ''mg'', reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical [[charged particle]]s of charge ''q'' in a uniform [[electric field]] of magnitude ''E'', where ''mg'' is replaced with the [[electrostatic force]] ''qE''.  Equating these two expressions yields a formula for the diffusivity, independent of ''mg'' or ''qE'' or other such forces:

:<math>\frac{\overline{x^2}}{2t}=D=\mu k_{\rm B}T=\frac{\mu RT}{N_\text{A}}=\frac{RT}{6\pi\eta rN_\text{A}}.</math>

Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of [[Boltzmann constant|Boltzmann's constant]] as ''k''<sub>B</sub> = ''R'' / ''N''{{sub|A}}, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant ''R'', the temperature ''T'', the viscosity ''η'', and the particle radius ''r'', the Avogadro constant ''N''{{sub|A}} can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by [[J. J. Thomson]]<ref name=Electricity>
{{cite book
 |last=Thomson |first=J. J.
 |year=1904
 |title=Electricity and Matter
 |url=https://archive.org/details/electricitymatte00thomuoft |pages=[https://archive.org/details/electricitymatte00thomuoft/page/80 80–83]
 |publisher=Yale University Press
}}</ref> in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a [[concentration gradient]] given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's Constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".<ref name=Electricity/>

An identical expression to Einstein's formula for the diffusion coefficient was also found by [[Walther Nernst]] in 1888<ref>
{{cite journal
 |last=Nernst |first=Walther
 |year=1888
 |title=Zur Kinetik der in Lösung befindlichen Körper
 |language=de
 |journal=[[Zeitschrift für Physikalische Chemie]]
 |volume=9 |pages=613–637
}}</ref> in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the [[Friction|frictional force]] and the velocity to which it gives rise. The former was equated to the [[Van 't Hoff factor|law of van 't Hoff]] while the latter was given by [[Stokes's law]]. He writes <math>k'=p_0/k</math> for the diffusion coefficient ''k′'', where <math>p_0</math> is the osmotic pressure and ''k'' is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the [[ideal gas law]] per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's.<ref>
{{cite book
 |last=Leveugle |first=J.
 |year=2004
 |title=La Relativité, Poincaré et Einstein, Planck, Hilbert
 |page=181
 |publisher=Harmattan
}}</ref> The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the [[mean free path]].<ref>
{{cite book
 |last=Townsend |first=J.E.S.
 |year=1915
 |title=Electricity in Gases
 |url=https://archive.org/details/electricityinga00towngoog |page=254
 |publisher=Clarendon Press
}}</ref>

At first, the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted.<ref>See P. Clark 1976, p. 97</ref> But Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the [[Kinetic theory of gases|kinetic theory of heat]]. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of [[thermal equilibrium]]. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the [[Second law of thermodynamics|second law]] of [[thermodynamics]] as being an essentially statistical law.<ref>See P. Clark 1976 for this whole paragraph</ref>

[[File:Brownian Motion.ogv|thumb|320px|Brownian motion model of the trajectory of a particle of dye in water.]]

===Smoluchowski model===
[[Marian Smoluchowski|Smoluchowski]]'s theory of Brownian motion<ref>
{{cite journal
 |last=Smoluchowski |first=M. M.
 |year=1906
 |title=Sur le chemin moyen parcouru par les molécules d'un gaz et sur son rapport avec la théorie de la diffusion
 |language=fr
 |trans-title=On the average path taken by gas molecules and its relation with the theory of diffusion
 |url=https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up
 |journal=[[Bulletin International de l'Académie des Sciences de Cracovie]]
 |page=202
}}</ref> starts from the same premise as that of Einstein and derives the same probability distribution ''ρ''(''x'', ''t'') for the displacement of a Brownian particle along the ''x'' in time ''t''. He therefore gets the same expression for the mean squared displacement: <math>\overline{(\Delta x)^2}</math>. However, when he relates it to a particle of mass ''m'' moving at a velocity <math>u</math> which is the result of a frictional force governed by Stokes's law, he finds
:<math>\overline{(\Delta x)^2}=2Dt=t\frac{32}{81}\frac{u^2}{\pi\mu a}=t\frac{64}{27}\frac{\frac{1}{2}u^2}{3\pi\mu a},</math>
where ''μ'' is the viscosity coefficient, and <math>a</math> is the radius of the particle. Associating the kinetic energy <math>u^2/2</math> with the thermal energy ''RT''/''N'', the expression for the mean squared displacement is 64/27 times that found by Einstein. The fraction 27/64 was commented on by [[Arnold Sommerfeld]] in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."<ref>See p. 535 in {{Cite journal
 |last=Sommerfeld |first=A.
 |year=1917
 |title=Zum Andenken an Marian von Smoluchowski |language=de
 |trans-title=In Memory of Marian von Smoluchowski
 |journal=[[Physikalische Zeitschrift]]
 |volume=18 |issue=22 |pages=533–539
}}</ref>

Smoluchowski<ref>
{{cite journal
 |last=Smoluchowski |first=M. M.
 |year=1906
 |title=Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles
 |url=https://archive.org/stream/bulletininternat1906pols#page/577/mode/2up
 |language=fr
 |trans-title=Test of a kinetic theory of Brownian motion and turbid media
 |journal=[[Bulletin International de l'Académie des Sciences de Cracovie]]
 |page=577
}}</ref> attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal.
If the probability of ''m'' gains and ''n''&nbsp;−&nbsp;''m'' losses follows a [[binomial distribution]],

:<math>P_{m,n}=\binom{n}{m} 2^{-n},</math>

with equal ''a priori'' probabilities of 1/2, the mean total gain is

:<math>\overline{2m-n}=\sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^n \left [ \left (\frac{n}{2} \right )! \right ]^2}.</math>

If ''n'' is large enough so that Stirling's approximation can be used in the form
:<math>n!\approx\left(\frac{n}{e}\right)^n\sqrt{2\pi n},</math>
then the expected total gain will be{{citation needed|date=July 2012}}
:<math>\overline{2m-n}\approx\sqrt{\frac{2n}{\pi}},</math>
showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass ''M'' is surrounded by lighter particles of mass ''m'' which are traveling at a speed ''u''. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be ''mu''/''M''. This ratio is of the order of 10<sup>−7</sup>&nbsp;cm/s. But we also have to take into consideration that in a gas there will be more than 10<sup>16</sup> collisions in a second, and even greater in a liquid where we expect that there will be 10<sup>20</sup> collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 10<sup>8</sup> to 10<sup>10</sup> collisions in one second, then velocity of the Brownian particle may be anywhere between 10 and 1000&nbsp;cm/s. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, ''U'', which depends on the collisions that tend to accelerate and decelerate it. The larger ''U'' is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, <math>MU^2/2</math>, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, <math>mu^2/2</math>.

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.<ref>
{{cite journal
 |last=von Smoluchowski |first=M.
 |year=1906
 |title=Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen
 |language=de
 |journal=[[Annalen der Physik]]
 |volume=326 |issue=14 |pages=756–780
 |bibcode=1906AnP...326..756V
 |doi=10.1002/andp.19063261405
|url=https://zenodo.org/record/1424073
 }}</ref> The model assumes collisions with ''M''&nbsp;≫&nbsp;''m'' where ''M'' is the test particle's mass and ''m'' the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of Δ''V''. If ''N''<sub>R</sub> is the number of collisions from the right and ''N''<sub>L</sub> the number of collisions from the left then after ''N'' collisions the particle's velocity will have changed by Δ''V''(2''N''<sub>R</sub>&nbsp;−&nbsp;''N''). The [[multiplicity (mathematics)|multiplicity]] is then simply given by:

:<math> \binom{N}{N_{\rm R}} = \frac{N!}{N_{\rm R}!(N-N_{\rm R})!}</math>

and the total number of possible states is given by 2<sup>''N''</sup>. Therefore, the probability of the particle being hit from the right ''N<sub>R</sub>'' times is:

:<math>P_N(N_{\rm R})=\frac{N!}{2^NN_{\rm R}!(N-N_{\rm R})!}</math>

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible Δ''V''s instead of always just one in a realistic situation.

===Other physics models using partial differential equations===
The [[diffusion equation]] yields an approximation of the time evolution of the [[probability density function]] associated to the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on [[Langevin equation|short]] timescales.

The time evolution of the position of the Brownian particle itself is best described using [[Langevin equation]], an equation which involves a random force field representing the effect of the [[thermal fluctuations]] of the solvent on the particle.

The displacement of a particle undergoing Brownian motion is obtained by solving the [[diffusion equation]] under appropriate boundary conditions and finding the [[root mean square|rms]] of the solution. This shows that the displacement varies as the square root of the time (not linearly), which explains why previous experimental results concerning the velocity of Brownian particles gave nonsensical results. A linear time dependence was incorrectly assumed.

At very short time scales, however, the motion of a particle is dominated by its inertia and its displacement will be linearly dependent on time: Δ''x'' = ''v''Δ''t''. So the instantaneous velocity of the Brownian motion can be measured as ''v'' = Δ''x''/Δ''t'', when Δ''t'' << ''τ'', where ''τ'' is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with [[optical tweezers]]) was measured successfully.<ref>{{cite journal
 |last      = Li
 |first     = Tongcang
 |last2     = Kheifets
 |first2    = Simon
 |last3     = Medellin
 |first3    = David
 |last4     = Raizen
 |first4    = Mark
 |year      = 2010
 |title     = Measurement of the instantaneous velocity of a Brownian particle
 |journal   = [[Science (journal)|Science]]
 |volume    = 328
 |issue     = 5986
 |pages     = 1673–1675
 |bibcode   = 2010Sci...328.1673L
 |doi       = 10.1126/science.1189403
 |pmid      = 20488989
 |url       = http://chaos.utexas.edu/wp-uploads/2010/06/science.1189403v1.pdf
 |archive-url = https://wayback.archive-it.org/all/20110331172407/http://chaos.utexas.edu/wp-uploads/2010/06/science.1189403v1.pdf
 |url-status  = dead
 |archive-date = 2011-03-31
 |citeseerx = 10.1.1.167.8245
}}</ref> The velocity data verified the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann velocity distribution]], and the equipartition theorem for a Brownian particle.

===Astrophysics: star motion within galaxies===
In [[stellar dynamics]], a massive body (star, [[black hole]], etc.) can experience Brownian motion as it responds to [[gravitational|gravitational forces]] from surrounding stars.<ref name=Merritt>
{{cite book
 |last=Merritt |first=David
 |year=2013
 |title=Dynamics and Evolution of Galactic Nuclei
 |page=575
 |publisher=Princeton University Press
 |isbn=9781400846122
 |ol=16802359W
}}</ref> The rms velocity ''V'' of the massive object, of mass ''M'', is related to the rms velocity <math>v_\star</math> of the background stars by
:<math> MV^2 \approx m v_\star^2 </math>
where <math>m\ll M</math> is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both <math>v_\star</math> and ''V''.<ref name=Merritt/> The Brownian velocity of [[Sagittarius A*|Sgr A*]], the [[supermassive black hole]] at the center of the [[Milky Way galaxy]], is predicted from this formula to be less than 1&nbsp;km ''s''<sup>−1</sup>.<ref name="Reid">
{{cite journal
 |last1=Reid |first1=M. J.
 |last2=Brunthaler |first2=A.
 |year=2004
 |title=The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A*
 |journal=[[The Astrophysical Journal]]
 |volume=616 |issue=2 |pages=872–884
 |arxiv=astro-ph/0408107
 |bibcode=2004ApJ...616..872R
 |doi=10.1086/424960
}}</ref>

==Mathematics==
{{Main|Wiener process}}
[[File:2D Random Walk 400x400.ogv|thumb|right|300px|An animated example of a Brownian motion-like [[random walk]] on a [[torus]]. In the [[scaling limit]], random walk approaches the Wiener process according to [[Donsker's theorem]].]]
In [[mathematics]], Brownian motion is described by the '''Wiener process''', a continuous-time [[stochastic process]] named in honor of [[Norbert Wiener]]. It is one of the best known [[Lévy process]]es ([[càdlàg]] stochastic processes with [[stationary process|stationary]] [[independent increments]]) and occurs frequently in pure and applied mathematics, [[economy|economics]] and [[physics]].
[[File:Wiener process 3d.png|thumb|A single realisation of three-dimensional Brownian motion for times 0&nbsp;≤&nbsp;''t''&nbsp;≤&nbsp;2]]

The Wiener process ''W<sub>t</sub>'' is characterized by four facts:{{Citation needed|date=February 2018}}
# ''W''<sub>0</sub> = 0
# ''W<sub>t</sub>'' is [[almost surely]] continuous
# ''W<sub>t</sub>'' has independent increments
# <math>W_t-W_s\sim \mathcal{N}(0,t-s)</math> (for <math>0 \leq s \le t</math>).
<math>\mathcal{N}(\mu, \sigma^2)</math> denotes the [[normal distribution]] with [[expected value]] ''μ'' and [[variance]] ''σ''<sup>2</sup>. The condition that it has independent increments means that if <math>0 \leq s_1 < t_1 \leq s_2 < t_2</math> then <math>W_{t_1}-W_{s_1}</math> and <math>W_{t_2}-W_{s_2}</math> are independent random variables.

An alternative characterisation of the Wiener process is the so-called ''Lévy characterisation'' that says that the Wiener process is an almost surely continuous [[martingale (probability theory)|martingale]] with ''W''<sub>0</sub> = 0 and [[quadratic variation]] <math>[W_t, W_t] = t</math>.

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent <math>\mathcal{N}(0, 1)</math> random variables. This representation can be obtained using the [[Karhunen–Loève theorem]].

The Wiener process can be constructed as the [[scaling limit]] of a [[random walk]], or other discrete-time stochastic processes with stationary independent increments. This is known as [[Donsker's theorem]]. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [[neighborhood (mathematics)|neighborhood]] of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is [[scale invariance|scale invariant]].

The time evolution of the position of the Brownian particle itself can be described approximately by a [[Langevin equation]], an equation which involves a random force field representing the effect of the [[thermal fluctuations]] of the solvent on the Brownian particle. On long timescales, the mathematical Brownian motion is well described by a Langevin equation. On small timescales, [[inertia]]l effects are prevalent in the Langevin equation. However the mathematical ''Brownian motion'' is exempt of such inertial effects. Inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular.{{Clarify|date=April 2010}} so that simply removing the [[inertia]] term from this equation would not yield an exact description, but rather a singular behavior in which the particle doesn't move at all.{{Clarify|date=April 2010}}

===Statistics===
The Brownian motion can be modeled by a random walk.<ref>
{{cite book
 |last=Weiss |first=G. H.
 |year=1994
 |title=Aspects and applications of the random walk
 |publisher=North Holland
 |isbn=
}}</ref> Random walks in porous media or fractals are anomalous.<ref>
{{Cite book
 |last1=Ben-Avraham |first1=D.
 |last2=Havlin |first2=S.
 |year=2000
 |title=Diffusion and reaction in disordered systems
 |page=
 |publisher= Cambridge University Press
 |isbn=
}}</ref>

In the general case, Brownian motion is a [[Markov process|non-Markov random process]] and described by [[Stochastic calculus|stochastic integral equations]].<ref name="mor-skr2a">
{{cite journal
 |last=Morozov |first=A. N.
 |last2=Skripkin |first2=A. V.
 |year=2011
 |title=Spherical particle Brownian motion in viscous medium as non-Markovian random process
 |journal=[[Physics Letters A]]
 |volume=375 |issue=46 |pages=4113–4115
 |bibcode=2011PhLA..375.4113M
 |doi=10.1016/j.physleta.2011.10.001
}}</ref>

===Lévy characterisation===
The French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] proved the following theorem, which gives a necessary and sufficient condition for a continuous '''R'''<sup>''n''</sup>-valued stochastic process ''X'' to actually be ''n''-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

Let ''X''&nbsp;=&nbsp;(''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub>) be a continuous stochastic process on a [[probability space]] (Ω,&nbsp;Σ,&nbsp;'''P''') taking values in '''R'''<sup>''n''</sup>. Then the following are equivalent:

# ''X'' is a Brownian motion with respect to '''P''', i.e., the law of ''X'' with respect to '''P''' is the same as the law of an ''n''-dimensional Brownian motion, i.e., the [[push-forward measure]] ''X''<sub>∗</sub>('''P''') is [[classical Wiener measure]] on ''C''<sub>0</sub>([0,&nbsp;+∞);&nbsp;'''R'''<sup>''n''</sup>).
# both
## ''X'' is a [[martingale (probability theory)|martingale]] with respect to '''P''' (and its own [[natural filtration]]); and
## for all 1&nbsp;≤&nbsp;''i'',&nbsp;''j''&nbsp;≤&nbsp;''n'', ''X''<sub>''i''</sub>(''t'')''X''<sub>''j''</sub>(''t'')&nbsp;−''δ''<sub>''ij''</sub>''t'' is a martingale with respect to '''P''' (and its own [[natural filtration]]), where ''δ''<sub>''ij''</sub> denotes the [[Kronecker delta]].

===Spectral content===
The spectral content of a stochastic process <math>X_t</math>can be found from the [[power spectral density]], formally defined as

<math>S(\omega)=\lim_{T\to\infty}\frac{1}{T}\mathbb{E}\left\{\left|\int^T_0
e^{i \omega t}X_t dt\right|^2\right\}</math>,

where <math>\mathbb{E}</math> stands for the [[expected value]]. The power spectral density of Brownian motion is found to be<ref>{{Cite book|title=Fundamentals of Noise and Vibration Analysis for Engineers by M. P. Norton|last=Karczub|first=D. G.|last2=Norton|first2=M. P.|date=2003|language=en|doi=10.1017/cbo9781139163927|isbn=9781139163927}}</ref>

<math>S_{BM}(\omega)=\frac{4 D}{\omega^2}</math>.

where <math>D</math> is the [[diffusion coefficient]] of <math>X_t</math>. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e.,

<math>S^{(1)}(\omega,T)=\frac{1}{T}\left|\int^T_0 e^{i \omega t}X_t dt\right|^2</math>,

which for an individual realization of a Brownian motion trajectory,<ref name=":0">{{Cite journal|last=Krapf|first=Diego|last2=Marinari|first2=Enzo|last3=Metzler|first3=Ralf|last4=Oshanin|first4=Gleb|last5=Xu|first5=Xinran|last6=Squarcini|first6=Alessio|date=2018|title=Power spectral density of a single Brownian trajectory: what one can and cannot learn from it|url=http://stacks.iop.org/1367-2630/20/i=2/a=023029|journal=New Journal of Physics|language=en|volume=20|issue=2|pages=023029|doi=10.1088/1367-2630/aaa67c|issn=1367-2630|arxiv=1801.02986|bibcode=2018NJPh...20b3029K}}</ref> it is found to have expected value <math>\mu_{BM}(\omega,T)</math>

<math>\mu_{BM}(\omega,T)=\frac{4 D}{\omega^2}\left[1-\frac{\sin\left(\omega T\right)}{\omega T}\right]</math>

and [[variance]] <math>\sigma_{BM}^2(\omega,T)</math><ref name=":0" />

<math>\sigma_S^2(f,T)=\mathbb{E}\left\{\left(S^{(j)}_T(f)\right)^2\right\}-\mu_S^2
(f,T)
=\frac{20 D^2}{f^4}\Bigg[1-\Big(6-\cos\left(f T\right)\Big)\frac{2\sin\left(
fT\right)}{5fT}
+\frac{\Big(17-\cos\left(2fT\right)-16\cos\left(f T\right)\Big)}{10 f^2 T^2}
\Bigg]</math>.

For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density <math>S(\omega)</math>, but its coefficient of variation <math>\gamma=\sqrt{\sigma^2}/\mu</math> tends to <math>\sqrt{5}/2</math>. This implies the distribution of <math>S^{(1)}(\omega,T)</math>is broad even in the infinite time limit.

===Riemannian manifold===
[[File:BMonSphere.jpg|thumb|Brownian motion on a sphere]]
{{Technical|section|date=June 2011}}

The [[Infinitesimal generator (stochastic processes)|infinitesimal generator]] (and hence characteristic operator) of a Brownian motion on '''R'''<sup>''n''</sup> is easily calculated to be ½Δ, where Δ denotes the [[Laplace operator]]. In [[image processing]] and [[computer vision]], the Laplacian operator has been used for various tasks such as blob and [[edge detection]]. This observation is useful in defining Brownian motion on an ''m''-dimensional [[Riemannian manifold]] (''M'',&nbsp;''g''): a '''Brownian motion on''' ''M'' is defined to be a diffusion on ''M'' whose characteristic operator <math>\mathcal{A}</math> in local coordinates ''x''<sub>''i''</sub>, 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''m'', is given by ½Δ<sub>LB</sub>, where Δ<sub>LB</sub> is the [[Laplace–Beltrami operator]] given in local coordinates by

:<math>\Delta_{\mathrm{LB}}=\frac{1}{\sqrt{\det(g)}} \sum_{i=1}^m \frac{\partial}{\partial x_i} \left(\sqrt{\det(g)} \sum_{j=1}^m g^{ij} \frac{\partial}{\partial x_j} \right),</math>

where [''g''<sup>''ij''</sup>]&nbsp;=&nbsp;[''g''<sub>''ij''</sub>]<sup>−1</sup> in the sense of [[Invertible matrix|the inverse of a square matrix]].

==Narrow escape==
The [[narrow escape problem]] is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle ([[ion]], [[molecule]], or [[protein]]) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a [[singular perturbation]] problem.

==See also==
{{div col|colwidth=30em}}
* [[Brownian bridge]]: a Brownian motion that is required to "bridge" specified values at specified times
* [[Brownian covariance]]
* [[Brownian dynamics]]
* [[Brownian motion of sol particles]]
* [[Brownian motor]]
* [[Brownian noise]] ([[Martin Gardner]] proposed this name for sound generated with random intervals. It is a pun on Brownian motion and [[white noise]].)
* [[Brownian ratchet]]
* [[Brownian surface]]
* [[Brownian tree]]
* [[Brownian web]]
* [[Rotational Brownian motion]]
* [[Free will in antiquity#Epicurus|Clinamen]]
* [[Complex system]]
* [[Continuity equation]]
* [[Diffusion equation]]
* [[Geometric Brownian motion]]
* [[Itō diffusion]]: a generalisation of Brownian motion
* [[Langevin equation]]
* [[Lévy arcsine law]]
* [[Local time (mathematics)]]
* [[Many-body problem]]
* [[Marangoni effect]]
* [[Nanoparticle tracking analysis]]
* [[Narrow escape problem]]
* [[Osmosis]]
* [[Random walk]]
* [[Schramm–Loewner evolution]]
* [[Single particle tracking]]
* [[Statistical mechanics]]
* [[Surface diffusion]]: a type of constrained Brownian motion.
* [[Thermal equilibrium]]
* [[Thermodynamic equilibrium]]
* [[Tyndall effect]]: physical chemistry phenomenon where particles are involved; used to differentiate between the different types of mixtures.
* [[Ultramicroscope]]
{{div col end}}

==References==
{{Reflist|30em}}
* [[Lester Eli Dubins]] & Gideon Schwarz (1965). ''On Continuous Martingales''. Proceedings of the National Academy of Sciences.

==Further reading==
* {{cite journal |last=Brown |first=Robert |year=1828 |title=A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies |journal=[[Philosophical Magazine]] |volume=4 |issue=21 |pages=161–173 |url=http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf|doi=10.1080/14786442808674769 }} Also includes a subsequent defense by Brown of his original observations, ''Additional remarks on active molecules''.
* {{cite journal |last1=Chaudesaigues |first1=M. |year=1908 |title=Le mouvement brownien et la formule d'Einstein |trans-title=Brownian motion and Einstein's formula |language=French |journal=[[Comptes Rendus]] |volume=147 |issue= |pages=1044–6}}
* {{cite book |last=Clark |first=P. |year=1976 |chapter=Atomism versus thermodynamics |title=Method and appraisal in the physical sciences |editor-last=Howson |editor-first=Colin |publisher=Cambridge University Press |isbn=978-0521211109}}
* {{cite journal |last=Cohen |first=Ruben D. |year=1986 |title=Self Similarity in Brownian Motion and Other Ergodic Phenomena |url=http://rdcohen.50megs.com/BrownianMotion.pdf |journal=[[Journal of Chemical Education]] |volume=63 |issue=11 |pages=933–934|bibcode=1986JChEd..63..933C |doi=10.1021/ed063p933 }}
* {{cite book |last=Einstein |first=A. |title=Investigations on the Theory of Brownian Movement |location=New York |publisher=Dover |year=1956 |isbn=978-0-486-60304-9 |url=https://archive.org/details/investigationson00eins |access-date=6 January 2014 |url-status=dead |df=dmy-all |url-access=registration }}
* {{cite journal |last=Henri |first=V. |year=1908 |title=Études cinématographique du mouvement brownien |trans-title=Cinematographic studies of Brownian motion |journal=[[Comptes Rendus]] |issue=146 |pages=1024–6 |language=fr}}
* [[Lucretius]], ''On The Nature of Things'', translated by [[William Ellery Leonard]]. (''[http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=785 on-line version]'', from [[Project Gutenberg]]. See the heading 'Atomic Motions'; this translation differs slightly from the one quoted).
* [[Edward Nelson|Nelson, Edward]], (1967). ''Dynamical Theories of Brownian Motion''. [https://web.math.princeton.edu/~nelson/books/bmotion.pdf (PDF version of this out-of-print book, from the author's webpage.)]  This is primarily a mathematical work, but the first four chapters discuss the history of the topic, in the era from Brown to Einstein.
* {{cite journal |author1=Pearle, P. |author2=Collett, B. |author3=Bart, K. |author4=Bilderback, D. |author5=Newman, D. |author6=Samuels, S. |year=2010 |title=What Brown saw and you can too |journal=[[American Journal of Physics]] |volume=78 |issue=12 |pages=1278–1289 |bibcode=2010AmJPh..78.1278P |doi=10.1119/1.3475685 |arxiv=1008.0039 }}
* {{cite journal |last=Perrin |first=J. |year=1909 |title=Mouvement brownien et réalité moléculaire |trans-title=Brownian movement and molecular reality |journal=[[Annales de chimie et de physique]] |series=8th series |volume=18 |pages=5–114}}
** See also Perrin's book "Les Atomes" (1914).
* {{cite journal |last=von Smoluchowski |first=M. |year=1906  |title=Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen |journal=[[Annalen der Physik]] |volume=21 |issue=14 |pages=756–780|url=http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer |doi=10.1002/andp.19063261405 |bibcode=1906AnP...326..756V}}
* {{cite book |last=Svedberg |first=T. |year=1907 |title=Studien zur Lehre von den kolloiden Losungen}}
* [[Thorvald N. Thiele|Theile, T. N.]]
** Danish version: "Om Anvendelse af mindste Kvadraters Methode i nogle Tilfælde, hvor en Komplikation af visse Slags uensartede tilfældige Fejlkilder giver Fejlene en ‘systematisk’ Karakter".
** French version: "Sur la compensation de quelques erreurs quasi-systématiques par la méthodes de moindre carrés" published simultaneously in ''Vidensk. Selsk. Skr. 5. Rk., naturvid. og mat. Afd.'', 12:381–408, 1880.

==External links==
{{Commons category|Brownian motion}}
{{wikisource|A brief account of microscopical observations made on the particles contained in the pollen of plants}}
* [https://web.archive.org/web/20010222031055/http://www.bun.kyoto-u.ac.jp/~suchii/einsteinBM.html Einstein on Brownian Motion]
* [http://physerver.hamilton.edu/Research/Brownian/index.html Discusses history, botany and physics of Brown's original observations, with videos]
* [http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/ "Einstein's prediction finally witnessed one century later"] : a test to observe the velocity of Brownian motion
* [https://www.youtube.com/watch?v=Xscn-QSmFo4 Brownian Motion on YouTube]

{{Einstein}}
{{Fractals}}

{{Authority control}}

{{DEFAULTSORT:Brownian Motion}}
[[Category:Statistical mechanics]]
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[[Category:Fractals]]
[[Category:Colloidal chemistry]]
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