Mean free path

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In molecular physics, mean free path is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles.

It is defined by

[math]\ell = \frac{\mu}{p} \sqrt{\frac{\pi k_\text{B} T}{2 m}},[/math]

or alternately,

[math]\ell = \frac{\mu}{p} \sqrt{\frac{\pi RT}{2 M}},[/math]

where:

[math]\ell[/math] is the mean free path, [math]m[/math] is the mass of the particle [math]M[/math] is the molar mass of the substance [math]\mu[/math] is the gas's viscosity [math]R[/math] is the universal gas constant Kb is Boltzmann constant, T is temperature P is pressure

Application

Mean free path determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer the pressure wave.

In air at STP, mean free path is about 68 nm. Together with 340 m/s sound speed, this gives a maximum of 5 GHz of frequency. If there is a species which can hear this high of a sound, the maximum EDO which can be applied will be defined by the species' lowest frequency, or the place where the difference between rhythm and pitch occurs.

If the beat rate is less than the lowest frequency,beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair.

If humans could hear to the mean free path limit, this would limit our perception range to [math]\frac{5\cdot10^9 \ln{2}}{16}[/math] or about 216,608,494-EDO. This corresponds to a step size of about 5.5 x 10-6 cents, which is roughly the size of the unnoticeable rascal comma, [-7470 2791 1312⟩. Playing in a temperament which makes a distinction for intervals this small is physically impossible without beating interference on all notes.

However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters.

In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice. Theoretically, an alien civilization with their "ears" designed to hear through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".