Section 41–1 Equipartition of energy

Brownian molecular motion / Mirror wobbling / Johnson noise

 

In this section, Feynman discusses three related phenomena: the random motion of a suspended particle (classical Brownian motion), the wobbling of a mirror due to radiation pressure (rotational Brownian motion), and the Johnson noise of an electrical resistor (Brownian motion of electrons). They can be understood as three manifestations of thermal fluctuation—governed by the Equipartition Theorem and Fluctuation-Dissipation Theorem. Thus, a fitting title for the section might be “Three Types of Brownian Motion,” instead of simply “Equipartition of energy.” It is worth mentioning that the explanation of classical Brownian motion was first established by Einstein in his 1905 “Miracle Year” paper, On the Movement of Small Particles Suspended in a Stationary Liquid, Required by the Molecular-Kinetic Theory of Heat.

 

1. Brownian molecular motion

“This was later proved to be one of the effects of molecular motion, and we can understand it qualitatively by thinking of a great push ball on a playing field, seen from a great distance, with a lot of people underneath, all pushing the ball in various directions. We cannot see the people because we imagine that we are too far away, but we can see the ball, and we notice that it moves around rather irregularly. We also know, from the theorems that we have discussed in previous chapters, that the mean kinetic energy of a small particle suspended in a liquid or a gas will be 3kT/2 even though it is very heavy compared with a molecule (Feynman et al., 1963, p. 41-1).”

Brownian motion is the continuous, rapid, and irregular zigzag movement of particles (such as pollen grains) suspended in a fluid (liquid or gas). This phenomenon serves as direct evidence for the physical reality of atoms, confirming that fluids are composed of perpetually moving molecules.

We can define Brownian motion by three interconnected features:

(1) Random motion: The suspended particles undergo continuous, chaotic, and non-directional translational and rotational movements. Its path is described as a random walk because the direction of motion is unpredictable.

(2) Thermal fluctuations: The motion or collisions arise from thermal fluctuations instead of simply absolute temperature. The temperature sets the strength of those fluctuations, so higher temperature means more vigorous Brownian motion.

(3) Diffusion: The particle’s motion is quantitatively related to diffusion. The mean square distance of the particle is directly proportional to the observation time and it is related to the diffusion coefficient (Einstein, 1905).

In short, Einstein’s theory allows for the calculation of fundamental constants, such as Avogadro’s number and the size of molecules, by measuring the observable particle movements.

 

“The Brownian movement was discovered in 1827 by Robert Brown, a botanist. While he was studying microscopic life, he noticed little particles of plant pollens jiggling around in the liquid he was looking at in the microscope, and he was wise enough to realize that these were not living, but were just little pieces of dirt moving around in the water. In fact he helped to demonstrate that this had nothing to do with life by getting from the ground an old piece of quartz in which there was some water trapped. It must have been trapped for millions and millions of years, but inside he could see the same motion. What one sees is that very tiny particles are jiggling all the time (Feynman et al., 1963, p. 41-1).”

 

In De Rerum Natura (about 60 BCE), Lucretius described the motion of dust particles in the sunlight as evidence of atomic collisions (Powles, 1978). Although this is not exactly the Brownian motion in the modern sense, it is an early recognition that “random walk” can be explained by the presence of invisible microscopic particles. In 1827, Robert Brown established the phenomenon through meticulous microscopic observations of pollen grains suspended in water and explained that the motion was not biological but physical in origin. Almost 80 years later, Einstein (1905) and Smoluchowski (1906) provided the theoretical foundation of Brownian motion quantitatively in terms of molecular collisions, thereby offering decisive evidence for the atomic nature of matter. However, Jean Baptiste Perrin was awarded the Nobel Prize for Physics in 1926 for his experimental work on Brownian motion. 

 

2. Mirror wobbling

“What is the mean-square angle over which the mirror will wobble? Suppose we find the natural vibration period of the mirror by tapping on one side and seeing how long it takes to oscillate back and forth, and we also know the moment of inertia, I (Feynman et al., 1963, p. 41-2).”

 

When a mirror is suspended freely—typically by a thin fiber or wire—it exhibits an irregular, continuous wobbling motion even in the absence of external, visible disturbances. This mirror wobbling is the rotational equivalent of classical translational Brownian motion and serves as a direct, macroscopic demonstration of thermal energy fluctuations. The phenomenon is characterized by:

(1) Random Angular displacement: The mirror undergoes small, continuous, and unpredictable angular displacements. This erratic oscillation is readily observed as a “random walk" in the reflected light beam.

(2) Thermal origin: The random motion originates from the fluctuating net torque applied to the mirror. This torque is caused by incessant, unequal bombardment from the surrounding air molecules, whose energy is determined by the absolute temperature.

(3) Fluctuation-Dissipation: The amplitude and frequency of the wobble are determined by the balance between the fluctuating thermal torque (fluctuation) and the mechanical forces opposing it (dissipation). This involves the mirror’s moment of inertia, the stiffness of the fiber (restoring force), and the damping from the surrounding air.

In essence, mirror wobbling is the random, thermally driven angular motion of a suspended mirror, observable as the jitter of its reflected beam—a macroscopic manifestation of microscopic molecular agitation.

 

It should be worth mentioning that the mirrors in the interferometers have the function to split the incoming laser beam into two perpendicular beams. The mirrors are made of fused silica (very pure glass) and suspended by glass fibers, but they still have thermal noise from (translational and rotational) Brownian motion. Today’s mirror stabilization control injects harmful noise, constituting a major obstacle to sensitivity improvements, e.g., in the Laser Interferometer Gravitational-Wave Observatory (LIGO). This would affect accuracy in the detection of black holes, where gravitational waves emerge from events like black hole mergers produce minuscule changes in the length of an interferometer's arms, often smaller than a proton's diameter. If the random, thermal vibrations of the mirrors are larger than the gravitational wave signal, they will obscure it, rendering the event undetectable.

 

“We know the formula for the kinetic energy of rotation—it is given by Eq. (19.8): T=½Iω². That is the kinetic energy, and the potential energy that goes with it will be proportional to the square of the angle—it is V=½αθ². But, if we know the period t₀ and calculate from that the natural frequency ω₀=2π/t₀, then the potential energy is V=½Iω₀²θ² (Feynman et al., 1963, p. 41-2).”

 

Feynman’s brief explanations of a rotational harmonic oscillator could be unpacked as follows:

1. Rotational Kinetic Energy

The kinetic energy T for rotational motion is given by T=½Iω², where I is the moment of inertia and ω is the angular velocity.

 

2. Restoring torque

In a torsional system, the restoring torque t is proportional to the angular displacement q. This means: t = -kq where k is the torsional constant (a measure of the stiffness of the system).

 

3. Potential Energy of a Torsional Oscillator

The potential energy V stored in this twist is analogous to that of a spring and is given by: V=½kθ². (In Feynman’s text, this is written as V=½αθ², where α is equivalent to k.

 

4. Natural angular frequency 

The system undergoes simple harmonic motion with a period t₀. The natural angular frequency ω₀ (in radians per second) is related to the period by: ω₀ = 2π/t₀.

 

5. Equation of motion of a torsional oscillator

For a torsional oscillator, the equation of motion is: Id²q/dt² = -kq.

This simplifies to: d²q/dt² = -(k/I)q which is the standard form for simple harmonic motion with angular frequency ω₀ = Ö(k/I). Solving for k: k = Iω₀²

 

6. Substituting Torsional constant into Potential Energy

Substituting this into the potential energy formula: V = ½kθ² = ½Iω₀²θ²  

The constant α in the initial potential energy expression is commonly known as the torsional constant k.

 

3. Johnson noise

“So now we can design circuits and tell when we are going to get what is called Johnson noise, the noise associated with thermal fluctuations! Where do the fluctuations come from this time? They come again from the resistor—they come from the fact that the electrons in the resistor are jiggling around because they are in thermal equilibrium with the matter in the resistor, and they make fluctuations in the density of electrons (Feynman et al., 1963, p. 41-2).”

 

Johnson noise (or thermal noise) is the random, electrically measurable voltage (or current) fluctuation produced by a resistor at thermodynamic equilibrium. It arises from the universal requirement of thermal equilibrium: any system that absorbs energy must also emit it to maintain a constant temperature.

Three Perspectives on the Phenomenon

Dynamical Perspective (Brownian Motion of Electrons): The noise manifests as the Brownian motion of electrons (charge carriers) within the resistor. The electrons are constantly being scattered by thermal vibrations of the lattice atoms, causing their velocity to fluctuate randomly. This random motion of charge creates the tiny, ever-changing voltage fluctuations across the resistor's terminals.

Thermodynamic Perspective (Black-Body Analogy): The resistor behaves like a one-dimensional black-body radiator, emitting and absorbing thermal radiation in the form of fluctuating electromagnetic fields. At thermal equilibrium, the strength of these fluctuations is precisely balanced, which ensures zero net energy flow.

Electrodynamical Perspective (Nyquist's Theorem): The noise is governed by Nyquist's (1928) theorem, an example of the Fluctuation-Dissipation Theorem, states that the mean-square noise voltage is directly proportional to the absolute temperature and the dissipation (resistance) of the system: <V²> = 4kTRΔf. In this equation, k is the Boltzmann constant, T is the absolute temperature, R is the resistance, and Δf is the bandwidth over which the noise is measured.

In short, we can unify the concepts of Brownian motion and Johnson noise under the Fluctuation-Dissipation Theorem. However, Johnson noise is not just a technical limitation in sensitive electrical circuits, it is fundamentally a manifestation of the intrinsic thermal fluctuations present in any resistive material.

 

“Let P(ω)dω be the power that the generator would deliver in the frequency range dω into the very same resistor. Then we can prove (we shall prove it for another case, but the mathematics is exactly the same) that the power comes out P(ω)dω = (2/π)kTdω, and is independent of the resistance when put this way (Feynman et al., 1963, p. 41-3).”

 

Feynman’s equation P(ω)dω = (2/π)kTdω  is potentially confusing. Below is a brief note on the conversions (with units) step-by-step, so that the relations between voltage spectral density, power spectral density, and units/conventions are clearer.

 

1) Voltage Spectral Density (Source Noise)

Based on the Johnson–Nyquist noise formula, a resistor R at temperature T exhibits an open‑circuit mean‑square voltage (voltage noise) of

 <V²(f)> = 4 kT R       (Units: V²/Hz)

(Unit check: kT has units J = V·A·s = V²·s / Ω, so kT R→ V²·s. Dividing by s (1/Hz) yields V²/Hz.)

 

2) Power delivered to a matched load (per Hz)

When the source and load resistances are equal (impedance‑matched), the available thermal‑noise power is maximized. By voltage division, half the open‑circuit voltage appears across the load, so the mean‑square voltage across the load is   <V_L²>_rms = <V²(f)>/4.

 

Power spectral density delivered to the load (W/Hz): P(f) = <V²(f)>/4R = 4 kT R/4R = kT

 

Thus, the available noise power per unit bandwidth is independent of R and equal to kT (the “Johnson noise power formula”).

 

3. Power Density per Angular Frequency (ω)

Power spectral densities expressed per hertz ( ) and per radian‑per‑second ( ) are related by P(ω) dω = P(f) df ,      df/dω = 1/2π

                             

Hence, P(ω) = P(f)(df/dω) = kT/2π

 

Conclusion

The correct power delivered by the resistor into a matched load in the angular frequency range ω is:   P(ω) dω = (kT/2π) dω

Feynman's expression P(ω) dω = (2/π)kT dω is larger by a factor of 4. The discrepancy could arise from omitting the factor 1/4 that comes from the matched‑load voltage division. Importantly, the power delivered to a matched load is a universal quantity defined solely by temperature and the Boltzmann constant, independent of the resistor's specific value R.

 

Key Takeaways: Three Types of Thermal Fluctuation

These three distinct phenomena—translational, rotational, and electrical fluctuations—are unified by the Equipartition Theorem, demonstrating how thermal energy, kT, dictates the average energy stored in every degree of freedom, regardless of the system's nature (mechanical or electrical).

Fluctuation–Dissipation Theorem (FDT): All three examples serve as evidence for the FDT, which states that the magnitude of thermal fluctuations (the jiggling, wobbling, or noise) is directly proportional to the system's ability to dissipate energy (viscosity or resistance) and the absolute temperature.

 

Macroscopic Proof of Atomism: Collectively, these "three types of Brownian motion" provide compelling, measurable proof that matter and energy are composed of discrete, constantly moving microscopic entities, turning atomic theory from a hypothesis into an established, quantitative physical fact.

 

The Moral of the Lesson:

The random motion of smoke particles—readily visible in a sunlit beam of air—is another classic example of Brownian motion. While there is no historical account of Einstein having a flash of insight about Brownian motion while watching his pipe smoke, it would not be surprising if he reflected on such everyday phenomena. Einstein was known to be a dedicated pipe smoker throughout his life and was famously quoted as asserting:

“I believe that pipe smoking contributes to a somewhat calm and objective judgment in all human affairs (Oeijord, 2011).”

He died in 1955 at age 76 from a ruptured abdominal aortic aneurysm (Tesler, 2020), a condition for which smoking is now a well-established risk factor (Aune et al., 2018). In the end, Einstein’s relationship with smoking stands as a humbling reminder that even the most extraordinary physicist remains subject to the ordinary physical consequences of human habit.

Source: www.amazon.com

 

Review Questions:

1. How would you define Brownian motion?

2. How would you explain the wobbling of a freely suspended mirror?

3. How would you explain Johnson noise?

References:

Aune, D., Schlesinger, S., Norat, T., & Riboli, E. (2018). Tobacco smoking and the risk of abdominal aortic aneurysm: a systematic review and meta-analysis of prospective studies. Scientific reports, 8(1), 14786.

Einstein, A. (1905). On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat (English translation, 1956). Investigations on the Theory of the Brownian Movement.

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.

Nyquist, H. (1928). Certain topics in telegraph transmission theory. Transactions of the American Institute of Electrical Engineers, 47(2), 617-644.

Oeijord, N. K. (2011). The General Genetic Catastrophe: On the Discovery and the Discoverer. iUniverse.

Powles, J. G. (1978). Brownian motion-June 1827 (for teachers). Physics Education, 13(5), 310.

Smoluchowski, M. (1906), Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Annalen der Physik, 21(14), 756–780.

Tesler, U. F. (2020). A history of cardiac surgery: an adventurous voyage from antiquity to the artificial heart. Cambridge Scholars Publishing.

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.