Duke University
The Zigs and Zags of Brownian Motion
Justin Garfinkle
Math 89s: Math of the Universe
Professor Bray
2/22/16
Brownian motion is the quintessential example of the way that physical science intersects with other fields. Brownian motion moves (in Brownian motion) through botany,physics, finance, poetry and philosophy. Theories that have been derived from Brownian motion have brought many benefits throughout the various fields, but also faced objections and challenges.
Brownian motion is the zig-zag motion of particles observed by Scottish botanist Robert Brown in 1827 when he was studying grains of pollen (Clarkia) suspended in water under a microscope. Brown did not know what caused the motion. He initially believed that the motion was due to the male sex cells of the plant (Nott). This was part of Brown’s effort to distinguish the movement of male parts of plants, which he believed would be active, from the female part of plants. Brown ruled out movement by a life form through repeating his experiment with inorganic matter, however. Brown controlled for various factors that could explain the motion such as water current. He was surprised that there wasobservable movement of inorganic matter in still liquid, including chips of glass, metal, and a small piece of the Sphinx(Nott). Brown concluded that the motion was in everything, whether living, dead, or never alive, as long as the particles were small enough (Nott). Brownian motion became a term to describe the physical phenomenon that minute particles immersed in a fluid will experience random motion. The characteristics of Brownian motion included that a given particle would be equally likely to move in any direction, further motion seemed unrelated to past motion, and the motion never stopped (Britannica). When Brown died in 1858, “Brownian motion” was a well-known term, but the reason for the motion had not yet been discovered (Chowdhury).
Human nature searches for truth through science but challenges that truth through skepticism. Over 100 years after Brown had initially recorded his observation of the “Brownian motion” of pollen particles, his observations were challenged in the zigzag of history. In 1991, D.H. Deutsch published an article that called into question whether it was actually possible that Brown observed Brownian motion (Deutsch). In 1992, through repetition of the experiment with the original microscope, Brownian motion was clearly observed with Clarkia pollen (Ford).
Science builds on itself, which is the key to greater discovery. In 1889, G.L. Gouy observed that Brownian movement appeared more rapid for smaller particles and lower viscosity. Gouy’s observations were confirmed in 1900 by F.M. Exner, who also observed increased motion at elevated temperatures. Exner was the first to make quantitative studies of the dependence of Brownian motion on particle size and temperature(Chowdhury). Soon, Brownian motion became a term used to describe both the physical phenomenon of the random movement of particles and the mathematical models used to describe this physical phenomenon. Brownian motion is a stochastic process because there is a random probability pattern that can be analyzed statistically but not predicted precisely.
In1905,the young German physicistAlbert Einsteintook an interest analyzing why particles would move in the stochastic manner observed by Brown. At the time, Einstein was an unknown patent clerk attending graduate school, yet he published five papers that year that revolutionized the way many people thought about the mechanics of the universe. This explosion of thought in such a short period of time is unmatched and incomparable throughout history, and it has since come to be known as Einstein’s “Annus Mirabilis,” or miracle year(Stachel).
As a general trend in his 1905 publications, Einstein tried to unite two schools of thought that previously were seen as separate or contradicting. His papers on special relativity, for example, united the ideas of space and time to show that everything in the universe, even time, is a matter of perspective. Additionally, his paper on the photoelectric effect provided a statistical analysis to show that light consisting of electromagnetic waves sometimes behave as particles, known as photons, by using an established wave length formula for distribution of heat radiation (Cassidy). Einstein’s paper on photons led him to receive the Nobel Prize in 1921. Given the importance of each of the papers published in 1905, sometimes his analysis of Brownian motion goes under the radar. However, both Einstein’s doctoral thesis that dealt with Brownian motion andhis other paper on the subject were significant scientific contributions. Einstein used Brownian motion to show that matter was discrete and made of atoms, not continuous (Holt). Einstein’s research united atomic and kinetic theory and created a model of random movements applicable to a broad range of scientific areas, even today (Holt).
In the introduction to Einstein’s paperon Brownian motion called “On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat”, Einstein wrote: “It is possible that the motions to be discussed here are identical with the so-called ‘Brownian molecular motion’; however, the data available to me on the latter are so imprecise that I could not form a definite opinion on this matter” (Einstein). Nonetheless, Einstein’s paper on Brownian motion definitely buildson the observations of Brown, Gouy, and Exner. His paper stated that if his predictions were proven correct, it would prove the molecular-kinetic theory of heat and provide proof of the existence of atoms and molecules (Einstein). In fact, Einstein’s formulas for Brownian motion were able to provide support for the existence of molecules and atoms, estimation of the size of molecules and atoms, and evidence for microscopic fluctuations of properties of matter. Einstein’s statistical models were also able to determine the number of particles in one mole of a substance, known as Avogadro’s number. The search for Avogadro’s number “was akin to a search for the Holy Grail among atomists” (Nott, p.41)
Einstein theorized thata small particle in still water, such as the pollen grain observed by Brown, would move randomly in the water due to a perpetual multitude of collisions between the pollen grain and the water molecules. The force of the molecules hitting the particle caused the motion of the particle. The fluctuations in the position and movement of the molecules would not be symmetrical, thus causing an unbalanced force on the visible particle and causing it to move in the random zig-zag motion observed by Brown. The collisions of theparticle with the molecules would move the object in random directions and at random times, depending on the size and placement of the molecules. Einstein was able to establish predictive statistical models that could be used to determine the dimension of these unobservable molecules (Cassidy). In addition, the ability to make these predictions provided proof that the molecules actually exist (Cassidy).
According to the kinetic theory of gases first developed by James Clerk Maxwell in 1860, the average kinetic energy at which the molecules of a substance are moving depends only upon temperature. Since Exner had observed that particles in Brownian motion correlated to temperature, Einstein was able to explore if Brownian motion equations also could be used to corroborate kinetic theory. Since the average kinetic energy depends only on temperature, increases in the average kinetic energy from heat cause the molecules comprising particular matter to become more diffuse and thus move more freely.Einstein’s formula was able to draw on this to predict theanticipated changes in motion patterns of the observable particle as the temperature increased and the particle hit against themore diffuse molecules in the host matter. Specifically, Einstein was able to compute the probability of a particle moving a certain distance in any given direction during a particular time interval based on how diffuse the medium was in which the particle was moving.Einstein’s formula showed the link between the random activity of the observable particle and the diffusion of the many molecules (Chowdhury). This quantitative description of Brownian motion was evidence of the molecular-kinetic theory of heat. In addition, Einstein derived an equation that showed that the rate an object moved in a substance depended on the number of molecules in a mole of fluid in which the object was suspended and, from this, the size of the molecules could be calculated (Koberlein).
In his analysis of Brownian motion, Einstein used osmosis to study the dynamic equilibrium that could be established between opposing forces. Osmosis is the movement of solvent molecules through a semipermeable membrane from higher concentration areas to lower concentration areas in order to equalize the solution. He reasoned that dissolved particles should behave the same with regards to osmosis as suspended particles do in connection with Brownian motion, since they differ only in dimensions. Although this was a valid method to derive certain information that he was trying to establish, it was not necessary to use osmosis to achieve this result (Nott).
Einstein’s paper on Brownian motionprovided formulas to predict movement of observable particles in still liquid that were verified by experiments conducted by French physicist Jean-Baptiste Perrin in 1908. Perrin spent months making his experiment as precise as possible (Nott, p. 43). Unfortunately, Avogadro’s number obtained through the experiments of Perrin initially were not the same as the value that had been obtained by Einstein(Straumann). In 1910, Einstein wrote to his former student, Ludwig Hopf, and asked for him to check Einstein’s work for calculation errors. In 1911, Einstein was able to publish correction of his work based on an error discovered by Hopf, and also to communicate the result to Perrin. The statistical formula developed by Einstein was now consistent with the experimental data obtained by Perrin (Straumann, p.11). Perrin’s work showed that atoms and molecules exist as actual physical entities (Britannica). In 1926, Perrin received the Nobel Prize for Physics for his work in verifying Einstein’s analysis.
The importance of Einstein’s work on Brownian motion should be considered in its historical context. When Einstein wrote his paper on Brownian motion in 1905, it was not settled science that molecules existed. Before Einstein, nobody had found a way to measure or observe molecules in any way, which thus left their existence up for debate. In 1897, Austrian physicist Ernst Mach had stated that he did not believe that atoms exist because there was no way to observe or measure them (Yourgrau). Russian-German chemist Wilhelm Ostwald, a highly respected scientist at that time, argued for the theory of energeticism, that the world was made up solely of energy, and not atoms. He claimed that atoms could not exist because it contradicted the second law of thermodynamics presented by Sir Isaac Newton. (Cassidy). This law states that natural processes are irreversible, but physical shifts in atomic alignment should have been able to go in both directions. Australian physicist Ludwig Boltzmann tried to reconcile the existence of atoms with the law of thermodynamics by theorizing that it was possible for a natural process such as melting to reverse, just incredibly unlikely. Those opposed to atomism still were not convinced, though, as there still was no tangible proof that atoms existed. However, following the verification of Einstein’s theory of Brownian motion, which reconciled these discrepancies,most of the doubters in the scientific community were forced to admit that they had been wrong (Blackmore). It was the “precision of Einstein’s results” that convinced the scientific community of the true existence of atoms (Koberlein). In his “Autobiographical Notes” in 1949, Einstein wrote “The antipathy of these scholars toward atomic theory can indubitably be traced back to their positivistic philosophical attitude. This is an interesting example of the fact that even scholars of audacious spirit and fine instinct can be obstructed in the interpretation of facts by philosophical prejudices” (Einstein “Autobiographical Notes”). Einstein was sending a message that in the zigzag of the development of collective human knowledge, scientists should not stop at mere sensory experience in the search for greater truth.
Brownian motion continues to be studied by the scientific community today. It provides “versatility in explaining a wide range of biological processes that occur at the molecular level” (Lyshevski). In addition, engineered nanoscale objects can be propelled by Brownian motion in the development of technological advancements (Drexler). Brownian motion is also studied in connection with controlling the intensity of noise. Experiments have shown that the signal-to-noise ratio vanishes in the absence of noise, rises with the increase of noise, but then decreases with a further increase in noise because of the randomization caused by the noise. This means that noise at a certain level, known as Brown noise, can have a constructive effect of enhancing a signal over a lower noise intensity. This has application in electrical engineering (Chowdhury). Optical microscopy is also “a wonderland dominated by Brownian motion” (Chowdhury, p.14).
Although Einstein made a significant contribution in connection with Brownian motion that is still relevant today, parts of his theory have beenrefined by modern scientists. Diffusion is critical to many processes such as water purification and drug delivery so it continues to be studied (Kloeppel). Einstein’s paper on Brownian motionincluded a graph showing the probability of a particle moving a particular distance in a given direction over a specific time interval would result in a typical “bell” (or Gaussian) curve(Britannica). According to Steve Garnick, a Professor of Engineering at University of Illinois, “Einstein had it right – almost” (Kloeppel). According to Professor Garnick, with the ability to measure small distances more precisely, “we have found that we can have extremes much farther than previously imagined” (Kloeppel quoting Garnick). Tracking motion of 100-nanometer colloidal beads moving by Brownian motion in tiny tubes of lipid molecules, the researchers found that at the extremes, the movement was “exponential” rather than Gussian (Kloeppel). This recent discovery changes expectations regarding diffusion and will enable adjustments in calculations where needed.
The idea of Brownian motion is not limited to the physical sciences. Five years before Einstein applied the stochastic process to thermodynamics, Louis Bachelier applied it in the area of finance (Holt). In 1900, the Ph.D. thesis of French mathematician Louis Bachelier called "Théorie de la Spéculation" (Annales de l'Ecolenormalesuperiure) (“Theory of Speculation”) applied the concept of Brownian motion to pricing for stocks and bonds. Bachelier’s work was the precursor to mathematical finance. Although Bachelier wrote his thesis five years before Einstein’s paper on Brownian motion, he also drew on the physical diffusion of heat as an analogy for his financial model. Bachelier explored why speculative assets such as stocks and bonds moved in certain ways by studying the movement of bond prices on the Paris Bourse. In applying the movement of bond prices to Brownian motion, the bonds were Brown’s pollen particles and the speculators were the molecules. In a dense substance, a particle will not move because all of the molecules hitting it on all sides with equal force. If there is dense knowledge of future stock movement (such as knowledge that a price of a stock is expected to spike at a certain time in the future), then speculators trading on the knowledge will smooth out the spike. This is now known as the efficient market hypothesis. Actual fluctuations will be based on factors that are not knowable from past history, and will therefore appear as the zigzag random movement of the pollen particle in Brownian motion. Although the price movement of a given stock or bond on a given day is random and thus impossible to predict, Bachelier charted the price movements over time in the classic bell curve (Holt). As discussed above, Einstein also used a bell curve to show the predictable randomness of small physical particle fluctuations in still liquid. Based on the financial collapse of the stock and option markets in 2008, it seems that financial markets do not follow the bell curve at the extremes any more than the University of Illinois team found to be the case for very small particles in still liquid.
In 1973, Fischer Black and Myron Scholes built upon the work of Bachelier in their paper called "The Pricing of Options and Corporate Liabilities", published in theJournal of Political Economy. Their paper put forward a hedging strategy that would eliminate the risk of buying an option by trading in the underlying asset in a particular way. Robert Merton published a paper expanding on the mathematical model for options pricing. In 1997, Merton and Scholes received the Nobel Memorial Prize in Economic Sciences for the option pricing model. These models are now used broadly for derivative pricing and risk management. The models operate within the context of the future volatility of the underlying asset, which cannot be known when the option is being priced. One commentator has said that the Black-Scholes equation “became the Black Hole equation, sucking money out of the universe in an unending stream” when the sub-prime mortgage market collapsed (Stewart).