Molecular Noise and "Blackbody" Radiation

The subject of this section is the background noise level created by the random motion of gas molecules, in the absence of any other source of sound. My original motivation for this study was to determine if the threshold of human hearing is limited by molecular noise at the eardrum. It is not, but it is close.

The problem is defined as follows: a piston is at one end of a tube. Gas molecules exert pressure on the piston, which can be separated into a static DC term (atmospheric pressure), and variable AC terms (noise). The objective is to quantify these terms.

Approaches

The two approaches presented here are: (1) bounce "billiard balls" off of the piston; and (2) the "blackbody" method. These two methods can be categorized as treating sound as if it consists of particles, or of waves. Modern physics teaches that everything (light, electrons, billiard balls, etc.) has both a wave and a particle nature; for an analysis it is usually necessary to focus on one aspect or the other. The billiard ball approach is obviously in the particle camp. The "blackbody" method treats sound as waves. It would be great if both approaches gave the same answer, but they don't. For reasons discussed in more detail below, I believe the billiard ball approach gives the correct answer.

The billiard ball approach assumes that pressure is the result of individual molecular collisions, each collision causing a impulsive force. Thus a graph of pressure vs. time consists of a sequence of narrow spikes, and the pressure can theoretically drop to zero between the spikes.

The blackbody approach assumes that sound inside a cavity exists as discrete wave modes. The modes act like a collection of independent harmonic oscillators. According to statistical mechanics, each mode will on average contain kT Joules of energy, which can increase or decrease by energy quanta equal to the mode frequency times Planck's constant. The sound pressure is computed from a combination of these modes. A graph of pressure vs. time in this case is a noisy, but continuous, signal. This is exactly the same way that electromagnetic blackbody radiation is analyzed, the main difference being that the objective in that case is to quantify radiation from a cavity, rather than pressure inside the cavity.

The problem is formulated in terms of the mean and variance of the force against a piston of area A. The force is integrated over a time interval τ, equivalent to setting the bandwidth Δf=1/τ. The molecular weight is m, gas density ρo, and variance of one component of molecular velocity σ2. The piston is immovable.

Billiard ball approach

The detailed analysis for this case is presented in two separate sections: an analysis of a generic sequence of impulses, and then the application to the specific problem at hand, molecular noise. The result is a force variance of

The noise spectrum is the same as the spectrum of a single impulse, i.e. flat, out to a very high frequency. A quantum mechanics solution of a particle colliding with an obstacle indicates that the force is exerted over a time interval approximately equal to the time it takes the particle to travel one molecular diameter (see Quantum Mechanical Particles). At the average molecular speed a molecule travels one molecular diameter in 7x10-13 seconds, indicating that the upper frequency is on the order of 1012 Hz.

"Blackbody" approach

This case is also analyzed in a separate section. In this case the force variance is

This is 1.62 times the billiard ball result. Both of these results are for a monatomic molecule; both approaches predict a lower noise value for a diatomic molecule. Diatomic noise is lower by a factor of 0.917 based on the blackbody analysis, vs. 0.9565 for the billiard ball approach.

Here the pressure is a combination of sine waves at various discrete frequencies, plus a DC term. So the noise is a combination of continuous sine waves, and the spectrum consists of spikes; exactly the opposite of the billiard ball case. This is one major discrepancy. Furthermore, if the tube is made longer or wider, the frequencies change. In the billiard ball approach molecular statistics are totally independent of the size and shape of the tube.

The above blackbody result assumes that the piston occupies the entire cross section of the tube. The frequency spectrum is flat out to a frequency of 8.6x1010 Hz. In certain cases when the piston does not occupy the entire cross section of the tube, the frequency spectrum is proportional to the square of the frequency. This is clearly a very serious inconsistency. Finally, the result can also depend on the position of the piston within the tube. So there are lots of problems with the "blackbody" approach. More on this below.

Molecular Noise Level at the Eardrum

As discussed in detail by Hartmann, the ear acts as if sound is filtered in a "critical band." The ear is most sensitive around 4kHz, where the equivalent rectangular bandwidth of this filtering effect is 456 Hz, and the hearing threshold is about -3 dB SPL. Everest gives a value of A=80 square millimeters for the area of the eardrum. My guess is that the effective area is smaller; I don't know by how much, but I am going to use 40 square millimeters. The pressure is doubled to account for noise from both sides of the eardrum. Using the diatomic value computed via molecular statistics, the result is a threshold of -6 dB

The noise is generated by molecules right at the eardrum. The minimum audible pressure at the eardrum (MAP) is different than the minimum audible field (MAF) pressure of an incoming sound wave. Killon gives a value of 13 dB SPL for the MAP. This indicates that the threshold of human hearing is 19 dB higher than the molecular noise at the eardrum. There could be a significant increase in the noise level if the effective area of the eardrum is much smaller than its physical area.

There will also be a noise contribution from molecular collisions with the oval window. Its area is about 27 times smaller, increasing the noise, but there is a factor of 15 increase in signal pressure between the eardrum and oval window. This would raise the threshold another few dB.

In any case it appears that noise from air molecules is not the limiting factor in human hearing, but it is pretty close, especially considering the overall 100-120 dB dynamic range of human hearing.

The Hypothesized Flaw in the Blackbody Analysis

Representing a sound wave by a continuous mathematical function is an artificial construct that may, or may not, accurately reflect physical reality. A "real" sound wave consists of the motion of a set of discrete molecules. In most cases the behavior can be modeled with great accuracy by a continuous mathematical function, but not always.

It is also true that quantum mechanics represents molecules themselves as waves, solutions of the Schrödinger equation. But quantum mechanics also includes the correspondence principle, which says that a large collection of molecules at room temperature will behave like discrete particles, for all practical purposes.

In general, sound can be represented as a spectrum of plane waves, as discussed in the section on wave spectra. In a cavity, according to conventional sound theory, the continuous spectrum degenerates to the set of discrete frequencies given by equation (2) in the blackbody analysis section, which are a function of the cavity dimensions.

Consider a rectangular cavity with perfectly rigid walls, containing N molecules. According to quantum mechanics it is impossible to simultaneously know the positions and velocities exactly, but here we will assume that we do know the location vectors {ri} and velocity vectors, {vi}, i=1...N, of the molecules. Suppose we define the amplitude of a plane wave in the cavity traveling in the direction of a propagation vector k by

where |k|=ω/c. Further suppose the molecular velocities consist of a systematic component u(r,t) plus a random component, such that

where E{-} denotes the expected value. Then

and

By construction, if

then E{a(k)}=uo, and

Therefore in the limit of an infinite number of molecules, the definition of wave amplitude by equation (3) corresponds exactly to the conventional amplitude uo, as long as uo>0. With uo=0 the expected value of the amplitude of any particular wave is zero. But the variance, given by equation (8), is non-zero. The most significant aspect of this result is that it is true for any wave. The wave doesn't have to satisfy the boundary condition at all! Note that the molecular velocity statistics are exactly the same everywhere in the cavity, including right next to the surface of a wall.

My conclusion is that the imposition of a specific boundary condition is not valid in the case of interest, where the molecular motion is completely random.

This doesn't quite wrap up the story. I will end this section with questions rather than answers. In any finite region, any continuous function can be represented by a Fourier series. So even if you totally forget about boundary conditions, you should still be able to mathematically represent any wave by a set of modes. But are these modes then real physical harmonic oscillators?

If there is a flaw for sound waves, what about electromagnetic waves? As noted above, in this case the objective is to evaluate radiation from a cavity, rather than pressure inside the cavity. Certainly there is considerable experimental evidence that the predictions are correct, and based on classical electromagnetic theory, there is no flaw. But can one compute pressure based on photon impacts, and does the result agree with wave theory?

Note: this section was modified September 2004, including the addition of the final subsection above, and modified again in June 2006.

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Acknowledgment

Thanks to Jont Allen for bringing the Nyquist approach to my attention, and for several interesting exchanges related to this subject.