Brownian motion

Brownian motion (named after the Scottish botanist Robert Brown) is the seemingly random movement of particles suspended in a fluid (i.e. a liquid such as water or air) 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. However, movements in share prices may arise due to unforeseen events which do not repeat themselves, and physical and economic phenomena are not comparable.^{[citation needed]}

Brownian motion is among the simplest of the continuous-time stochastic (or random) processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This 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.^{[clarification needed]}

History

Reproduced from the book of Jean Baptiste Perrin, *Les Atomes*, three tracings of the motion of colloidal particles of radius 0.53 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 µm).^[1]

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 alcohol in 1785. Nevertheless Brownian motion is traditionally regarded as discovered by the 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 (in his 1905 paper) and Marian Smoluchowski (1906) who independently 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.

Specifically, Einstein predicted that Brownian motion of a particle in a fluid at a thermodynamic temperature T is characterized by a diffusion coefficient $D=k_{B}T/b$ , where k_B is Boltzmann's constant and b is the linear drag coefficient on the particle (in the Stokes/low-Reynolds regime applicable for small particles).^[2] As a consequence, the root mean square displacement in any direction after a time t is ${\sqrt {2Dt}}$ .^[2]

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.^[why?] But Einstein's predictions were finally confirmed in a series of experiments carried out by Chaidesaigues in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the 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 as being an essentially statistical law.^[3]

Intuitive metaphor

Consider a large balloon of 10 meters in diameter. Imagine this large balloon in a football stadium. 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 towards the left and the right are imbalanced in favor of the left; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion of the balloon. If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.

Considering Brown's pollen particle moving randomly in water: we know that a water molecule is about 0.1 by 0.2 nm in size, whereas a pollen particle is roughly 25 µm in diameter, some 250,000 times larger. So the pollen particle may be likened to the balloon, and the water molecules to the fans except that in this case the balloon is surrounded by fans. The Brownian motion of a particle in a liquid is thus due to the instantaneous imbalance in the combined forces exerted by collisions of the particle with the much smaller liquid molecules (which are in random thermal motion) surrounding it.

An animation of the Brownian motion concept is available as a Java applet.

Theory

1D Smoluchowski model

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.^[4] The model assumes collisions with M ≫ m where M is the test particles mass and m the mass of one of the individual particles composing the fluid. If initially the test particle has velocity V and the fluid particle velocity v then after one collision the test particle's velocity will have increased by roughly $\Delta V\approx (m/M)v$ ^[5]. It is assumed that the particle collisions are confined to one dimension and that its equally probable for the test particle to be hit from the left as from the right. Its also assumed that every collision always imparts the same magnitude of $\Delta V$ . If $N_{R}$ is the number of collisions from the right and $N_{L}$ the number of collisions from the left then after N collisions the particle's velocity will have changed by $\Delta V(2N_{R}-N)$ . The multiplicity is then simply given by:

{\frac {N!}{N_{R}!(N-N_{R})!}}

and the total number of possible states is given by $2^{N}$ . Therefore the probability of the particle being hit from the right $N_{R}$ times is:

P_{N}(N_{R})={\frac {N!}{2^{N}N_{R}!(N-N_{R})!}}

From this probability certain statistical properties can be ascertained. One property that is often of interest is the average total velocity change given by:

\Delta V\langle |2N_{R}-N|\rangle =\sum _{N_{R}={\frac {N}{2}}}^{N}2\,\Delta V(2N_{R}-N)P_{N}(N_{R})={\frac {\Delta VNN!}{2^{N}\left(\left({\frac {N}{2}}\right)!\right)^{2}}}

and in the limit of large N:

\Delta V\langle |2N_{R}-N|\rangle \approx \Delta V{\sqrt {\frac {2N}{\pi }}}.

Unfortunately 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 cannot be made. For example, the assumption that on average there 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 $\Delta V$ s instead of always just one in a realistic situation.

Modelling using differential equations

The equations governing Brownian motion relate slightly differently to each of the two definitions of Brownian motion given at the start of this article.

Mathematics

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 processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics.

The Wiener process $W_{t}$ is characterized by three facts:

$\ W_{0}=0$
$\ W_{t}$ is almost surely continuous
$\ W_{t}$ has independent increments with distribution $W_{t}-W_{s}\sim {\mathcal {N}}(0,t-s)$ (for $0\leq s\leq t$ ).

${\mathcal {N}}(\mu ,\sigma ^{2})$ denotes the normal distribution with expected value μ and variance σ². The condition that it has independent increments means that if $0\leq s_{1}\leq t_{1}\leq s_{2}\leq t_{2}$ then $W_{t_{1}}-W_{s_{1}}$ and $W_{t_{2}}-W_{s_{2}}$ 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 with $W_{0}=0$ and quadratic variation $[W_{t},W_{t}]=t$ .

A third characterization is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ${\mathcal {N}}(0,1)$ 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 of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is 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, inertial 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^{[clarification needed]}, 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^{[clarification needed]}.

Physics

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.

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 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.

Lévy characterization

The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous Rⁿ-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 = (X₁, ..., X_n) be a continuous stochastic process on a probability space (Ω, Σ, P) taking values in Rⁿ. 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_∗(P) is classical Wiener measure on C₀([0, +∞); Rⁿ).
both
1. X is a martingale with respect to P (and its own natural filtration); and
2. for all 1 ≤ i, j ≤ n, X_i(t)X_j(t) −δ_ijt is a martingale with respect to P (and its own natural filtration), where δ_ij denotes the Kronecker delta.

Riemannian manifold

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 (and hence characteristic operator) of a Brownian motion on Rⁿ 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 ${\mathcal {A}}$ in local coordinates x_i, 1 ≤ i ≤ m, is given by ½Δ_LB, where Δ_LB is the Laplace–Beltrami operator given in local coordinates by

\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),

where [g^ij] = [g_ij]⁻¹ in the sense of the inverse of a square matrix.

Gravitational motion

In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars.^[6] The rms velocity $V$ of the massive object, of mass $M$ , is related to the rms velocity $v_{\star }$ of the background stars by

MV^{2}\approx mv_{\star }^{2}

where $m\ll M$ 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 $v_{\star }$ and $V$ .^[6] The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s⁻¹.^[7]

Popular culture

Science fiction author Douglas Adams raised the visibility of Brownian Motion in popular culture when he included it in his description of a Finite Improbability Generator in his work The Hitchhiker's Guide to the Galaxy. Said generator required a source of truly random motion and this was provided by an atomic vector plotter suspended in "a nice hot cup of tea".

Fantasy author Terry Pratchett references Brownian motion in several of his Discworld books

Notes

^ Perrin, 1914, p. 115
^ ^a ^b S. Chandrasekhar, "Stochastic problems in physics and astronomy," Reviews of Modern Physics vol. 15, pp. 1–89 (1943).
^ See P. Clark 1976 for this whole paragraph
^ Smoluchowski, M. (1906), "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen", Annalen der Physik, 326 (14): 756–780, doi:10.1002/andp.19063261405
^ This follows simply from conservation of momentum and energy.
^ ^a ^b Merritt, D.; Berczik, P.; Laun, F. (February 2007), "Brownian motion of black holes in dense nuclei", The Astronomical Journal, 133: 553–563 {{citation}}: Unknown parameter |month= ignored (help)
^ Reid, M. J.; Brunthaler, A. (December 2004), "The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A*", The Astrophysical Journal, 616: 872–884 {{citation}}: Unknown parameter |month= ignored (help)

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. (PDF version of original paper including a subsequent defense by Brown of his original observations, Additional remarks on active molecules.)
Chaudesaigues, M. (1908) 'Le mouvement brownien et le formula d'Einstein' Comptes Rendus, 147 pp 1044–6
Clark, P. (1976) 'Atomism versus thermodynamics' in Method and appraisal in the physical sciences, Colin Howson (Ed), Cambridge University Press 1976
Einstein, A. (1905), "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.", Annalen der Physik, 17: 549–560 {{citation}}: External link in |title= (help)
Einstein, A. "Investigations on the Theory of Brownian Movement". New York: Dover, 1956. ISBN 0-486-60304-0 [1]
Henri, V(1908) 'Etudes cinematographique du mouvement brownien' Comptes Rendus 146 pp 1024–6
Lucretius, 'On The Nature of Things.', translated by William Ellery Leonard. (on-line version, from Project Gutenberg. see the heading 'Atomic Motions'; this translation differs slightly from the one quoted).
Nelson, Edward, Dynamical Theories of Brownian Motion (1967) (PDF version of this out-of-print book, from the author's webpage.)
J. Perrin, "Mouvement brownien et réalité moléculaire". Ann. Chim. Phys. 8ième série 18, 5–114 (1909). See also Perrin's book "Les Atomes" (1914).
Ruben D. Cohen (1986) "Self Similarity in Brownian Motion and Other Ergodic Phenomena", Journal of Chemical Education 63, pp. 933–934 [2]
Smoluchowski, M. (1906), "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen", Annalen der Physik, 21: 756–780 {{citation}}: External link in |title= (help)
Svedberg, T. Studien zur Lehre von den kolloiden Losungen 1907
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

[1] Perrin, 1914, p. 115

[Chandrasekhar43-2] S. Chandrasekhar, "Stochastic problems in physics and astronomy," Reviews of Modern Physics vol. 15, pp. 1–89 (1943).

[3] See P. Clark 1976 for this whole paragraph

[4] Smoluchowski, M. (1906), "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen", Annalen der Physik, 326 (14): 756–780, doi:10.1002/andp.19063261405

[5] This follows simply from conservation of momentum and energy.

[Merritt-6] Merritt, D.; Berczik, P.; Laun, F. (February 2007), "Brownian motion of black holes in dense nuclei", The Astronomical Journal, 133: 553–563 {{citation}}: Unknown parameter |month= ignored (help)

[Reid-7] Reid, M. J.; Brunthaler, A. (December 2004), "The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A*", The Astrophysical Journal, 616: 872–884 {{citation}}: Unknown parameter |month= ignored (help)

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