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).
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