Section 41–2 Thermal equilibrium of radiation

Oscillator (idealization) / Damping (approximation) / Blackbody radiation

 

In this section, Feynman discusses a charged oscillator (idealization), its damping force (approximation), and blackbody radiation that are related to the derivation of Rayleigh-Jean’s law of radiation. Essentially, the derivation is based on the assumption of the charged oscillator in a box and the approximation method involving damping force, which results in the incorrect law for blackbody radiation. Thus, the section could be titled “A derivation of Rayleigh-Jean’s law” instead of “Thermal equilibrium of radiation.” Although Feynman’s approach differs substantially from Rayleigh’s (1900) paper titled Remarks upon the law of complete radiation, both arrive at the same result.

 

1. Oscillator (Idealization)

“Suppose we have a charged oscillator like those we were talking about when we were discussing light, let us say an electron oscillating up and down in an atom. If it oscillates up and down, it radiates light. Now suppose that this oscillator is in a very thin gas of other atoms, and that from time to time the atoms collide with it. Then in equilibrium, after a long time, this oscillator will pick up energy such that its kinetic energy of oscillation is ½kT, and since it is a harmonic oscillator, its entire energy will become kT… (Feynman et al., 1963, p. 41-3).”

 

Feynman derives Rayleigh-Jeans Law of radiation by using three classical assumptions: (1) Idealized Atom: An atom is modeled as a negatively charged electron bound to a fixed positive nucleus by a massless, perfect spring, forming a classical harmonic oscillator. (2) Ideal cavity: The radiation is confined within a cavity whose perfectly reflecting, conducting walls allow the system to reach thermodynamic equilibrium. (3) Classical Equipartition Theorem: Each classical harmonic oscillator has two quadratic degrees of freedom (one for the electric field and one for the magnetic field energy). These assumptions result in the same formula as Rayleigh-Jeans Law, which diverges at high frequencies—ultraviolet catastrophe—thereby signaling a failure of classical physics.

 

Historically, Rayleigh assumed a cubical cavity filled with electromagnetic radiation in thermal equilibrium with its walls. He treated the radiation as a system of standing electromagnetic wave and counted the number of allowed vibration modes within a small frequency interval (dn). Based on his method, he concluded that the number of modes per unit volume was proportional to the square of the frequency, N(n) µ n². Next, he applied the classical Equipartition Theorem, which states that every mode must possess an average energy of kT. Multiplying the density of modes by the average energy per mode (U(n) = N(n) ´ kT), he obtained the spectral energy density, U(n) µ n²kT. This formula, also known as the Rayleigh-Jeans Law, only agrees with experimental data at low frequencies. Compared with Feynman’s derivation, which models matter–radiation interaction using a damped charged oscillator, Rayleigh’s classical wave-based method is conceptually simpler and mathematically more direct.

 

2. Damping (Approximation)

“Thus we first calculate the energy that is radiated by the oscillator per second, if the oscillator has a certain energy. (We borrow from Chapter 32 on radiation resistance a number of equations without going back over their derivation.) The energy radiated per radian divided by the energy of the oscillator is called 1/Q (Eq. 32.8): 1/Q=(dW/dt)/ω₀W. Using the quantity γ, the damping constant, this can also be written as 1/Q=γ/ω₀, where ω₀ is the natural frequency of the oscillator—if gamma is very small, Q is very large (Feynman et al., 1963, p. 41-4).”

 

It is important to recognize that Feynman’s derivation relies on two substitutions that are conceptually different in purpose and interpretation: ω ® ω₀  and ω₀ ® ω.

 

ω®ω₀ (Replace ω by ω₀)

Feynman’s derivation employs two main approximations to simplify the mathematics and ensure the final integral tractable, followed by a third, fundamentally fatal assumption that leads to the Rayleigh–Jeans law.

A.    The Narrow Peak (Resonance) Approximation

This approximation simplifies the analysis of the scattering cross-section by using the high quality factor (Q) of the charged oscillator.

Assumption: The oscillator scatters incoming radiation very strongly only when the incident frequency (ω) is extremely close to its natural resonant frequency (ω₀). Since Q is very large (~10⁸), the scattering cross-section (s_s) is sharply peaked in a very narrow interval around ω₀.

Simplification: Near resonance, the denominator in s_s can be approximated as

ω²-ω₀² = (ω+ω₀)(ω-ω₀) » 2ω₀(ω-ω₀)

Purpose: This simplifies the scattering cross-section s_s into the form of a resonance curve, allowing the integral to be evaluated analytically.

 

B.     The Smooth Spectrum (Constant Intensity) Approximation

This approximation simplifies the integration of the energy balance equation.

Assumption: The spectral intensity of black-body radiation, I(ω)—the unknown quantity being solved for—varies very slowly across the extremely narrow frequency range where the scattering cross-section s_s is significant.

Simplification: Because the resonance peak is so narrow (due to the high Q factor), the function I(ω) can be treated as a constant value, I(ω₀), and pulled outside the integral sign:

òI(ω) s_s(ω) dω » I(ω₀) ò s_s(ω) dω

Purpose: This allows I(ω₀) to be factored out of the integral and expressed it in terms of known constants (k, T, g) and a standard integral of the form ò dx/(x²+a²) = p/a.

C.    The Fatal Approximation: Classical Equipartition

Beyond these approximations lies the crucial classical assumption that ultimately invalidates the final formula.

Assumption: Every standing-wave mode (degree of freedom) of the electromagnetic field possesses an average energy kT, as stated by the classical equipartition theorem.

Why it fails:

The fundamental problem in this approximation is to assume the charge oscillator can possess a continuous range of energies, rather than being restricted to discrete energy levels.

 

Summary

The first two approximations are physically reasonable relying on resonance and constancy arguments. The breakdown arises from the third approximation—classical equipartition—which incorrectly links the possible vibrational (electromagnetic) modes to thermodynamics. This fundamental flaw necessitated Planck’s introduction of energy quantization and marked the collapse of classical physics in the description of blackbody radiation.

“Then we substitute the formula (41.6) for gamma (do not worry about writing ω₀; since it is true of any ω₀, we may just call it ω) and the formula for I(ω) then comes out I(ω) = ω²kT/π²c² (Feynman et al., 1963, p. 41-5).”

 

ω₀®ω (Why "We May Just Call It ω")

After solving the integral, Feynman obtains I(ω₀) = ω₀²kT/π²c², he then says, "since it is true of any ω₀, we may just call it ω." In a sense, this means that the oscillator with natural frequency ω₀ served as a “probe” to measure the energy density at its resonant frequency.

(In principle, one can construct oscillators with any arbitrary resonant frequency, the formula must hold for all ω.) More important, Feynman later emphasizes an consistency check on the derivation itself: “The charge of the oscillator, the mass of the oscillator, all properties specific to the oscillator, cancel out, because once we have reached equilibrium with one oscillator, we must be at equilibrium with any other oscillator of a different mass...” The complete disappearance of the oscillator’s specific properties in the intensity formula indicates the universality of the derived classical black-body radiation law. However, the functional form of Feynman’s result is identical to that derived from Rayleigh’s method—a clear indication that both share the same flaw in the classical limit. This is another reason why Feynman “did not worry” about his derivation; he could have worked backward from the known formula of Rayleigh.

 

Note: The apparent discrepancy between the Rayleigh-Jeans law and Feynman's formula arises from the different, though physically equivalent, quantities using distinct mathematical conventions. In addition, the Rayleigh-Jeans law itself appears in several equivalent forms, depending on whether it is expressed in terms of frequency or angular frequency, intensity or energy density, and per unit volume or per unit solid angle. Once these definitional and normalization differences are accounted for, Feynman’s formula is seen to be consistent with the Rayleigh-Jeans law. 

 

3. Blackbody radiation

“It is called the blackbody radiation. Black, because the hole in the furnace that we look at is black when the temperature is zero (Feynman et al., 1963, p. 41-6).”

 

Historically, the concept of the blackbody was introduced by Gustav R. Kirchhoff in 1860, who defined it as an object that absorbs all incident electromagnetic radiation (a perfect absorber). Kirchhoff suggested:

“If a volume is enclosed by bodies of the same temperature and rays cannot penetrate those bodies, then each bundle of rays inside this volume has the same quality and intensity it would have had if it had come from a completely black body of the same temperature, and is therefore independent of the constitution and the shape of these bodies and is determined by the temperature alone (Hoffmann, 2009, p. 36-37).”

This statement summarizes Kirchhoff's Law of thermal radiation: the spectral distribution of the radiation within a closed cavity at thermal equilibrium is universal and depends only on the absolute temperature. However, the so-called blackbody becomes highly luminous when heated, glowing red, yellow, or white depending on its temperature. Astrophysical objects like the Sun are considered near-ideal blackbody radiators because their characteristic continuous emission spectrum is defined solely by their temperature, a property unrelated to their visual color. Strictly speaking, no perfect blackbody exists in nature, and the term blackbody is, therefore, something of a misnomer. Because the phenomenon fundamentally concerns the thermal radiation spectrum rather than the visual appearance of an object, many physicists today prefer the term thermal radiation or cavity radiation.

 

Note: In Griffiths’ (2017) words: “… This is called blackbody radiation. It’s a real misnomer – the Sun is, in this sense, a ‘blackbody’! It should be called ‘thermal radiation,’ because it is due to the random thermal motion of the charged particles (especially electrons) of which the object is composed.”

 

“Of course we know this is false. When we open the furnace and take a look at it, we do not burn our eyes out from x-rays at all. It is completely false. Furthermore, the total energy in the box, the total of all this intensity summed over all frequencies, would be the area under this infinite curve. Therefore, something is fundamentally, powerfully, and absolutely wrong. Thus was the classical theory absolutely incapable of correctly describing the distribution of light from a blackbody, just as it was incapable of correctly describing the specific heats of gases. Physicists went back and forth over this derivation from many different points of view, and there is no escape. This is the prediction of classical physics. Equation (41.13) is called Rayleigh’s law, and it is the prediction of classical physics, and is obviously absurd (Feynman et al., 1963, p. 41-6).”

 

The Rayleigh-Jeans Law suffered from critical failures that revealed the limits of classical physics. Firstly, it was unable to predict the characteristic peak in a blackbody's emission spectrum, contradicting the established experimental observation that radiance reaches a maximum at a specific wavelength. Secondly, its predictions deviated severely from empirical data at high frequencies, being accurate only in the long-wavelength, infrared regime. The most profound failure, known as the ultraviolet catastrophe, was its prediction that energy output would diverge to infinity at short wavelengths, implying that every object should radiate an infinite amount of ultraviolet and higher-frequency light, a result that was both physically impossible and in direct violation of the principle of energy conservation.

 

Key Takeaways:

Why the Beach is Safe (Absence of the Ultraviolet Catastrophe)

An explanation to relax safely on the beach is contributed by Planck’s quantum hypothesis in 1900, which solved the classical physics problem known as the Ultraviolet Catastrophe.

1. The Classical Prediction: The Rayleigh-Jeans Law predicted that the energy emitted by a blackbody (e.g., the Sun) should increase infinitely as the wavelength of the light decreases (i.e., moving into the UV and X-ray regions).
2. The Ultraviolet Catastrophe: If this classical prediction were true, the sun would emit most of its energy as lethal, high-frequency, short-wavelength radiation (ultraviolet, X-rays, and Gamma rays). The intense high-energy radiation would instantly incinerate all life on Earth, making the beach—and the planet—uninhabitable and unenjoyable.
3. The quantum solution: Planck’s quanta restrict the amount of high-frequency radiation that can be produced at a given temperature, allowing the black-body radiation curve to fall sharply in the ultraviolet region.

In short, the Ultraviolet Catastrophe does not occur because energy is quantized (Planck's Law). Importantly, the Sun's energy peak is within the visible and infrared spectrum—not the high-energy UV range—a crucial fact that permits life to flourish and people to enjoy the beach.

 

The Moral of the Lesson:

In his autobiography, Feynman mentions about enjoying the beach in the afternoon in Brazil...

In Feynman’s (1985) words: When I got to the center, we had to decide when I would give my lectures--in the morning, or afternoon. Lattes said, "The students prefer the afternoon."

“But the beach is nice in the afternoon, so why don't you give the lectures in the morning, so you can enjoy the beach in the afternoon.”

"But you said the students prefer to have them in the afternoon."

"Don't worry about that. Do what’s most convenient for you! Enjoy the beach in the afternoon."

So I learned how to look at life in a way that's different from the way it is where I come from. First, they weren't in the same hurry that I was. And second, if it's better for you, never mind! So I gave the lectures in the morning and enjoyed the beach in the afternoon. And had I learned that lesson earlier, I would have learned Portuguese in the first place, instead of Spanish……

 

One day, about 3:30 in the afternoon, I was walking along the sidewalk opposite the beach at Copacabana past a bar. I suddenly got this treMENdous, strong feeling: "That's just what I want; that'll fit just right. I'd just love to have a drink right now!" I started to walk into the bar, and I suddenly thought to myself, "Wait a minute! It's the middle of the afternoon. There's nobody here, There's no social reason to drink. Why do you have such a terribly strong feeling that you have to have a drink?"--and I got scared. I never drank ever again, since then. I suppose I really wasn't in any danger, because I found it very easy to stop. But that strong feeling that I didn't understand frightened me. You see, I get such fun out of thinking that I don't want to destroy this most pleasant machine that makes life such a big kick. It's the same reason that, later on, I was reluctant to try experiments with LSD in spite of my curiosity about hallucinations (p. 204).”

 

This passage from Feynman's autobiography contains at least two distinct lessons, both contributing to a central moral about self-awareness, intellectual integrity, and optimizing one's life experience. In essence, the moral is: Live intentionally. Whether it's a societal expectation or a sudden craving, pause and ask "Why?" Choose your actions based on understanding and reason, not on unexamined habit or impulse. This disciplined self-awareness is what allows you to preserve your greatest asset—your conscious, thinking self—and maintain your freedom to enjoy life fully.

 

Feynman was enjoying the beach by “absorbing” Vitamin D without ultraviolet catastrophe.

Malibu, 1950 (Source: No Ordinary Genius: The Illustrated Richard Feynman, p. 88)

 

Is a Physicist's Beach Day an Act of Faith?

One might ask why a physicist, who understands the sun's damaging UV radiation, would willingly bask in it. The answer is a calculated balance of biological need and physical trust.
The compelling reason is Vitamin D. The body uses UV-B photons as a catalyst to produce this crucial hormone, supporting skeletal and immune health. Without it, no amount of sunscreen or caution matters for long-term wellness.

The permission comes from Planck. Our field tells us the solar spectrum is benign at its core, governed by quantum mechanics that prevent an ultraviolet energy overflow. We enjoy the beach with faith that the physics is sound and the risk manageable.

In short, the physicist enjoys the beach with informed intentionality: to absorb a precise band of solar radiation for biological benefit, while aware of the dangers of its higher-energy UV components.

 

References:

Feynman, R. P. (1985). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

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.

Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Pearson.

Hoffmann, D. (2009). Black Body. In Compendium of Quantum Physics (pp. 36-39). Berlin, Heidelberg: Springer Berlin Heidelberg.

Rayleigh, L. (1900). Remarks upon the law of complete radiation. Phil. Mag, 49, 539.

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.

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