Everyday entropy: Boltzmann’s door

Planck and Einstein may not have appreciated the weight of the door that Boltzmann opened for them, but they fully appreciated the passageway he left behind.  Both of their nobel prize works departed through it.  It allowed Einstein, in particular, to describe the invisible behaviour of light quanta as point particles in space in a mathematically rigorous framework.
Concerning an Heuristic Point of View Toward the Emission and Transformation of Light, by A. Einstein, Bern, 17 March 1905.  http://www.esfm2005.ipn.mx/ESFM_Images/paper1.pdf
“If we confine ourselves to investigating the dependence of the entropy on the volume occupied by the radiation, and if we denote by S0 the entropy of the radiation at volume v0, we obtain
S − S0 = (E/βν) ln (v/v0).
This equation shows that the entropy of a monochromatic radiation of sufficiently low density varies with the volume in the same manner as the entropy of an ideal gas or a dilute solution. In the following, this equation will be interpreted in accordance with the principle introduced into physics by Herr Boltzmann, namely that the entropy of a system is a function of the probability its state…
If the entropy of monochromatic radiation depends on volume as though the radiation were a discontinuous medium consisting ofenergy quanta of magnitude Rβν/N, the next obvious step is to investigate whether the laws of emission and transformation of light are also of such a nature that they can be interpreted or explained by considering light to consist of such energy quanta.”
On the centenary of the death of Ludwig Boltzmann, Carlo Cercignani examines the immense contributions of the man who pioneered our understanding of the atomic nature of matter. The man who first gave a convincing explanation of the irreversibility of the macroscopic world and the symmetry of the laws of physics was the Austrian physicist Ludwig Boltzmann, who tragically committed suicide 100 years ago this month. One of the key figures in the development of the atomic theory of matter, Boltzmann’s fame will be forever linked to two fundamental contributions to science. The first was his interpretation of ‘entropy’ as a mathematically well-defined measure of the disorder of atoms. The second was his derivation of what is now known as the Boltzmann equation, which describes the statistical properties of a gas as made up of molecules. The equation, which described for the first time how a probability can evolve with time, allowed Boltzmann to explain why macroscopic phenomena are irreversible. The key point is that while microscopic objects like atoms can behave reversibly, we never see broken coffee cups reforming because it would involve a long series of highly improbable interactions – and not because it is forbidden by the laws of physics.
Happy centenary, photon, by Anton Zeilinger, Gregor Weihs, Thomas Jennewein and Markus Aspelmeyer, Nature 433, 230-238 (20 January 2005)

The way that Einstein arrives at the photon concept in his seminal paper “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (“On a heuristic aspect concerning the production and transformation of light”) is, contrary to widespread belief, not through the photoelectric effect. Instead, Einstein compares the entropy of an ideal gas filling a given volume with the entropy of radiation filling a cavity. The logarithmic dependence on the volume of the entropy of the gas can easily be understood by referring to the connection between entropy and probability suggested by Boltzmann. Because it is less probable that the gas particles will occupy a smaller volume, such a state has a higher order, and hence lower entropy. Interestingly, for the case of radiation filling a cavity, Einstein merely uses the Wien black-body radiation density, which is known to be correct only for high radiation frequencies.

Einstein’s crucial insight comes when he observes that the entropy of light in a cavity varies in exactly the same way with the volume of the cavity as the entropy of a gas. On the basis of this observation, he suggests that light also consists of particles which he calls light quanta. He clearly states that this is only a heuristic point of view and not a logically binding conclusion. Only in the last chapter (of eight) of the paper does Einstein finally get to the photoelectric effect by asking where quanta of light might have implications. He notes that it would naturally explain why the wavelength of light emitted in photo-luminescence is always larger than that of the absorbed light. This is because a single particle of light is absorbed and unless additional energy is supplied, the energy of the emitted particles of light in general is lower.

Einstein’s Revolutionary Light-Quantum Hypothesis, by Roger H. Stuewer

Einstein gave two arguments for light quanta, a negative and a positive one. His negative argument was the failure of the classical equipartition theorem, what Paul Ehrenfest later called the “ultraviolet catastrope.” 5 His positive argument proceeded in two stages. First, Einstein calculated the change in entropy when a volume Vo filled with blackbody radiation of total energy U in the Wien’s law (high-frequency) region of the spectrum was reduced to a subvolume V. Second, Einstein used Boltzmann’s statistical version of the entropy to calculate the probability of finding n independent, distinguishable gas molecules moving in a volume Vo at a given instant of time in a subvolume V. He found that these two results were formally identical, providing that U = n(Rβ/N)ν, where R is the ideal gas constant, β is the constant in the exponent in Wien’s law, N is Avogadro’s number, and ν is the frequency of the radiation. Einstein concluded: “Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as if it consisted of mutually independent energy quanta of magnitude Rβν/N.” 6 Einstein cited three experimental supports for his light-quantum hypothesis, the most famous one being the photoelectric effect, which was discovered by Heinrich Hertz at the end of 18867 and explored in detail experimentally by Philipp Lenard in 1902.

Einstein cited three experimental supports for his light-quantum hypothesis, the most famous one being the photoelectric effect, which was discovered by Heinrich Hertz at the end of 1886 and explored in detail experimentally by Philipp Lenard in 1902. Einstein wrote down his famous equation of the photoelectric effect, in the Patent Office in Bern, Switzerland.  Πe = (R/N)βν – P, where Π is the potential required to stop electrons (charge e) from being emitted from a photosensitive surface after their energy had been reduced by its work function P. It would take a decade to confirm this equation experimentally. Einstein also noted, however, that if the incident light quantum did not transfer all of its energy to the electron, then the above equation would become an inequality: Πe < (R/N)βν – P.

We see, in sum, that Einstein’s arguments for light quanta were based upon Boltzmann’s statistical interpretation of the entropy. He did not propose his light-quantum hypothesis “to explain the photoelectric effect,” as physicists today are fond of saying. As noted above, the photoelectric effect was only one of three experimental supports that Einstein cited for his light-quantum hypothesis, so to call his paper his “photoelectric-effect paper” is completely false historically and utterly trivializes his achievement. In January 1909 Einstein went further by analyzing the energy and momentum fluctuations in black-body radiation.  He now assumed the validity of Planck’s law and showed that the expressions for the mean-square energy and momentum fluctuations split naturally into a sum of two terms, a wave term that dominated in the Rayleigh-Jeans (low frequency) region of the spectrum and a particle term that dominated in the Wien’s law (high frequency) region.

This constituted Einstein’s introduction of the wave-particle duality into physics.  Einstein presented these ideas again that September in a talk he gave at a meeting of the Gesellschaft Deutscher Naturforscher und Ärzte in Salzburg, Austria.  During the discussion, Max Planck took the acceptance of Einstein’s light quanta to imply the rejection of Maxwell’s electromagnetic waves which, he said, “seems to me to be a step which in my opinion is not yet necessary.”  Johannes Stark was the only physicist at the meeting who supported Einstein’s light-quantum hypothesis.  In general, by around 1913 most physicists rejected Einstein’s light-quantum hypothesis, and they had good reasons for doing so.

First, they believed that Maxwell’s electromagnetic theory had to be universally valid to account for interference and diffraction phenomena. Second, Einstein’s statistical arguments for light quanta were unfamiliar to most physicists and were difficult to grasp. Third, between 1910 and 1913 three prominent physicists, J.J. Thomson, Arnold Sommerfeld, and O.W. Richardson, showed that Einstein’s equation of the photoelectric effect could be derived on classical, non-Einsteinian grounds, thereby obviating the need to accept Einstein’s light-quantum hypothesis as an interpretation of it.  Fourth, In 1912 Max Laue, Walter Friedrich, and Paul Knipping showed that X rays can be diffracted by a crystal, which all physicists took to be clear proof that they were electromagnetic waves of short wavelength. Finally, in 1913 Niels Bohr insisted that when an electron underwent a transition in a hydrogen atom, an electromagnetic wave, not a light quantum, was emitted.



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