Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often itself called "Brownian motion", even in mathematical sources.

This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by 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 and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).

This motion is named after the Scottish botanist 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 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision of Henri Poincaré. Then, in 1905, theoretical physicist Albert Einstein published a paper in which he modelled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.

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. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter".

The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to 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 sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).

The Roman philosopher-poet Lucretius' scientific poem On the Nature of Things (c. 60 BC) has a remarkable description of the motion of dust particles in verses 113-140 from Book II. He uses this as a proof of the existence of atoms:

Although the mingling, tumbling motion of dust particles is caused largely by macroscopic air currents and convection, the glittering, microscopic jiggling motion of small dust particles is caused chiefly by true Brownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example".

The formal scientific discovery of this phenomenon is credited to the botanist Robert Brown in 1827. Brown was studying plant reproduction when he observed pollen grains of the plant Clarkia pulchella in water under a simple microscope. These grains contain minute particles on the order of 1/4,000 of an inch (6.4 microns) in size. He observed these particles executing a continuous, jittery motion. By repeating the experiment with particles of inorganic matter, such as glass and rock dust, he was able to rule out that the motion was life-related, although its physical origin was yet to be explained.

The mathematics of much of stochastic analysis, including the mathematics of Brownian motion, was introduced by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented an innovative probabilistic analysis of the stock and option markets. However, this pioneering mathematical work connecting random walks to continuous time was largely unknown until the 1950s.

The early 20th century saw the theoretical formalization of Brownian motion bridging the gap between thermodynamics and atomic theory:

• Albert Einstein (in one of his 1905 papers) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their physical existence was still fiercely debated by scientists. Einstein proved the mathematical relation between the probability distribution of a Brownian particle and the macroscopic diffusion equation.
• These predictive equations describing Brownian motion were subsequently verified by the meticulous experimental work of Jean Baptiste Perrin in 1908, leading to his Nobel prize and settling the atomic debate.
• Norbert Wiener gave the first complete and rigorous mathematical analysis of the phenomenon in 1923, leading to the underlying mathematical concept being permanently called a Wiener process.

Condensed: the full section is in Wikipedia.

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. 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. In accordance to Avogadro's law, this volume is the same for all ideal gases, namely 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro constant or as Avogadro's number (approximately 6.02×10 mol), 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 molar mass of the gas by the Avogadro constant.

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. 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 collisions per second.

Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution: \[\rho(x,t) = \frac{N}{\sqrt{4\pi Dt}} \exp{\left(-\frac{x^2}{4Dt}\right)}.\] This expression (which is a normal distribution with the mean \(\mu = 0\) and variance \(\sigma^2 = 2Dt\) usually called Brownian motion \(B_t\)) allowed Einstein to calculate the 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 \[\mathbb{E}{\left[x^2\right]} = 2 D t.\] 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. 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.

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 the Avogadro 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.

Condensed: the full section is in Wikipedia.

Smoluchowski's theory of Brownian motion, later contextualized in comprehensive reviews of stochastic physics, 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 axis in time t. He therefore gets the same expression for the mean squared displacement: \(\mathbb{E}{\left[(\Delta x)^2\right]}\). However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds \[\mathbb{E}{\left[(\Delta x)^2\right]} = 2Dt = t \frac{32}{81} \frac{mu^2}{\pi \mu a} = t\frac{64}{27} \frac{\frac{1}{2}mu^2}{3 \pi \mu a}\] where μ is the viscosity coefficient, and a is the radius of the particle.

Associating the kinetic energy \(mu^2/2\) with the thermal energy \(RT/N\), the expression for the mean squared displacement is 64/27 times that found by Einstein. This discrepancy arises from differing theoretical approaches: Einstein assumed Stokes drag applied directly to the macroscopic drift velocity, while Smoluchowski performed a more detailed kinematic collision analysis but introduced a slight calculation variance when averaging over the Maxwellian velocity distribution. 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."

Smoluchowski 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 exactly equal. To address this paradox, he relies on the inevitability of statistical fluctuations:

• If the probability of m gains and \(n - m\) losses follows a binomial distribution,

\[P_{m,n} = \binom{n}{m} 2^{-n}\] with equal a priori probabilities of 1/2, the mean total gain is \[\mathbb{E}{\left[2m-n\right]} = \sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^{n+1} \left [ \left (\frac{n}{2} \right )! \right ]^2}\]

• If n is large enough so that Stirling's approximation can be used in the form \(n! \approx \left(\frac{n}{e}\right)^n \sqrt{2\pi n}\), then the expected total absolute gain representing the net drift can be approximated. This expected gain will be:

\[\mathbb{E}{\left[2m - n\right]} \approx \sqrt{\frac{n}{2\pi}}\] showing that the net displacement increases proportionally to the square root of the total population of collision events.

Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Smoluchowski reasons that the mechanics of these interactions produce a macroscopically observable effect:

• The particle collisions are confined to one dimension along a single axis.
• It is equally probable for the test particle to be hit from the left as from the right.
• Every collision always imparts the exact same discrete magnitude of velocity change, \(\Delta V\).

Condensed: the full section is in Wikipedia.

The diffusion equation yields an approximation of the time evolution of the probability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using the Langevin equation, an equation that involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle. At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated using Brownian dynamics in place of Langevin dynamics.

In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity V of the massive object, of mass M, is related to the rms velocity \(v_\star\) of the background stars by \[MV^2 \approx m v_\star^2\] 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. 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.

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 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 four facts:

1. W₀ = 0
2. W_t is almost surely continuous
3. W_t has independent increments
4. \(W_t-W_s\sim \mathcal{N}(0,t-s)\) (for \(0 \leq s \le t\)).

\(\mathcal{N}(\mu, \sigma^2)\) denotes the normal distribution with expected value μ and variance σ. The condition that it has independent increments means that if \(0 \leq s_1 < t_1 \leq s_2 < t_2\) then \(W_{t_1}-W_{s_1}\) and \(W_{t_2}-W_{s_2}\) are independent random variables. In addition, for some filtration \(\mathcal{F}_t\), \(W_t\) is \(\mathcal{F}_t\) measurable for all \(t \geq 0\).

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 with W₀ = 0 and quadratic variation \([W_t, W_t] = t\).

A third characterisation 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 Kosambi-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. A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian motion."

The Brownian motion can be modeled by a random walk.

In the general case, Brownian motion is a Markov process and described by stochastic integral equations.

The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous R-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 R. Then the following are equivalent:

1. 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, ∞); R).
2. 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_ij t is a martingale with respect to P (and its own natural filtration), where δ_ij denotes the Kronecker delta.

The spectral content of a stochastic process \(X_t\) can be found from the power spectral density, formally defined as \[S(\omega)=\lim_{T\to\infty}\frac{1}{T}\mathbb{E}\left\{ \left|\int^T_0 e^{i \omega t} X_t dt \right|^2\right\},\] where \(\mathbb{E}\) stands for the expected value. The power spectral density of Brownian motion is found to be \[S_{BM}(\omega) = \frac{4 D}{\omega^2}.\] where D is the diffusion coefficient of X_t. 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., \[S^{(1)}(\omega,T) = \frac{1}{T}\left|\int^T_0 e^{i \omega t}X_t dt\right|^2 ,\] which for an individual realization of a Brownian motion trajectory, it is found to have expected value \(\mu_{BM}(\omega,T)\) \[\mu_\text{BM}(\omega,T) = \frac{4 D}{\omega^2} \left[1 - \frac{\sin\left(\omega T\right)}{\omega T}\right]\] and variance \(\sigma_\text{BM}^2(\omega,T)\) \[\sigma_S^2(f,T) = \mathbb{E}\left\{\left(S^{(j)}_T(f)\right)^2\right\}-\mu_S^2(f,T) =\frac{20 D^2}{f^4}\left[ 1-\Big(6-\cos\left(f T\right)\Big) \frac{2\sin\left(f T\right)}{5fT} +\frac{\Big(17-\cos\left(2fT\right) - 16\cos\left(f T\right)\Big)}{10 f^2 T^2} \right].\]

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

Brownian motion is usually considered to take place in Euclidean space. It is natural to consider how such motion generalizes to more complex shapes, such as surfaces or higher dimensional manifolds. The formalization requires the space to possess some form of a derivative, as well as a metric, so that a Laplacian can be defined. Both of these are available on Riemannian manifolds.

Riemannian manifolds have the property that geodesics can be described in polar coordinates; that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space.

The infinitesimal generator (and hence characteristic operator) of Brownian motion on Euclidean R is ⁠1/2⁠Δ, where Δ denotes the Laplace operator. Brownian motion on an m-dimensional Riemannian manifold (M, g) can be defined as diffusion on M with the characteristic operator given by ⁠1/2⁠Δ_LB, half the Laplace-Beltrami operator Δ_LB.

One of the topics of study is a characterization of the Poincaré recurrence time for such systems.

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.