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 3/2kT 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 molecular-kinetic theory of heat, 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 a different convention for defining P(ω) or 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.

 

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.