Brownian motion

On the atomic scale, all matter is characterized by incessant motion even at the lowest temperatures. One consequence of this motion is the random "jiggling" of microscopic particles suspended in a liquid, first reported by Robert Brown. The statistical properties of Brownian motion provide important information about the distribution of molecular speeds.

Overview

Brownian motion is the name given to the unceasing random motion of microscopic particles suspended in a liquid or gas. Although the trajectories of the individual particles exhibiting this random motion are by definition unpredictable, the statistical properties of Brownian motion are very well defined. The interpretation of Brownian motion as resulting from the collision of these particles with atoms or molecules of the surrounding medium and the resulting quantitative explanation of the statistical properties of Brownian motion played an important role in establishing the modern theory of the atomic structure of matter.

One of the most striking claims of the atomic theory of matter is that the atoms or molecules that make up any piece of matter are in a state of constant random motion, with the amount of motion, as measured by the average molecular speed, depending on the temperature.

At any instant, any object much larger than a simple molecule will be in collision with numerous molecules of the surrounding fluid, which together will exert on the object a net force that varies rapidly in both magnitude and direction. For objects big enough to be seen by the naked eye, the effect of this random force is too small to be seen. For objects the size of a pollen grain or a bacterial cell, however, the effect of the random force is an incessant "jiggling motion" visible under a good light microscope.

The effect that has come to be known as Brownian motion was first noted in 1827 by the English botanist, Robert Brown, who was studying pollen grains under a newly improved microscope. Although the effect was first observed in matter with an organic origin, subsequent observations by Brown and others of the same behavior in a wide range of materials proved that the motion had nothing to do with the origin of the material in a living organism and was rather a universal characteristic of the behavior of matter on a microscopic scale.

While the overall path of any one particle executing Brownian motion will not necessarily look like that of any other particle executing Brownian motion under the same conditions, study of a great number of paths reveals a number of important statistical properties.

For example, if one were to record the distance a Brownian particle had moved from its starting point at the end of one second, two seconds, three seconds, and so on, and then repeat this sequence of observations for many identical particles, one would find that the average distance of the collection of particles from their starting points--taking both distance and direction into account--remained zero. This result is not particularly surprising, however, since the direction of the net force exerted on the particle at any instant is equally likely to lie in any direction. If, instead, one takes the average of the squares of the displacements of the particles from their starting points--the mean square displacement--one finds that the mean square displacement increases in direct proportion to the time elapsed. The square root of this sum of squares is termed the "root-mean-square displacement" and, since it is the square root of the sum of positive numbers, it will be a positive number, unless all the particle displacements are zero at the same time. The root-mean-square displacement of a particle undergoing Brownian motion is proportional to the square root of the elapsed time. Thus, if particles of a particular size in a particular fluid are found to achieve a root-mean-square displacement of 1 micron after one second, it will require four seconds to find a root-mean-square displacement of 2 microns and nine seconds to find a root-mean-square displacement of 3 microns.

The kinetic energy, or energy of motion, of a particle is equal to one-half the mass of the particle times the square of its speed. One of the most important discoveries of statistical mechanics, the branch of physics that studies the behavior of systems of large numbers of particles, is that the absolute temperature of a substance is a direct measure of the average kinetic energy of the atoms or molecules of which it is composed. The fact that this is so even for a mixture of molecules of different sizes is an illustration of the principle of equipartition of energy. The average kinetic energy of translational motion for particles in a fluid is proportional to three-halves of the absolute temperature times the Boltzmann constant, a constant that appears very small when expressed in standard metric units, fourteen trillionths of a trillionth of a joule per degree on the absolute temperature scale. In almost all important respects, the particle executing Brownian motion can be treated as one extremely large "molecule" in thermal equilibrium with the molecules of the surrounding liquid or gas. It will have the same average kinetic energy, although since its mass is so much greater than that of the surrounding molecules, its mean square velocity will be correspondingly less.

Since it is the velocity of the molecules that collide with the Brownian particle that determine the changes in its motion, it is not surprising that the mean square displacement of the particle is proportional to the absolute temperature. In fact, it has been found that the mean square displacement for a Brownian particle can be described by a simple formula that involves only Boltzmann's constant, the absolute temperature, the elapsed time, and a single number describing the "drag" of the fluid on the particle.

In addition to its importance as a physical process, Brownian motion has provided a special challenge to mathematicians. Brownian motion is an example of the type of process called a "random walk" in the theory of probability, which also describes many of the characteristics of games of chance. The geometrical description of the path of a Brownian particle is indeed complex. If one takes a motion picture of a particle undergoing Brownian motion, one obtains on film a sequence of "snapshots" of the particle's position a small fraction of a second apart. When this film is projected on a screen, the position of the particle is seen to jump from place to place. The viewer might imagine that the particle moving along straight line segments between the points on the screen is being seen. The more advanced viewer might guess that the particle is actually moving along a smooth, but very complex curve between the individual snapshots, but this would be incorrect. Taking a motion picture with a high-speed camera (many more frames per second) yields a film that when projected in a slow motion mode (fewer frames per second) displays the same sort of disconnected motion as the earlier film.

Indeed, it is found that there is no smoothness to the path of a Brownian particle on any practical timescale. Rather, the motion is, in a statistical sense, "self-similar," in that motion pictures of the motion taken at different speeds and at magnifications inversely proportional to the square root of the time between frames could not be distinguished when projected at a common speed.

The paths taken by particles executing Brownian motion are examples of a class of geometrical objects for which Benoit B. Mandelbrot coined the term "fractal" in the 1970's.

Fractals are sets of points in space that, in a sense, are best described as having a geometric dimension that is not a whole number. In elementary geometry, points have a dimension of zero, lines and line segments a dimension of one, planes a dimension of two, and solids a dimension of three. Combining a few objects of the same dimension does not result in a different dimension for the object. Thus, a combination of line segments, or a line with several kinks in it, is still a one-dimensional object. Questions arise, however, when a line has an infinite number of kinks in it, since such an object can be made to include almost all the points on a two-dimensional surface. It is possible to define a type of dimension, the fractal dimension, which is one, two, or three for the more familiar geometrical objects but is greater than one and less than two for this nearly space-filling curve. Many of the objects described by a fractional dimension in fractal geometry are self-similar in the same sense as Brownian paths. Although a Brownian path is one-dimensional in the conventional geometric sense, it will have a fractal dimension between one and two.

Applications

The discovery of Brownian motion is important in the history of science as providing direct experimental evidence that all matter consists of atoms and molecules and that these constituent particles are in constant motion. It has also provided a model for understanding other consequences of random motion on the atomic scale. One of the most important applications of the basic theory developed to understand Brownian motion was its use by the French physicist Jean-Baptiste Perrin to obtain the first experimental value of Avogadro's number.

One of the great discoveries of nineteenth century chemistry was the realization, now known as Avogadro's law, that equal volumes of pure gaseous substances at the same temperature and pressure contained the same number of molecules. Under standard conditions of temperature and pressure (that is, at 0 degrees Celsius and normal atmospheric pressure) 1 gram molecular weight, or "mole," of any gaseous substance occupies a volume of 22.4 liters. The number of molecules in this volume was termed Avogadro's number, although there was no obvious way to determine the appropriate numerical value. It was also discovered that at moderate temperatures and pressures, the volume occupied by 1 mole of gas responded to changes in temperature and pressure in essentially the same way. The resulting "ideal gas law" stated that for any gas, the product of the pressure and volume divided by the product of the number of moles and the volume and the absolute temperature is a constant (given in modern units as 8.3 joules per mole per degree). The kinetic theory of gases, which interpreted the pressure of a gas as resulting from the collisions of the gas molecules with the walls of the vessel containing the gas, required that the constant be equal to Boltzmann's constant times Avogadro's number, so that once Boltzmann's constant could be known, Avogadro's number could be calculated easily.

It had long been known that atmospheric pressure decreased with increasing altitude.

One of the simplest results of statistical mechanics was an equation relating the decrease in pressure with height for a column of pure gas to the ratio of the weight of the individual molecules to the product of Boltzmann's constant and the absolute temperature. Perrin's insight was that since the studies of Brownian motion showed that Brownian particles could be treated exactly as very large molecules in regard to their collisions with other molecules, the same formula would apply to their distribution in a column of liquid. Perrin studied the distribution of small particles of a natural rubber derivative in water and made a separate determination of their weight, correcting for the effect of fluid buoyancy. Since he could also measure the temperature of the water, the only unknown in the theoretical formula was Boltzmann's constant, which Perrin then calculated. On taking the ratio of the gas constant to Boltzmann's, Perrin thus obtained a value about 10 percent larger than the currently accepted value of 602 billion trillion.

The generation of Johnson (thermal) noise in electrical circuits is in many ways analogous to Brownian motion. Metals and semiconductors are able to conduct electricity because they include a number of electrons that are relatively free to move through the material.

These electrons will engage in collisions with one another and with the atoms making up the framework of the material. They will thus have an average kinetic energy proportional to the absolute temperature and will be moving in random directions. In a simple device such as a resistor, there will at any instant be a slight excess of electrons near one terminal and a slight deficiency near the other, resulting in a small and rapidly fluctuating voltage. The mean square voltage is proportional to the absolute temperature. The existence of this thermal noise is one reason why receivers used to detect very weak signals, such as those in radio telescopes, must be maintained at very low temperatures, a few degrees above absolute zero at the most.

Context

Although the hypothesis that all matter is composed of atoms dates back to the ancient Greeks, the formulation of the atomic theory as an integral part of a scientific world view dates back to the seventeenth century. Robert Boyle, an English physicist, was perhaps the first individual to employ the atomic hypothesis in the interpretation of experiments. In his New Experiments, Physico-Mechanical, Touching the Spring of the Air and Its Effects (1660), he concludes (correctly) that air is composed of several different types of particles and (incorrectly) that the resistance of air to compression arises in the forces of contact between the particles.

Shortly thereafter, Sir Isaac Newton employed an atomic model to describe the propagation of sound through air and advanced a number of explanations for optical phenomena based on the idea of light rays as streams of particles.

The chemical consequences of the atomic composition of matter did not, however, receive extensive study until the beginning of the nineteenth century, when the English physicist John Dalton began to study the combining weights of different elements and the French physicist and chemist Joseph-Louis Gay-Lussac and the Italian physicist and chemist Amedeo Avogadro published their observations about the volumes of gases entering into chemical combination.

Brown's first observation of Brownian motion was formulated in 1827. Theoretical understanding of atomic and molecular motion was lacking at this time, however, and almost eighty years had to elapse before the characteristics of Brownian motion could be adequately explained.

The latter part of the nineteenth century saw the development of a quantitative theory of atomic motion, beginning with the demonstration by the Englishman James Prescott Joule in 1845 that a fixed amount of mechanical energy produced a fixed increase in temperature of a body of water. By the beginning of the twentieth century, the English physicist James Clerk Maxwell had determined the detailed distribution of molecular speeds in a gas, the German physicist Ludwig Boltzmann had developed a probabilistic and molecular interpretation of entropy, and the American physicist Josiah Willard Gibbs had established general statistical methods for calculating the properties of bulk matter from the properties of the component molecules.

It may therefore be difficult to believe that the hypothesis that all matter is composed of atoms was open to question in the first years of the twentieth century, but such was the case. The German physicist and philosopher Ernst Mach had objected to the atom as the unnecessary introduction of a model into physical theory and even some chemists, including the German chemist Wilhelm Ostwald, who later won the 1909 Nobel Prize in chemistry, were reluctant to accept the idea that atoms were to be found in all states of matter. It remained for a series of papers by the German physicist Albert Einstein, in 1905 and 1906, that demonstrated how the statistical properties of Brownian motion could be calculated from the properties of the surrounding liquid to eliminate the last remnants of resistance to the atomic idea.

Principal terms

ABSOLUTE TEMPERATURE: the number of Celsius degrees above the absolute zero of temperature

AVOGADRO'S NUMBER: the number of molecules in 1 gram molecular weight, or mole, of a substance

BOLTZMANN'S CONSTANT: a physical constant that relates the average energy of motion of molecules to the absolute temperature

FRACTAL: a geometic object that can be described as having a dimension not equal to a whole number

IDEAL GAS LAW: the quantitative relationship between pressure, volume, temperature, and number of moles that is nearly obeyed by all normal gaseous substances under normal conditions

KINETIC ENERGY: the energy of motion of a particle or other object

KINETIC THEORY: the attempt to relate the physical properties of matter in bulk to the motion of the constituent atoms or molecules

MEAN: the arithmetic average of a series of numbers or measurements

MOLE: the amount of a pure chemical substance obtained by adding together the atomic weights in grams of the constituent elements

ROOT-MEAN-SQUARE: the square root of the arithmetic average of the squared values of a series of numbers or measurements

Bibliography

Amaldi, Ginestra. THE NATURE OF MATTER. Chicago: University of Chicago Press, 1961. This relatively brief and very readable volume covers the development of atomic theory from ancient Greek speculation to the subatomic zoo of the early 1960's. Brownian motion is discussed in relation to the determination of molecular masses and the kinetic theory of matter.

Boorse, Henry A., and Lloyd Motz. THE WORLD OF THE ATOM. New York: Basic Books, 1966. This two-volume compilation of landmark papers in the history of atomic theory includes both excerpts from Brown's original report on Brownian motion and Perrin's calculation of Avogadro's number. The latter is presented as a model of clarity in scientific writing.

Einstein, Albert, and Leopold Infeld. THE EVOLUTION OF PHYSICS. New York: Simon & Schuster, 1951. This totally nonmathematical overview of the development of physics by one of the most distinguished physicists of the twentieth century and a noted collaborator is accessible to readers with no special scientific training. Brownian motion is discussed as one of the last successes of the mechanical description of microscopic phenomena.

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. 3 vols. Reading, Mass.: Addison-Wesley, 1965. This comprehensive set of lectures by one of the leading theoretical physicists of the mid-twentieth century is an attempt to convey both modern and classical physics to beginning university students. Lecture 41 uses Brownian motion to introduce the idea of statistical distribution of energy and discusses several applications of this concept.

Mandelbrot, Benoit B. THE FRACTAL GEOMETRY OF NATURE. New York: W. H. Freeman, 1983. This extensively illustrated volume provides numerous examples of objects exhibiting the characteristics of self-similarity and fractional dimension. Brownian paths are discussed in detail at several points throughout this book.

Toulmin, Stephen, and June Goodfield. THE ARCHITECTURE OF MATTER. New York: Harper & Row, 1962. This entertaining historical study discusses both the atomic theory and the cell theory of living matter. Nearly half the book is devoted to the philosophical speculations and unsystematic discoveries of the period before Boyle.

Fractal

Chemical Formulas and Combinations

Diffusion in Gases and Liquids

The Behavior of Gases

The Atomic Structure of Liquids