Uncertainty

by David Lindley

  But Brown distrusted vague hypotheses of this sort. Observation, not speculation, was his forte. Testing pollen from other plants, he found that those grains danced about too. But then he examined minute fragments of leaves and stems and saw that those also jogged perplexingly about under his microscope’s gaze.

  His attention caught by this “very unexpected fact of seeming vitality,” Brown could not help but probe the matter further. He obtained dust from dried plant samples, some more than a century old. He scraped tiny fragments from a piece of petrified wood. All these tiny grains had once been living but were now long dead, any vital spark extinguished. Examined under a microscope, they also shimmied. He moved on to truly inorganic material, knocking tiny shards from a variety of rocks and a piece of ordinary window glass. They too jiggled about. To put the case to the ultimate test, he scratched powder from a piece of the Sphinx, to which, as a curator of the British Museum, he had easy access and which he presumably regarded as certifiably, unarguably dead, on account of its provenance.

  Placed in a drop of water under his microscope lens, ancient dust from the Sphinx danced about like everything else.

  Brown acknowledged that he was not the first to see things jiggling about under the microscope. A certain Mr. Bywater of Liverpool, he noted, had a few years earlier looked at fragments of both organic and inorganic materials and observed “the animated and irritable particles” that they all shed. But Brown, through a variety of ingenious and careful experiments, established that the ceaseless motion of all these tiny fragments was neither the “animalcular motion” Leeuwenhoek and others had seen nor movement produced by vibration or turbulence of the fluid suspensions, by the action of heat, or through electric or magnetic influences.

  This was contradictory and baffling. Dead particles of dust clearly couldn’t move of their own volition, nor was any external influence pushing them around. Yet move they all too plainly did. Brown himself made no attempt at an explanation. He was a careful descriptive botanist, not a philosopher of nature, and as Charles Darwin said, “much died with him, owing to his excessive fear of never making a mistake.”

  Faced with this impossible dilemma, science took the prudent course and ignored Brownian motion for decades. Its deep significance went unnoticed because the phenomenon was so far beyond scientific apprehension. There was no way even to begin to grasp what it meant. Anyone who used a microscope knew about Brownian motion, at least as a great nuisance, but few read carefully what Brown himself had said about it. Most botanists and zoologists persisted in the idea that it was a manifestation of vital spirits, conveniently ignoring Brown’s demonstration that inert particles jiggled about just as much. Or else they decided their specimens were buffeted by heat or vibrations or electrical disturbances, ignoring Brown’s experiments that ruled out those and other influences.

  It was not until after Brown’s death in 1858 that a few scientists began to see their way to an understanding of the phenomenon. As often happens in science, the observations could not be understood until there was at least the glimmering of a theory by which to understand them. The theory in this case was not a new idea but a very old one that science finally had the means to make sense of.

  The Greek thinker Democritus, who flourished around 400 B.C, believed that all matter was made of tiny, fundamental particles called atoms (from the Greek atomos, indivisible). No matter how prescient this notion seems in retrospect, it was really a philosophical conceit more than a scientific hypothesis. What atoms were, what they looked like, how they behaved, how they interacted—such things could only be guessed at. Modern interest in atoms revived first among the chemists. In 1803, John Dalton in England proposed that rules of proportion in chemical reactions—hydrogen and oxygen combining in a fixed ratio to produce water, for example—came about because atoms of chemical substances joined together according to simple numerical rules.

  Atoms didn’t gain credibility overnight. As late as 1860 an international conference convened in Karlsruhe, Germany, to debate the atomic hypothesis. The weight of opinion by now favored atoms, but there was significant dissent. Many distinguished chemists were happy to take laws of chemical combination as basic rules in their own right, and saw no reason to indulge in extravagant speculation about invisible particles.

  August Kekulé, the German chemist who famously devised the ring structure of the benzene molecule while dozing in his fireside armchair and dreaming about snakes catching their own tails, offered a more shaded opinion. He accepted the existence of the chemical atom, along the lines suggested by Dalton and others, and he noted that some physicists, for their own reasons, had also recently begun arguing for atoms. But were the chemist’s atom and the physicist’s atom the same thing? Kekulé thought not, or at least that any such judgment was premature.

  To the chemist, an atom was a thing of almost tactile qualities. It possessed in some way the characteristics of the substance it represented, and it could hook up with or detach from other atoms, according to their respective qualities. Chemists mostly imagined that atoms in bulk sat still, filling space like oranges in a crate.

  Physicists thought quite differently. Their atoms were tiny, hard pellets, flying about at high speed in mostly empty space, occasionally crashing into each other and bouncing off again. The role of these atoms was specific. Beginning about halfway through the nineteenth century, a number of mathematically inclined physicists began to pursue the idea that the frenetic motion of atoms could explain the hitherto mysterious phenomenon of heat. As atoms in a volume of gas gained energy, they would fly about faster, bump into each other more violently, and crash more forcefully into the walls of a container. This was why gas would expand when heated, and exert more pressure. In this so-called kinetic theory of heat, heat was nothing but the energy of atomic motion. The deeper implication was that large-scale physics of heat and gases ought to follow ineluctably from the small-scale behavior of atoms as they moved and collided in strict obedience to Newton’s laws of motion.

  Thus arose the reliable cliché of the atom as a tiny billiard ball, hard but inert, banging mindlessly about. Whether this had anything to do with chemistry was another question. Physicists allowed that the atoms of one gas might be lighter or heavier than those of another, but why gases had distinct chemical properties was none of their business.

  The atom, in short, was in these early days by no means a unifying hypothesis. If chemists and physicists had little to say to each other, still further excluded were the microscopists and biologists. Kinetic theory came with mathematical complexities that repelled all but a select few, while the typical mathematician, if even aware of Brownian motion, most likely assumed it was a trivial phenomenon of strictly botanical interest.

  Nevertheless, a connection awaited discovery. A first hint came from Ludwig Christian Wiener, who spent most of his life teaching mathematics and geometry at German universities. In 1863, after conducting experiments that confirmed everything Brown had long ago found, Wiener felt able to publish an intriguing if speculative suggestion. If the liquid in which Brownian particles jiggle about was in reality a welter of furious atoms, then these atoms would buffet the suspended particles from all sides. The erratic but incessant agitation of the invisible atoms, he argued, would cause the larger visible particles to stagger unpredictably about.

  In keeping with the tangled history of this subject, Wiener’s daring proposal attracted next to no interest.

  It fell to a series of French and Belgian Jesuit priests to keep digging for a scientific account of Brownian motion. During the nineteenth century many clergymen maintained an active and useful interest in the observational and collecting sciences: botany, geology, zoology, and so on. The clerical connection comes up in Middlemarch, when the determinedly atheistic man of science Dr. Lydgate visits the agreeably untheological Reverend Fare-brother and finds that the clergyman has an impressive natural history collection, including specimens, books, and journals. Happy to encounter a f
ellow philosopher of nature, Lydgate offers to show Farebrother a few of his own items, in particular “Brown’s new thing—Microscopic Observations on the Pollen of Plants—if you don’t happen to have it already.”

  What’s more, Jesuits and many other churchmen had a surprisingly broad and rigorous education in philosophy, logic, and even mathematics. Such men were singularly well equipped to deal with problems that we might now call cross-disciplinary but which in those days were merely part of the broad enterprise called science. Mathematical physicists, by contrast, were by the latter half of the nineteenth century on their way to becoming a breed apart, inhabitants of their own recondite discipline, a realm that even those with an adequate amateur command of mathematics were increasingly shy of entering.

  This growing divide meant that by the end of the 1870s, a number of scientists perceived the correct qualitative explanation of Brownian motion but lacked the means to put their hypothesis into convincingly quantitative terms. It’s strangely difficult to find anyone willing to take credit for seeing the answer. In an 1877 issue of the London Monthly Microscopical Journal, for example, we find Father Joseph Delsaulx, S.J., attributing to an unnamed colleague the suggestion that Brownian motion results from the constant agitation of small particles by the atoms or molecules that make up a liquid. (Chemists had by this time established the distinction between atoms, which were fundamental, and molecules, which were combinations of atoms.)

  Three years later, writing for the Revue des Questions Scientifiques, Father J. Thirion, S.J., mentions that he had seen some years before a similar proposal jotted down in the unpublished laboratory notes (!) of “Fr. Carbonelle, a savant well known to our readers, to whom another of our colleagues, Fr. Renard, had shown for the first time the curious movement of libelles.” These libelles, Thirion is helpful enough to explain, are microscopic dark spots seen within little pockets of liquid trapped in samples of quartz. They are in fact tiny bubbles of gas caught within these liquid inclusions, and they jiggle about in the now-familiar fashion. Father Delsaulx also refers to libelles, and adds that since quartz is known to be very old, this is a case of Brownian motion that must have been going on unabated for millions of years. Clearly, he says, no external cause can be responsible. What Father Renard showed to Father Carbonelle must be the result, Father Delsaulx affirms, of molecules bouncing endlessly around.

  The reverend gentlemen were on the right track, but their lack of mathematical sophistication prevented them from going much further. Delsaulx suggested vaguely that the observed amplitude of Brownian motion—how far and fast a particle travels on each zig or zag—must have something to do with what he called the “law of large numbers.” A molecule of liquid, it was clear by this time, is far too small for a single impact with a Brownian particle to cause any observable motion. Rather, the molecules crash constantly into the particle from all sides, but not quite uniformly. Variations in the impacts on different sides of the Brownian particles make it jiggle about; at the same time, the greater the number of molecules involved, the more their random impacts would tend to cancel each other out, making the motion smaller. The “law of large numbers,” by which Delsaulx evidently means to imply some sort of statistical reasoning, should in principle connect the magnitude of Brownian motion with the size and number and speed of molecules in the liquid. More than this he could not say.

  Ten years later a nonclerical French scientist, Louis-Georges Gouy, performed a series of careful experiments on Brownian motion, which he nicely described as “une trépidation constante et caractéristique.” He commented that even now, sixty years after Brown’s decisive work, the general opinion seemed to be that the motion was “an accident produced by some external agitation.” But as he goes on to say (repeating what Brown and Wiener and various Jesuits had already said), this was clearly not the case. His experiments established—yet again—that the motion occurs for all kinds of particles in any kind of liquid. He could not find any kind of particle that did not jiggle about. He concluded, as many had done by now, that molecular activity was the cause.

  But he ventured a little further. He first assured his readers that Brownian motion was not, as some had suggested, a kind of perpetual motion, which the recently formulated laws of thermodynamics clearly forbade. As molecules bounce around, he explained, they crash into each other, exchanging energy, some going a little slower, some a little faster—but always sharing the same total amount of energy. No problem there. Then he noted that recent estimates put the typical speed of molecules in the region of 100 million times greater than the speed at which Brownian particles appeared to move. Again, no problem: the “law of large numbers” would take care of that. But like Father Delsaulx, he could offer no specific calculation relating the size of a particle, the number of molecules in it, or the number of times it got hit by liquid molecules to the way it moved.

  Brownian motion, evidently, was a statistical phenomenon: the unpredictable, apparently random jiggling of tiny particles reflected in some way the average or aggregate motion of unseen molecules. It might not be possible to explain in minute detail just why a Brownian particle moved as it did, in its erratic way, but the broad parameters of its movement ought to follow from some suitable statistical measure of the motion of unseen molecules.

  But the few early investigators who perceived this connection lacked the means to make their theorizing mathematically precise. And perhaps because they could offer no specific calculation, they failed to see the conceptual puzzle that arose. If the underlying motion of molecules followed orderly Newtonian rules of cause and effect, of perfect predictability, how could it give rise to a phenomenon that appeared to demonstrate the workings of chance? That conundrum, however, was precisely what the more sophisticated advocates of kinetic theory soon found themselves obliged to contend with.

  Chapter 2

  ENTROPY STRIVES TOWARD A MAXIMUM

  Writing about Brownian motion in 1889, Louis-Georges Gouy expressed some perplexity that “this phenomenon seems to have barely attracted the attention of physicists.” Among those who had failed to grasp its significance, he claimed, was the Scottish physicist James Clerk Maxwell, arguably the most eminent theorist of the nineteenth century, who had apparently believed that if “submitted to a more powerful microscope…the [Brownian particles] will demonstrate only more perfect repose.” Better optics, in other words, would make this nuisance go away.

  Unfortunately Gouy, as was often the case in those days, gave no source for Maxwell’s alleged statement, and it remains unclear even now whether his accusation is just. Certainly, though, nothing in Maxwell’s published works hints that he saw in Brownian motion any clue to the molecular composition of gases and liquids. His omission is all the more surprising since it was Maxwell who first used a statistical technique to solve a problem in physics and who later helped develop the kinetic theory of heat to a fine mathematical pitch.

  As far back as the middle of the seventeenth century, Blaise Pascal, Pierre de Fermat, and others had worked out simple laws of mathematical probability for various card and dice games, but it was a long time before such ideas left the gambling parlors. In 1831, the Belgian mathematician Adolphe Quetelet tabulated crime rates in France according to the age, sex, and education of the perpetrator, the climate at the crime location, and the time of year when the crime occurred. This heralded, for good or ill, the widespread application of statistical methods in demographics and the social sciences.

  About thirty years later Maxwell, influenced by his reading of Quetelet, came up with an ingenious way to prove that the rings of Saturn must be made of dust. Picturing the rings as aggregations of small particles controlled by Saturn’s gravity, Maxwell set up a statistical description that allowed the particles to have a range of sizes and orbital speeds. Applying standard mechanical analysis to this model, he proved that if the rings were to maintain their shape over long periods of time, the particle sizes must fall within some limited range.

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bsp; Shortly afterward, Maxwell saw that he could use similar methods to describe the speeding and colliding atoms that constituted a volume of gas. It was in understanding the nature of heat that physicists first found themselves obliged to tackle seriously questions of statistics and probability. But from the outset, there was something disquieting, almost self-contradictory, about this venture.

  If heat is simply the collective bustle of atoms, then the physics of heat must ultimately follow from Newton’s laws of motion applied to those atoms. Atomic collisions ought to be as calculable as caroms on the billiard table, in which case the behavior of heat ought to be similarly predictable. This vision of scientific omniscience, in which every particle in the universe must follow strict and rational laws, is captured in the famous words of the Marquis de Laplace, one of the leading eighteenth-century developers of Newtonianism at its mathematically splendid best:

  We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.

  Nothing could be uncertain: that was the crucial point. Mangling the words of another Frenchman, we can say: Tout comprendre c’est tout prédire, to understand all is to predict all. From such grandiose contentions derive all the familiar clichés about the world as machine, the universe as clockwork, and all of science as ultimately deterministic and inexorable.

  On the other hand, as physicists quickly realized, any hope of actually calculating the individual behavior of every last atom or molecule in a volume of gas was not just unattainable but positively absurd. (By the latter half of the nineteenth century, scientists had a pretty good idea of how tiny and therefore how numerous molecules must be. A laboratory flask filled with water contains a trillion trillion of them.) In order to accomplish anything practical by way of theorizing about large crowds of atoms and molecules, physicists would have to resort to statistical descriptions of their behavior and set aside the utopian goal of perfect knowledge. Nowhere did this compromise show up more disturbingly than in the notorious second law of thermodynamics—the one about entropy, and the contest between order and disorder.