Brownian motion
Brownian motion is the random motion of particles suspended in a medium. The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often 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. 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.
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, 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 to Brownian motion.
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 air currents, the glittering, 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 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 microscope. These grains contain minute particles on the order of 1/4000th of an inch in size. He observed these particles 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 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 analysis of the stock and option markets. However this work was largely unknown until the 1950s.
Albert Einstein provided an explanation of Brownian motion in terms of atoms and molecules at a time when their existence was still debated. Einstein proved the relation between the probability distribution of a Brownian particle and the diffusion equation. These equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908, leading to his Nobel prize. Norbert Wiener gave the first complete and rigorous mathematical analysis in 1923, leading to the underlying mathematical concept being called a Wiener process.
The instantaneous velocity of the Brownian motion can be defined as, when, where is the momentum relaxation time.
In 2010, the instantaneous velocity of a Brownian particle was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.
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, 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 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 space as a random variable with some probability density function . Further, assuming conservation of particle number, he expanded the number density 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 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 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, and is the absolute temperature.
File:Brownian motion gamboge.jpg|thumb|Perrin examined the equilibrium of granules of gamboge, a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air.
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.
The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson 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".
An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by Stokes's law. He writes for the diffusion coefficient, where is the osmotic pressure and 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. 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.
Confirming Einstein's formula experimentally proved difficult.
Initial attempts by Theodor Svedberg in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula. Victor Henri in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain. 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 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.