The unravelling of the mystery of Brownian motion a century ago by Einstein brought to an end all debate on whether atoms really existed or they were merely mental constructs.
IMAGINE an enormous pumpkin floating in the vacuum of outer space in the middle of a dense swarm of the tiniest of mustard seeds that ceaselessly bounce randomly off each other and the pumpkin. What kind of motion would the pumpkin exhibit?
This scenario is not just a bit of idle speculation. Scaled down appropriately in space and time, this is a crucial question about how matter comprising an extremely large number of individual atoms behaves under certain conditions. The mustard seed is replaced by a molecule of water, a tiny object about a ten-billionth of a metre in size. (A billion is a thousand million.) The pumpkin could be a grain of pollen, about a millionth of a metre in size, a giant in relative terms. Typically, the mass of a pollen grain would be something like a hundred billion billion times that of a molecule. Examining under a microscope the motion of pollen grains suspended in water, this was how the Scottish botanist Robert Brown, in 1827, made the first systematic experimental study of what has come to be known as Brownian motion. The "ceaseless and chaotic dance" of the pollen grains was a truly astonishing phenomenon. Although the phenomenon had been observed even earlier, it was not clear whether any `life-force' or biological effects were involved. Brown's observations ruled out any such biological origin of the motion.
In the years that followed, it became clear that internal motions in the liquid were somehow responsible for the motion. But there was no satisfactory explanation either for the apparently perpetual motion of the Brownian particle, or for the irregularity of its trajectory. The conundrum was unravelled by the 26-year-old Albert Einstein in his doctoral dissertation and in the seminal papers he wrote on it in 1905, the annus mirabilis of physics. Like the other two major themes (the light quantum and special relativity) upon which Einstein's golden touch fell in this miraculous year, this work, too, was truly the key to a vast empire of knowledge.
Going back to our analogy, the pumpkin does move under the constant buffeting of the mustard seeds, contrary to what we might have guessed at first sight. But surely it would never really get very far, because the seeds hit it randomly from all directions, and the pumpkin is an enormous object compared to the seeds? Wrong again. The almost imperceptible effects of the tiny but numerous and incessant collisions would cause it to tremble and jiggle about in a highly irregular fashion. These jiggles would occur on a hierarchy of scales: in principle, the graph of its path would be extremely jagged, and would remain so even if the resolution used in sketching the path were made finer. When averaged over all possible directions of motion, of course, the different displacements of the pumpkin from its starting point would tend to neutralise each other - because a step in any direction is as likely as an equally long step in the opposite direction. There can therefore be no particular direction, relative to its starting point, in which the pumpkin is more likely to find itself than any other direction, at any given time. But, given sufficient time, the pumpkin might find itself at quite a distance from where it started - in precisely the same way as it is possible to lose a substantial fortune over a period of time by continually placing small "can't-lose-much" bets, as many have found to their chagrin! In fact, it turns out that the average value of the square of the pumpkin's distance from the start increases steadily, being exactly proportional to the time for which we follow its motion. Einstein's deep insight lay in recognising that this average or mean squared distance, rather than the velocity of the particle, was the quantity to be studied and measured in such random motion.
Incredibly enough, in the century that has elapsed since then, it has turned out that the mathematical analysis of this random, irregular motion is the fundamental paradigm for a staggeringly large number of phenomena in subjects ranging from astronomy through economics and meteorology to zoology!
Einstein's own interest in the problem arose from a deep-seated conviction that atoms were real, that their sizes and properties could be experimentally determined, and that their populations in ordinary bits of matter, albeit astronomically large, could be estimated reliably. Using ingenious arguments involving kinetic theory and the dynamics of random molecular motion on the one hand, and heat and thermodynamics on the other, Einstein correctly explained the phenomenon. In the process, he showed how its observation could be used to determine the typical sizes of molecules - about a ten-billionth of a metre, as we have already mentioned. It could also be used to determine their number in a piece of matter under so-called `standard' conditions of temperature and pressure - known as Avogadro's number, this is an astronomically large quantity, about a hundred thousand billion billion. Einstein's predictions were tested and thoroughly vindicated within a few years after they were made. This marked the end of all debate on whether atoms really existed or they were merely convenient mental constructs, once and for all. The famous physicist Richard Feynman once opined that if humanity were to face cataclysmic destruction, while being permitted to pass on a single piece of knowledge to survivors to enable them to build a civilisation afresh, that knowledge would have to be the fact that matter consisted of atoms. In the light of this profoundly insightful remark, the importance of Einstein's breakthrough cannot be overstated.
What is even more remarkable is that Einstein used his uncanny instinct for the physics of the problem to explain Brownian motion quantitatively, carefully avoiding the pitfalls arising from the subtleties of the random processes representing Brownian motion in the strict mathematical sense, even though he was not aware of the relevant rigorous mathematics itself at the time. With the passage of time, it has become possible to gauge Einstein's true strengths with something approaching dispassionate objectivity, on the basis of his total contribution to several areas of physics, such as statistical mechanics, quantum physics, relativity and gravitation. There can be no doubt that he had the most exceptionally deep insight into fundamental concepts in physics such as the role of fluctuations, symmetry, invariance and causality, among others.
How has the analysis of Brownian motion (or a random walk, as a version of it is called) and of its subsequent generalisations contributed to human knowledge? Even a minute part of the remarkably diverse list of applications that have emerged over the past one hundred years is quite astounding. The random processes representing Brownian motion and its off-shoots are relevant to the dynamics of objects as varied as polymers in solution, chemically reacting molecules, neutrons in a nuclear reactor, clouds, sand piles, avalanches, insect swarms, animal herds, and star clusters, to techniques of computation, to fundamental issues in the theory of probability, quantum mechanics and quantum field theory, to the fluctuations of the stock market, and so on, apparently endlessly. It is interesting to note that, in the case of Brownian motion, too, the development of the relevant underlying mathematics actually preceded its incorporation into the realm of physics. As we have said, Einstein himself seems to have been completely unaware of these mathematical results, and so these did not presage Einstein's work by any means. By 1900, fully five years ahead of Einstein, Louis Bachelier had worked out substantial parts of it in his doctoral thesis, in connection with an attempt to model the fluctuating prices of shares in the stock market. Although Bachelier's work did not receive the attention it deserved for some time, it was really the progenitor of many deep results in the theory of probability in the hands of brilliant mathematicians such as Norbert Wiener and A.N. Kolmogorov. (In fact, Brownian motion is also called the Wiener process in the mathematical literature.) A hundred years later, we have come full circle. The application of the theory of probability and random processes to precisely the same class of problems as Bachelier considered, namely, in finance, is a major current preoccupation.
A century after 1905, with the benefit of hindsight, what can we say about Einstein's scientific achievements in 1905, and how do these compare with other stupendous human achievements? One may accept the judgment made by Abraham Pais in his definitive biography of Einstein: "No one before or since has widened the horizons of physics in so short a time as Einstein did in 1905." However, it is almost impossible (and perhaps ultimately irrelevant) to try to make a comparison between the highest peaks of excellence when these are widely separated in time and circumstance. But human interest in records is insatiable, and leads us to ask: Can we identify the most intense and sustained mental effort by a single person leading to the most profound results? A unique answer cannot be given. Newton, Darwin and Einstein, each at the peak of his creativity, would certainly be in the exclusive club that we may accept, as a more meaningful compromise, in place of any single person. When they scaled their respective heights, they changed something forever. Their discoveries represent watershed events for the human race itself, as they separate distinct eras in humankind's understanding of the universe in which it lives, and of its place in it. It is comfortingly salubrious to ponder over the fact that these are watersheds in a far more profound sense than mere political events - however tumultuous the latter may appear to be when they occur, or even in the long run, for that matter. After all, Ozymandias wasn't Archimedes.
V. Balakrishnan, a theoretical physicist, is Professor at the Department of Physics, IIT Madras, Chennai.
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