Brownian motion
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{{nofootnote}}
:''This article is about the physical phenomenon; for the stochastic process, see [[Wiener process]]. For the sports team, see [[Brownian Motion (Ultimate)]]. For the mobility model, see [[Random walk]].''
[[Image:Brownian hierarchical.svg|thumb|Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors.]]
[[Image:Wiener process 3d.png|thumb|A single realization of three-dimensional Brownian motion for times 0 ≤ ''t'' ≤ 2.]]
'''Brownian motion''' (named in honor of the botanist [[Robert Brown (botanist)|Robert Brown]]) is the random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a [[particle theory]].
The mathematical model of Brownian motion has several real-world applications. An often quoted example is [[stock market]] fluctuations. Another example is the evolution of physical characteristics in the fossil record.{{Fact|date=February 2007}}
Brownian motion is among the simplest continuous-time [[stochastic process]]es, and it is a [[limit (mathematics)|limit]] of both simpler and more complicated stochastic processes (see [[random walk]] and [[Donsker's theorem]]). This [[Universality (dynamical systems)|universality]] is closely related to the universality of the [[normal distribution]]. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use.
==History==
[[Image:PerrinPlot2.gif|thumb|left|Reproduced from the book of Perrin, ''Les Atomes'', three tracings of the motion of colloidal particles of radius 0.53µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2µm).]]
The Roman [[Lucretius]]'s scientific poem [[On the Nature of Things]] (c. 60 BC) has a remarkable description of Brownian motion of dust particles. He uses this as a proof of the existence of atoms: "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.
[[Jan Ingenhousz]] had described the irregular motion of [[coal]] [[dust]] particles on the surface of [[ethanol|alcohol]] in 1785. Nevertheless Brownian motion is traditionally regarded as discovered by the botanist [[Robert Brown (botanist)|Robert Brown]] in 1827. It is believed that Brown was studying [[pollen]] particles floating in water under the microscope. He then observed minute particles within the vacuoles of the pollen grains executing a jittery motion. By repeating the experiment with particles of dust, he was able to rule out that the motion was due to pollen particles being 'alive', although the origin of the motion was yet to be explained.
The first person to describe the mathematics behind Brownian motion was [[Thorvald N. Thiele]] in 1880 in a paper on the method of least squares. 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. However, it was [[Albert Einstein]]'s (in his [[Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|1905 paper]]) and [[Marian Smoluchowski]]'s (1906) independent research of the problem that brought the solution to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules.
==Intuitive metaphor ==
Consider a large balloon of 10 meters in diameter. Imagine this large balloon in a football stadium or any widely crowded area. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted from the left side and the right side are imbalanced in favor of the left side; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion. If we look at this situation from above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.
Now return to Brown’s pollen particle swimming randomly in water. One molecule of water is about 0.1 to 0.2 nm, (a hydrogen-bonded cluster of 300 atoms has a diameter of approximately 3 nm) where the pollen particle is roughly 1 µm in diameter, roughly 10,000 times larger than a water molecule. So, the pollen particle can be considered as a very large balloon constantly being pushed by water molecules. The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted by the small liquid molecules on the particle.
An [http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html animation of the Brownian motion concept] is available as a [[Java applet]].
==Modelling using differential equations==
{{Confusing|date=September 2007}}
The equations governing Brownian motion relate slightly differently to each of the two definitions of ''Brownian motion'' given at the start of this article.
===Mathematical ===
<small>for main article, see [[Wiener process]].</small>
In [[mathematics]], the '''Wiener process''' is 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]] [[statistical independence|independent]] increments) and occurs frequently in pure and applied mathematics, [[economy|economics]] and [[physics]].
The Wiener process ''W''<sub>t</sub> is characterized by three facts:
#''W''<sub>0</sub> = 0
#''W''<sub>''t''</sub> is [[almost surely]] continuous
#''W''<sub>''t''</sub> has independent increments with distribution <math>W_t-W_s\sim \mathcal{N}(0,t-s)</math> (for 0 ≤ ''s'' < ''t'').
''N''(''μ'', ''σ''<sup>2</sup>) denotes the [[normal distribution]] with [[expected value]] ''μ'' and [[variance]] ''σ''<sup>2</sup>. The condition that it has independent increments means that if 0 ≤ ''s''<sub>1</sub> ≤ ''t''<sub>1</sub> ≤ ''s'' <sub>2</sub> ≤ ''t''<sub>2</sub> then ''W''<sub>''t''<sub>1</sub></sub> − ''W''<sub>''s''<sub>1</sub></sub> and ''W''<sub>''t''<sub>2</sub></sub> − ''W''<sub>''s''<sub>2</sub></sub> are independent random variables.
An alternative characterization of the Wiener process is the so-called ''Lévy characterization'' that says that the Wiener process is an almost surely continuous [[martingale (probability theory)|martingale]] with ''W''<sub>0</sub> = 0 and [[quadratic variation]] [''W''<sub>''t''</sub>, ''W''<sub>''t''</sub>] = ''t''.
A third characterization is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ''N''(0,1) 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]].
One property of a Brownian motion process is that the length of the process is infinite over any length of interval. This means that Brownian motion is purely an abstract concept since no particle can move an infinite distance in a finite length of time. It's important to remember that Brownian motion is used to approximate the movement of particles.
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. Note that inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular, 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.
===Physical Brownian theory===
The [[diffusion equation]] yields an approximation of the time evolution of the [[probability density function]] associated to the position of the particle undergoing a Brownian movement under the physical definition. The approximation is valid on short timescales (see [[Langevin equation]] for details).
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.
==The Lévy characterization==
The [[France|French]] [[mathematician]] [[Paul Pierre Lévy|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 an alternative definition of Brownian motion.
Let ''X'' = (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) be a continuous stochastic process on a [[probability space]] (Ω, Σ, '''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, +∞); '''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 ≤ ''i'', ''j'' ≤ ''n'', ''X''<sub>''i''</sub>(''t'')''X''<sub>''j''</sub>(''t'') −''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]].
==Brownian motion on a Riemannian manifold==
[[Image:BMonSphere.jpg|thumb|The characteristic operator of a Brownian motion is ½ times the Laplace-Beltrami operator. Here it is the Laplace-Beltrami operator on a 2-sphere.]]
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]]. This observation is useful in defining Brownian motion on an ''m''-dimensional [[Riemannian manifold]] (''M'', ''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 ≤ ''i'' ≤ ''m'', is given by ½Δ<sub>LB</sub>, where Δ<sub>LB</sub> is the [[Laplace operator#Laplace-Beltrami operator|Laplace-Beltrami operator]] given in local coordinates by
:<math>\Delta_{\mathrm{LB}} = \frac1{\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>] = [''g''<sub>''ij''</sub>]<sup>−1</sup> in the sense of [[Invertible matrix|the inverse of a square matrix]].
==Cultural uses==
The awareness of Brownian motion as a stochastic process is referred to in science fiction. In [[Douglas Adams]]'s ''[[The Hitchhiker's Guide to the Galaxy]]'', Brownian motion is used to create (or rather calculate) the [[Infinite Improbability Drive]] that powers the spaceship ''[[Heart of Gold (spaceship)|Heart of Gold]]''. The Brownian motion generator is a really hot cup of [[tea]]. In [[Murray Leinster]]'s short story, [[A Logic Named Joe]], the logic (computer) suggests building a perpetual motion machine using Brownian motion.
It also appears in other novels. In [[Julio Cortazar]]'s novel ''[[Rayuela]]'', Brownian motion is used to describe travelers in [[Paris]] at night.
It also appears in a famous/notorious essay by Constance Penley, "Brownian Motion: Women, Tactics, and Technology".
It also appears in [[John Crowley]]'s short SF story 'Snow'. Brownian Motion is used to explain why a device that stores memories at a molecular level can only recall stored memories in a random fashion.
==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Brownian bridge]]: a Brownian motion that is required to "bridge" specified values at specified times
* [[Brownian dynamics]]
* [[Brownian frontier]]
* [[Brownian motor]]
* [[Brownian ratchet]]
* [[Brownian tree]]
* [[Complex system]]
* [[Diffusion equation]]
* [[Itō diffusion]]: a generalization of Brownian motion
* [[Langevin equation]]
* [[Local time (mathematics)]]
* [[Osmosis]]
* [[Red noise]], also known as ''brown noise'' ([[Martin Gardner]] proposed this name for sound generated with random intervals. It is a pun on Brownian motion and [[white noise]].)
* [[Surface diffusion]] - a type of constrained Brownian motion.
* [[Tyndall effect]]: physical chemistry phenomenon where particles are involved; used to differentiate between the different types of mixtures.
* [[Ultramicroscope]]
</div>
==References==
* Brown, Robert, "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." Phil. Mag. 4, 161-173, 1828. [http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf (PDF version of original paper including a subsequent defense by Brown of his original observations, ''Additional remarks on active molecules''.)]
* {{Citation | author=Einstein, A. | title =[http://www3.interscience.wiley.com/homepages/5006612/549_560.pdf Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.] | journal =Annalen der Physik | volume =17 | pages =549-560| year=1905}}
* {{Citation | author=Smoluchowski, M. | title =[http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen] | journal =Annalen der Physik | volume =21 | pages =756-780 | year=1906}}
* Einstein, A. "Investigations on the Theory of Brownian Movement". New York: Dover, 1956. ISBN 0-486-60304-0 [http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins_brownian.pdf]
* 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.
* Nelson, Edward, ''Dynamical Theories of Brownian Motion'' (1967) [http://www.math.princeton.edu/~nelson/books.html (PDF version of this out-of-print book, from the author's webpage.)]
* Ruben D. Cohen (1986) “Self Similarity in Brownian Motion and Other Ergodic Phenomena,” ''Journal of Chemical Education 63'', pp. 933-934 [http://rdcohen.50megs.com/BrownianMotion.pdf]
* J. Perrin, Ann. Chem. Phys. '''18''', 1 (1909). See also book "Les Atomes" (1914).
*[[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).
==External links==
*[http://www.ap.stmarys.ca/demos/content/thermodynamics/brownian_motion/brownian_motion.html A page describing Brownian motion.]
*[http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/applet.html Brownian motion java simulation]
*[http://xxx.imsc.res.in/abs/physics/0412132 Article for the school-going child]
*[http://www.bun.kyoto-u.ac.jp/~suchii/einsteinBM.html Einstein on Brownian Motion]
*[http://arxiv.org/abs/0705.1951 Brownian Motion, "Diverse and Undulating"]
[[Category:Stochastic processes]]
[[Category:Fractals]]
[[Category:Statistical mechanics]]
[[Category:Colloidal chemistry]]
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