Section 40–6 The failure of classical physics

Maxwell’s (1860) paper / Freezes out / Harmonic oscillator

 

In this section, Feynman discusses Maxwell’s (1860) paper, which first revealed the limitation of classical physics in explaining the behavior of gases. He then extends the discussion by introducing the quantum model of the harmonic oscillator, showing how the “freezing out” of certain modes of motion at low temperatures resolves the discrepancies found in the classical kinetic theory pertaining to the specific heat ratios. Thus, the section could be titled “A Limitation of the Kinetic Theory of Gases (or Classical Equipartition Theorem),” highlighting one of the key failures of classical physics; other, historically significant failures include the ultraviolet catastrophe, the photoelectric effect, and the discrete spectral lines of atoms.

 

1. Maxwell’s (1860) paper:

“The first great paper on the dynamical theory of gases was by Maxwell in 1859. On the basis of ideas we have been discussing, he was able accurately to explain a great many known relations, such as Boyle’s law, the diffusion theory, the viscosity of gases, and things we shall talk about in the next chapter. He listed all these great successes in a final summary, and at the end he said, “Finally, by establishing a necessary relation between the motions of translation and rotation (he is talking about the ½kT theorem) of all particles not spherical, we proved that a system of such particles could not possibly satisfy the known relation between the two specific heats.” He is referring to γ (which we shall see later is related to two ways of measuring specific heat), and he says we know we cannot get the right answer (Feynman et al, 1963, p. 40-8).”

 

In his 1860 paper Illustrations of the Dynamical Theory of Gases, Maxwell considered only translational and rotational motions when applying the equipartition theorem to molecular gases. He derived a necessary relation between these two types of motion and concluded that non-spherical molecules could not satisfy the observed ratio of specific heats at constant pressure and at constant volume. However, Maxwell’s analysis did not include other possible degrees of freedom—vibrational, torsional (rotational), and electronic—which later proved essential for understanding the specific heat capacity. These contributions were developed later, e.g., through the work of Boltzmann and quantum theory, which clarified that at ordinary temperatures many of the molecules still move and rotate, but do not vibrate (“frozen out”).

 

To be precise, Maxwell’s 1860 paper is titled “Illustrations of the Dynamical Theory of Gases.” (The paper was published in 1860, but it was read at the Meeting of the British Association at Aberdeen on September 21, 1859.) This is different from “On the Dynamical Theory of Gases,” a title that belongs to Maxwell’s later 1867 publication. This distinction matters because the earlier paper introduced the foundations of the kinetic theory of gases and the Maxwellian velocity distribution, while the 1867 work further refined and extended these ideas especially different properties of gases (diffusion, viscosity, and conductivity). While Feynman’s pedagogical insights remain undiminished, his potentially misleading description serves as a gentle reminder that the history of science is built on a meticulous attention to detail.

 

“Ten years later, in a lecture, he said, ‘I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory’ (Feynman et al, 1963, p. 40-9).”

 

Feynman (1963) remarked, “Ten years later, in a lecture, he said, ‘I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory.’” However, Maxwell’s actual words were slightly different. On February 15, 1875, in his lecture On the Dynamical Evidence of the Molecular Constitution of Bodies delivered before the Chemical Society, Maxwell stated: “I must now say something about these internal motions, because the greatest difficulty which the kinetic theory of gases has yet encountered belongs to this part of the subject.” Feynman’s paraphrase captures the spirit but not the precise wording of Maxwell’s statement. Moreover, this lecture was delivered roughly fifteen years after Maxwell’s 1860 paper Illustrations of the Dynamical Theory of Gases, marking a shift in focus—from establishing the foundations of kinetic theory to confronting its conceptual challenge: the role of internal molecular motions.

 

2. Freezes out:

“How can we understand such a phenomenon? Of course that these motions ‘freeze out’ cannot be understood by classical mechanics. It was only understood when quantum mechanics was discovered (Feynman et al, 1963, p. 40-9).”

 

The concept of the “freezing out” of vibrational modes is based on the foundational work of Planck and Einstein, who recognized that the classical equipartition theorem fails at low temperatures. Planck (1900) introduced the idea that oscillators (or resonators) can only exchange energy in discrete amounts of hν, explaining blackbody radiation. Einstein (1907) applied this concept to the vibrations of atoms in solids, proposing that each atom behaves like a Planck’s resonator. He showed that when kT << hν, most resonators remain in their lowest energy state, causing the specific heat to drop below the values predicted by Dulong–Petit law. In short, Einstein explains the peculiar decrease in the specific heat of solids at low temperatures using Planck’s theory of radiation.

 

“About 1905, Sir James Hopwood Jeans and Lord Rayleigh (John William Strutt) were to talk about this puzzle again. One often hears it said that physicists at the latter part of the nineteenth century thought they knew all the significant physical laws and that all they had to do was to calculate more decimal places. Someone may have said that once, and others copied it. But a thorough reading of the literature of the time shows they were all worrying about something. Jeans said about this puzzle that it is a very mysterious phenomenon, and it seems as though as the temperature falls, certain kinds of motions ‘freeze out’ (Feynman et al, 1963, p. 40-9).”

 

Lord Rayleigh and Sir James Jeans contributed to the understanding of limitations of classical physics through their work on blackbody radiation, but neither said that certain kinds of motions “freeze out.” However, the word “freeze” or “frozen” is potentially misleading because it implies that motions stop entirely—like water turning into ice, but molecular motion never truly stops, even at absolute zero. More important, the molecules still have translational kinetic energy and rotational kinetic energy, but their vibrational kinetic energy is reduced. What actually “freezes out” could be the vibrational motion (or rotational motion), which is its capacity to absorb and store additional thermal energy in that particular mode. The atoms still remain in motion, but the number (or probability) of atoms that are vibrating is dependent on the available thermal energy or temperature.

 

3. Harmonic Oscillator:

“Now it turns out that for a harmonic oscillator the energy levels are evenly spaced… Now let us see what happens. We suppose we are studying the vibrations of a diatomic molecule, which we approximate as a harmonic oscillator… All oscillators are in the bottom state, and their motion is effectively ‘frozen’; there is no contribution of it to the specific heat (Feynman et al., 1963, p. 40-9).

 

Feynman idealizes a diatomic molecule as a harmonic oscillator (linear vibrator) captures the essential concept of discrete energy levels. However, this simple model comes with physical limitations. Strictly speaking, molecular vibrations are anharmonic; the restoring force weakens as atoms separate, which could be more accurately described by the Morse potential. As the molecule vibrates, its fluctuating bond length alters the moment of inertia, thereby shifting the rotational energy levels, which are also quantized. This vibration-rotation interaction produces the distinctive fine structure observed in molecular spectra—a phenomenon which the harmonic oscillator fails to predict. Despite the limitations of the model, it illustrates that vibrational degrees of freedom "freeze out" at room temperature—a key insight that cannot be explained by classical physics.

 

“If we change the temperature but still keep it very small, then the chance of its being in state E₁ = ℏω remains infinitesimal—the energy of the oscillator remains nearly zero; it does not change with temperature so long as the temperature is much less than ℏω. All oscillators are in the bottom state, and their motion is effectively “frozen”; there is no contribution of it to the specific heat (Feynman et al., 1963, p. 40-9).”

 

The Boltzmann factor (e^{-energy gap/kT}) provides the probability of a molecule being in a higher-energy state relative to its ground state. When the energy gap is large compared with kT, this exponential factor becomes very small. (Instead of saying all oscillators are in the bottom state, it is possible that some are vibrating.) In other words, the chance that a molecule actually occupies an excited vibrational state is significantly lower because the energy is quantized. As a result, the vibrational modes do not really contribute to the molecule’s heat capacity. In essence, the process of thermal excitation is inherently statistical, governed by the Boltzmann factor, which determines probabilities. The thermal energy available from random motion is proportional to kT, but there is also a temperature fluctuation among the large number of molecules. (This could be related to the concept of random walk and Brownian movement, which will be discussed in the next chapter.)

 

 

Key Takeaways:

The classical physicist treats energy like a generous bartender with an endlessly adjustable tap: you could pour any amount—a sip, a half-glass, a full pint—into a resonator (like a molecule). However, the quantum physicist enforces a strict house rule: energy must be taken in whole quanta, like drinking beer only in full pints. No half-pints. No sips.

This single rule changes everything:

• The "Freeze Out" as a quantum problem: At low temperatures, the available thermal energy kT might only be enough for a few “sips.” In this context, quantum mechanics doesn’t allow sips. It cannot accept the partial energy, so the vibrational mode remains dormant, "frozen out," and doesn't contribute to the heat capacity.
• The Boltzmann Factor as the Club Bouncer: The probability that a molecule has enough energy to “buy a pint” is given by the Boltzmann factor e^-DE/kT. If the energy gap DE is large compared with kT, the bouncer almost always turns you away.

In short, “freeze out” does not mean the atoms stop moving, but it is about a degree of freedom becoming thermally inaccessible due to quantum energy gaps. In other words, the vibrational states are quantized and need a certain minimum of energy before they can be excited.

 

The Moral of the Lesson (In Feynman’s spirit):

In his Nobel lecture, Feynman (1965) remarks:

“The harmonic oscillator is too simple; very often you can work out what it should do in quantum theory without getting much of a clue as to how to generalize your results to other systems. So that didn’t help me very much, but when I was struggling with this problem, I went to a beer party in the Nassau Tavern in Princeton.  There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think that a good place to discuss intellectual matters is a beer party. So, he sat by me and asked, what are you doing and so on, and I said, I’m drinking beer.”

      However, the real lesson is not "drink beer to solve physics," but rather: "Seek out relaxed, interdisciplinary, and social environments where unexpected conversations can happen." Valuable insight emerged not from alcohol itself but from an open, spontaneous exchange with a thoughtful colleague who happened to ask the right question at the right moment. It is also important to resist romanticizing alcohol as a catalyst for scientific creativity. Using beer drinking as a form of “self-sacrifice” for solving physics problems is not only misguided but potentially harmful. Any amount of alcohol consumption has been associated with an increased risk of various health problems, including liver disease, cardiovascular issues, certain cancers, and neurological damage.

 

Review Questions

1. Why is it important to distinguish between Maxwell’s “Illustrations of the Dynamical Theory of Gases” (1860) and “On the Dynamical Theory of Gases” (1867) when interpreting Feynman’s reference to Maxwell’s work? Explain how each paper contributed differently to the development of kinetic theory and why misattributing them may obscure the historical evolution of the subject.

2. What common misconception can the term “freeze out” create, and what is the more accurate physical interpretation of the phenomenon? Clarify why the term might imply that molecular motion stops, and describe what actually happens to a molecule’s ability to absorb energy in a given mode.

3. What are the main physical limitations of modeling a diatomic molecule as a simple harmonic oscillator? Discuss how real molecular vibrations deviate from perfect harmonic behavior without using the term “freeze out.”

 

References:

Einstein, A. (1907). Planck’s theory of radiation and the theory of specific heat. Ann. Phys, 22, 180-190.

Feynman, R. P. (1965). The Development of the Space-Time View of Quantum Electrodynamics. The Official Web Site of The Nobel Prize.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Maxwell, J. C. (1860). II. Illustrations of the dynamical theory of gases. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 20(130), 21-37.

Maxwell, J. C. (1867). On the Dynamical Theory of Gases. Philosophical Transactions of the Royal Society of London, 157, pp. 49-88.

Maxwell, J. C. (1875). On the dynamical evidence of the molecular constitution of bodies. Nature, 11(279), 357-359.

Planck, M. (1900). On the theory of the energy distribution law of the normal spectrum. Verh. Deut. Phys. Ges, 2(237), 237-245.