In other words, given a magical room with walls that produce no vibration and transmit zero vibration from the outside, and nothing on the inside except room temperature air, what would be the noise level in dB SPL (sound pressure level) from the thermal motion of the air itself? (Similar to the noise floor of electronics being determined by thermal noise in the conductors.) What is the quietest possible anechoic chamber? What is the noise floor of air?
For reference: Sound pressure is defined as the root-mean-square value of the instantaneous pressure, measured in pascal = N/m². SPL is the same number, but expressed in decibels relative to 20 µPa.
(I assume that it has a white spectrum, but I could be wrong. Thermal noise in electronics is white, but other types of electronic noise are pink, and blackbody thermal radiation has a bandpass spectrum.)
Here's an explanation in the context of underwater acoustics. Not sure how this applies to air:
Mellen (1952) developed a theoretical model for thermal noise based on classical statistical mechanics, reasoning that the average energy per degree of freedom is kT (where k is Boltzmann’s constant and T is absolute temperature). The number of degrees of freedom is equal to the number of compressional modes, yielding an expression for the plane-wave pressure owing to thermal noise in water. For non-directional hydrophones and typical ocean temperatures, the background level due to thermal noise is given by:
NL = −15 + 20 log f (in dB re 1 µPa)
where f is given in kHz with f >> 1, and NL is the noise level in a 1 Hz band. Note that thermal noise increases at the rate of 20 dB decade−1. There are few measurements in the high-frequency band to suggest deviations from the predicted levels.
Citation is R. H. Mellen, The Thermal-Noise Limit in the Detection of Underwater Acoustic Signals, J. Acoust. Soc. Am. 24, 478-480 (1952).
Best Answer
Apparently the search term I was missing was "Brownian motion". With that, I found several leads. They contradict each other somewhat, but I can at least post a partial answer:
Geisler - Sound to Synapse: Physiology of the Mammalian Ear:
Dallos - The Auditory Periphery Biophysics and Physiology:
There's another available with more details:
Harris - Brownian motion and the threshold of hearing:
2.5 kHz to 3.5 kHz is not the total bandwidth that would be picked up by a microphone, though.
Yost & Killian - Hearing Thresholds:
So −20 to −30 dB SPL.
Howard & Angus - Acoustics and Psychoacoustics:
I would still like to know:
Update
I believe I've found an answer in these two papers, though both have errors that make it difficult to be sure:
Harris's equation 1 is taken from Sivian-White, but seems erroneous. The original is dimensionally consistent, at least:
$$\overline P = \left [ \int^{f_2}_{f_1}{P_f}^2\cdot df \right ]^{1/2} = \left [ \frac{8 \pi \rho k T} {3c} ({f_2}^3-{f_1}^3)\right ]^{1/2}$$
where $\overline P$ is RMS pressure, $\rho$ is density of air, $k$ is Boltzmann's constant, $T$ is temperature, $c$ is speed of sound, and $f_1$ and $f_2$ are the bandwidth limits.
Sivian-White then calculate $5\times 10^{-5}$ bars for 1000–6000 Hz, which... also seems erroneous. That's equal to 5 Pa, or 108 dB SPL? If I calculate over the same range, I get 5.3×10−11 bars = 5.3 µPa = −11.6 dB SPL, which seems more reasonable.
Now Harris says:
When I use Sivian's equation and Harris' −3 dB with Harris' frequency band of 2500–3500 Hz, I get Harris' answer of 1.273×10−5 dyne/cm2 (= 1.27 μPa = −24 dB SPL) so it seems like I'm doing it right.
But they're interested only in whether the self-noise of air is close to the threshold of hearing in the most sensitive band. Calculating total SPL, in the same way but over 20 Hz to 20 kHz, I get 21.8 μPa, very close to 0 dB SPL. Coincidence?
This equation also lets us calculate the spectral density, which seems to be violet noise, increasing by 6 dB every octave, same as the underwater reference in the question: