The physics of sound in air, from a statistical point of view (part 2)

This is the second part of a study on the statistical nature of sound.

The incident velocity of the collision

Let's consider a particle moving along the x-axis towards the right, in an isolated system with zero average particle velocity. A single particle, moving towards the right, among several particles moving according to the statistical distribution of ideal gases shown in the previous post. Now let's attach the reference point to the particle. This will be equivalent to a situation where the particle stays still, and all the other particles follow a shifted velocity distribution. The probability distribution will be shifted by the velocity of the single particle, but in in the opposite direction.

Note that as we are now investigating things from the perspective of the particle, all collisions will be identical, no matter what the direction of the incoming particle is. Only the velocity of the incoming particle and the angle of incidence matters. This is illustrated in the following image; we can rotate the possible collisions coming from a certain direction any way we like, the situation stays symmetric as the particle doesn't have any velocity from the perspective of the particle.

Collisions only depend on velocity and angle of incidence of other particles if we consider the situation from the perspective of a single particle

Collisions only depend on velocity and angle of incidence of other particles if we consider the situation from the perspective of a single particle

Now we have two things to sort out: how fast will the incoming particle move, and what angle will it arrive from?

Almost all the information we require to solve this is shown in the shifted probability distribution. As an intuitive example: the further we shift the probability distribution to the left, the larger portion of the distribution will be to the left of the particle. In practice this means that when a single particle moves fast enough to the right, almost all particles it encounters will move to the left (relative to the moving particle).

The faster some other particle moves, the more probable it is that it will hit the static particle we're observing within a given time. This relationship is very nearly linear, which can be based on the following: the area a single particle, somewhere in the box, will cover in a given time is almost directly proportional to its speed. Thus the chance that the area will overlap the area required for a collision is also directly proportional to its speed (except at very small velocities, which we can safely ignore here, as the average distance between collisions is significantly larger than the area of one particle, these small velocity collisions happen very rarely and almost no kinetic energy is transferred when they happen). This probabilistic relationship is shown in the figure below. The figure shows a random particle and the probability that it will hit another particle, purely on the basis of its velocity. Note that this figure will approach infinity if there are an infinite number of velocities (the probability will continue rising forever with increasing velocity), so the statistical distribution will always need to be confined into some specific range of velocities.

The probability of another randomly distributed particle hitting the particle in the center, as a function of velocity

The relative probability of another randomly distributed particle hitting our particle, as a function of velocity

Note also that the statistical distribution shown to the left is purely based on velocity. It doesn't take into account the distribution of velocities (the chance that a random particle has a certain velocity, based on the Maxwell speed distribution). The distribution of the velocities is taken into account in the shifted probability distribution.

To get the probability distribution for how likely it is that a random particle with some velocity will hit our particle, we do the following: we multiply the shifted particle velocity distribution with the probability distribution based purely on velocity (and normalize this distribution so that the statistics are valid, but never mind this now). This can be seen, in the following figure, to show what intuitively can be expected; if the particle is moving to the right, it is much more likely that a collision will happen with a particle which is moving to the left relative to the particle. Note that this statistical distribution will look different depending on the velocity of the particle we are investigating!

Probability of a particle colliding with a static particle, given the velocity

Probability of a  random particle colliding with a static particle if a certain average particle velocity exists, given the velocity of the random particle

To be continued (some day)!


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