Scene 2. Graz and Vienna, 1870s
Ludwig Boltzmann (1844–1906) extends Maxwell’s work. In 1868, a year after Maxwell’s paper, he produces an expression for the distribution of energy among molecules of a gas valid for gases of any kind. To derive it, he relies on a key assumption known as the equipartition theorem, according to which a molecule stores energy by spreading it equally among all the avenues (‘degrees of freedom’) available to it. This work also includes a now-famous term, Boltzmann’s constant, now referred to as k, 1.38 × 10−23 joules/Kelvin. The result is a thoroughly statistical interpretation of thermodynamics. In 1872, Boltzmann generalizes this work still further, in a revolutionary paper with a banal title, ‘Further Researches on the Thermal Equilibrium of Gas Molecules.’ In it, he derives a function related to entropy, now called the H-function, whose value almost always increases with time until it reaches a maximum – an entirely novel and innovative approach to proving the second law, and in a way that explicitly demonstrates irreversibility, or how it increases with time. But this work is subject to friendly fire: from Thomson, in an 1874 paper that refers to Maxwell’s little creature as a ‘demon’ and, in 1876, from Boltzmann’s former mentor Josef Loschmidt, for not having eliminated certain puzzles involving the relation between the second law and the first. Even complex many-body systems, such as the deployment of the planets around the sun, are cyclical, with the same patterns eventually recurring, so why doesn’t this happen in thermodynamic systems? Also, if two gases mixed, following the H-curve, entropy increases – but if you then reverse the velocities of all the gas molecules, wouldn’t it then make the H-curve, the ‘arrow of time’, go the other way, violating his theorem? Boltzmann replies (1877) that when a big state corresponds to many equally probable little states, its probability is related to the number of little states. This all but forces big states to evolve in the direction of their more probable states. Boltzmann’s approach is an explicit probabilistic interpretation of entropy, introduces probability into electromagnetism, and proves the centrality of irreversibility to thermodynamics. Newton’s laws + objects made of myriads of pieces + laws of probability = the arrow of time. The forbidden becomes the highly unlikely. On large scales, you play dice, and statistics rule. In 1879, this work is extended by Boltzmann’s former teacher Stefan into the Stefan-Boltzmann law, a law relating the dependence of radiation on temperature in black bodies. But Boltzmann becomes vulnerable to depression late in life, from both personal and professional setbacks, and in 1906, on vacation near Trieste, he hangs himself while his wife and daughter are out swimming. On his tombstone is engraved his equation, not in the form he wrote it but as Max Planck would: S = k log W.
ACT FIVE
Berlin, 1890s
Scene 1. Berlin, early 1890s
Physicist Wilhelm Wien (1864–1928), an introvert who is repeatedly thwarted in his attempts to become a farmer like his parents, extends Boltzmann’s ideas about the second law of thermodynamics. He is at the Physikalisch-Technische Reichanstalt, the imperial bureau of standards, and he, along with several other scientists, is studying something called ‘black body radiation’ out of a mixture of theoretical interest and practical motives (the calibration of electric lamps). If you heat up a body that absorbs all the radiation that falls on it (a ‘black body’), it eventually begins to glow and emit light itself. According to classical mechanics, something in the body is like a resonator, acting like a miniature antenna, in picking up and giving off energy in the form of electromagnetic waves. It is as if the electric charges were located in springs of varying flexibilities, and they oscillated back and forth at a rate depending on the flexibility of the spring and with an intensity that depended on the temperature. Maxwell had fully explained the radiation’s creation, absorption, and propagation. Experimenters had measured this radiation and produced curves of wavelengths and intensities at each temperature. Using the Stefan-Boltzmann work, Wien writes a paper entitled, ‘A New Relationship Between the Radiation From a Black Body and the Second Law of Thermodynamics.’ The paper contains Wien’s law, which uses the second law to map radiation’s dependence on temperature at high temperatures: ‘In the normal emission spectrum from a black body each wave-length is displaced with a change of temperature in such a way that the product of temperature and wave-length remains constant.’2 It is called a ‘displacement law’, and is reformulated in 1896. That year, scientists at the bureau construct a special oven to measure the wavelengths of the radiation, focusing on the shorter wavelengths that are easier to detect. Wien’s law says the energy emitted increases with temperature, though the increase is not equally distributed across all wavelengths but shifts toward shorter wavelengths. But experimenters discover that as the energy curve is extended to longer and longer wavelengths, Wien’s law breaks down, though when the energy gets low enough, it is fully explained by classical theory. Because Wien’s law relies directly on the logical structure of classical physics, this is cause for concern.
Scene 2. Berlin, late 1890s
Max Planck (1858–1947), a reluctant revolutionary whose ‘pet subject’ was thermodynamics, is drawn to study black body radiation. In 1878, as a graduate student, he had come across a collection of Clausius’s papers, was entranced, and commenced a dissertation that critiqued the existing formulations of the second law. By then, thermodynamics was regarded as virtually complete, and not an exciting or even promising field for a young scientist. But Planck is temperamentally conservative, interested in cementing foundations, and something distresses him about Boltzmann’s statistical interpretation of the second law. Laws, he feels, should be absolute – no exceptions, however rare! – and the second law should be as universal as the first, not true by virtue of some statistical sleight-of-hand. In 1895, having moved to Berlin, Planck is goaded by his assistant Zermelo, who produces an argument that the second law can never be proven, and, furthermore, that truly irreversible processes are impossible because any mechanical system, however complex, must eventually return to a state near its initial one. Resolving the conflict between the probabilistic, irreversible second law and the ironclad, reversible Newtonian mechanics was ‘the most important [problem] with which theoretical physics is currently concerned’, Planck is moved to say. And black body radiation seemed to hold the key, for the answer might lie in the way the resonators absorb and emit energy. Berlin happens to be the centre of studies of black body radiation; Wien is there, as well as several experimenters. Perhaps he could take advantage of their work to show how to use just the theory of electromagnetism plus the laws of thermodynamics to explain the distribution of radiation in equilibrium. Planck begins by revising Boltzmann’s work to make it clearer, and reformulating Wien’s law in terms of frequency instead of wavelength, seeking to tie all the loose ends of thermodynamics, statistical mechanics, and electromagnetic theory together. And in 1897 Planck gives at the Prussian Academy the first of a series of talks, which end up spanning several years, entitled ‘On Irreversible Radiation Processes’, in which he hopes to solve what he calls ‘the fundamental task of theoretical physics’, the reconciliation of the two laws of thermodynamics. In the very first, he points out the urgent need to look at black body radiation because of a contradiction between the two laws of thermodynamics. The first law of thermodynamics, or ‘the principle of energy conservation’, holds that all effects like friction have to be reduced on a microscopic level to mechanical and reversible processes. But the second law of thermodynamics, or ‘the principle of the increase of entropy’, requires that ‘all changes in nature proceed in one direction.’ Reconciling this, he tells his listeners again, is ‘the fundamental task of theoretical physics.’ A few talks later, on October 19, 1900, he comes up with an empirical formula that spans both Wien’s law at high energies, and classical physics at low energies, as well as that awkward place in between where Wien’s law does not quite fit the experimental data. The formula involved ‘completely arbitrary expressions for the entropy’
, he says, based on the notion that the resonators could not oscillate at any old frequencies but only at specific ones, related to a number called h. Planck is, as usual, very cautious. ‘[A]s far as I can see at the moment’, he says, the work ‘fits the observational data published up to now as satisfactorily as the best equations put forward for the spectrum.’ He concludes, ‘I should therefore be permitted to draw your attention to this new formula which I consider to be the simplest possible.’3 With these words, he introduces – hesitatingly, even reluctantly – the idea of the quantum into physics. That night, one of the experimenters is moved to return to the lab and test Planck’s ‘new formula’, confirming it. Planck, excited, sets back to work, and ‘after a few weeks of the most strenuous labour of my life the darkness lifted and a new, unimagined prospect began to dawn’, namely, that he has indeed wrapped thermodynamics, electrodynamics, and classical mechanics in one package, explaining this last piece of the experimental puzzle. It is indeed a neat package, but it has unsuspected implications. Neither he, nor anyone else involved at the time, realizes that in completing the foundations of thermodynamics, they’ve given birth to an entirely new conception of energy, and come to the threshold of a radically new world.
EPILOGUE
Storm Clouds
In April 1900, Thomson gives a talk at the Royal Institution entitled ‘Nineteenth Century Clouds over the Dynamical Theory of Heat and Light.’4 The ‘beauty and clearness’ of the theory of heat and light – of thermodynamics and electromagnetism – had been a crowning achievement of nineteenth-century science, he says, but the triumph was ‘obscured by two clouds.’
One cloud was the difficulty of conceiving how the earth moves through the ether. Scientists at the beginning of the nineteenth century had said that the ether must pass through the atoms of solid bodies ‘like wind blowing through a grove of trees.’ But Maxwell had shown that the ether must be more like a liquid or elastic solid, exerting force against objects that move through it, implying that the earth’s motion with respect to the ether should be detectable. But, Thomson continues, Albert Michelson and Edward Morley had recently carried out an ‘admirable experiment’, flawless in design and execution, that appeared to rule this out. One way out had been independently devised by George FitzGerald and Hendrik Lorentz, who had shown that scientists could ‘save’ the ether if matter flowing through it slightly changed its dimension in the direction of motion – only one hundred-millionth part (the square of the ratio of the earth’s velocity around the sun to the velocity of light) would do it! But Thomson found this possibility, though ‘brilliant’, also bizarre. ‘I am afraid we must still regard Cloud No. 1 as very dense’, he concludes.
The second cloud had to do with the ‘equipartition theorem’, as articulated by Maxwell and Boltzmann, according to which molecules store energy by spreading it among all available paths. The theorem provided the explanation for well-known laws regarding the specific heat capacities of solids at high temperatures, but was also in severe contradiction with experimental results involving low temperature solids, gases, and metals. Given the stunning success of thermodynamics, scientists at the time found this discrepancy baffling and scrambled for an explanation. Thomson admits that he had none. He quotes the English physicist Lord Rayleigh, who had daringly said that he was awaiting some new principle that would provide ‘some escape from the destructive simplicity’ of the equipartition theorem. If such an escape should appear, Thomson says in concluding his talk, it would banish Cloud No. 2, which had ‘obscured the brilliance of the molecular theory of heat and light during the last quarter of the nineteenth century.’
Thomson could not know it, but these two nineteenth-century clouds shortly would develop into twentieth-century hurricanes: relativity and quantum mechanics.
Other versions of the drama of the second law of thermodynamics may differ in detail and scope, and number and size of roles. But the drama itself, I claim, is Shakespearean. The cast involves powerful human beings who dedicate themselves body and soul to their work. The action unfolds as these individuals are troubled – sometimes deeply and tragically – by differences between what they find and their expectations, and try to make greater sense of the world by intervening in it. Has any drama ever had such finely drawn and unique characters, or more profoundly reshaped our understanding of ourselves and the world?
Interlude
THE SCIENCE OF IMPOSSIBILITY
Almost every progress in science has been paid for by a sacrifice, for almost every new intellectual achievement previous positions and conceptions had to be given up. Thus, in a way, the increase of knowledge and insight diminishes continually the scientist’s claim on ‘understanding’ nature.
– Werner Heisenberg
Many principles of science have the following form: ‘If you do this, what will happen is that.’ Newton’s second law, for example, says that the acceleration of a particular mass will be proportional to the force applied to it. Such principles imply that certain effects are practically impossible. A small number of principles, however, belong to a different category. These say, in effect, ‘That cannot happen.’ Such principles imply that certain effects are physically impossible.
Notorious examples of the latter include the first and second laws of thermodynamics. Other examples include Heisenberg’s uncertainty principle and the relativity principles regarding the impossibility of recognizing absolute velocity and the prohibition against faster-than-light travel. Such principles often represent not ‘new physics’ but deductions from other principles. What is different about them is their form. And something about that form – asserting that something is physically impossible – tends to make scientists want to rebel.
The science of impossibility goes by several names. ‘Forget about it’ science is one; ‘no-way’ science is another. Half a century ago, the mathematician and historian of science Sir Edmund Whittaker referred to ‘Postulates of Impotence’, which assert ‘the impossibility of achieving something, even though there may be an infinite number of ways of trying to achieve it.’
‘A postulate of impotence’, Whittaker wrote, ‘is not the direct result of an experiment, or of any finite number of experiments; it does not mention any measurement, or any numerical relation or analytical equation; it is the assertion of a conviction, that all attempts to do a certain thing, however made, are bound to fail.’
Postulates of impotence thus resemble neither experimental facts that we find in experience, nor mathematical statements that are true by definition prior to all experience. Nevertheless, Whittaker continued, such postulates are fundamental to science. Thermodynamics, he said, may be regarded as a set of deductions from its postulates of impotence: the conservation of energy and of entropy. It may well be possible, he argued, that in the distant future each branch of science will be able to be presented, à la Euclid’s Elements, as grounded in its appropriate postulate of impotence.
But no-way science is important for another reason: it attracts contrarians. I am not talking about the endless attempts by frauds and naifs to get around the laws of thermodynamics by creating perpetual-motion machines. Rather, I mean serious scientists who find no-way science a challenge to devise loopholes. In seeking these loopholes, they end up clarifying the foundations of the field.
Contrarian science played a role in both the discovery and the interpretation of the uncertainty principle, for instance. In 1926, Werner Heisenberg was promoting his new matrix mechanics – a purely formal approach to atomic physics – by claiming that physicists had to abandon all hope of observing classical properties such as the position and momentum of atomic electrons, and indeed space and time. Pascual Jordan played the role of contrarian by devising a thought experiment to overcome Heisenberg’s claims. Suppose, Jordan said, one could freeze a microscope to absolute zero – then one could measure the exact position and momentum of the atom and its constituents. This seems to have inspired Heisenberg to think about the interaction between the
observing instrument and the observed situation, putting him on the path that shortly led to his articulation of the uncertainty principle. Jordan, the contrarian, forced Heisenberg to think operationally rather than philosophically, and to clarify the physics of the situation.
Afterward, Einstein famously played the contrarian role – with Niels Bohr as his principal adversary – by trying to devise clever ways of simultaneously determining the position and momentum of a particle. While all his attempts failed, the discussion it provoked did much to help physicists understand the nature and implications of quantum mechanics.
Another famous example of contrarian physics was Maxwell’s thought experiment involving a tiny creature who operates a small door in a partition inside a sealed box. By opening and shutting the door, the ‘demon’ – as it was later called – lets all the faster-moving molecules in one side of the partition, violating the second law of thermodynamics by getting heat to flow to that side. The discussion of this thought experiment helped to clarify the then-mysterious concepts of thermodynamics.
Heisenberg is surely overstating when he says that progress in science diminishes the scientist’s claim to understand nature: surely the advance of science is more a matter of developing more subtle and complex concepts that replace but also encompass the simpler existing ones. But these more subtle and complex concepts are often produced by those who are dissatisfied by the prospect of having to make the kind of sacrifice Heisenberg mentions.
A Brief Guide to the Great Equations Page 12
