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).Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. 43–48, 73–74. ISBN 0-08-016747-0.
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).
History
The Roman philosopher-poet Lucretius' scientific poem On the Nature of Things () 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 (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.
The instantaneous velocity of the Brownian motion can be defined as , when , where is the momentum relaxation time. Advancements in modern physics have allowed this to be directly measured: In 2010, the instantaneous velocity of a single Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully for the first time. The velocity data perfectly verified the Maxwell–Boltzmann velocity distribution and confirmed the equipartition theorem for a Brownian particle at microscopic timescales.
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. 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 22.414liters 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 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 1014 collisions per second.
He regarded the increment of particle positions in time 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 () with some probability density function (i.e., is the probability density for a jump of magnitude , i.e., the probability density of the particle incrementing its position from to in the time interval ). Further, assuming conservation of particle number, he expanded the number density (number of particles per unit volume around ) at time in a Taylor series,
where the second equality is by definition of . 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:
Where the coefficient after the Laplacian, the second moment of probability of displacement , is interpreted as mass diffusivity D:
Then the density of Brownian particles at point at time satisfies the diffusion equation:
Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution:
This expression (which is a normal distribution with the mean and variance usually called Brownian motion ) 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
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.
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 , where is the mass of the particle, is the acceleration due to gravity, and is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius is , where 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 distribution
where is the difference in density of particles separated by a height difference, of , is the Boltzmann constant (the ratio of the universal gas constant, , to the Avogadro constant, ), and is the absolute temperature.
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 law,
where . Introducing the formula for , we find that
In a state of dynamical equilibrium, this speed must also be equal to . Both expressions for are proportional to , reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical charged particles of charge in a uniform electric field of magnitude , where is replaced with the electrostatic force . Equating these two expressions yields the Einstein relation for the diffusivity, independent of or or other such forces:
Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as , 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 , the temperature , the viscosity , and the particle radius , the Avogadro constant can be determined.