Particle collisions are responsible for equipartition of energy, but are otherwise ignored in the sections regarding the physics of sound. This section briefly outlines a justification of this position, and demonstrates that collisions rapidly result in equal Gaussian velocity distributions for the three Cartesian velocity components. Results are also presented on the mean time between collisions and mean-free path between collisions.
Particles are modeled as elastic Newtonian billiard balls in this section. A short introduction to quantum mechanical particles and collisions can be found here.
First consider the collision of two spheres of equal mass, in two dimensions. The collision is assumed to be perfectly elastic. The velocities prior to collision are
where ax and ay are Cartesian unit vectors (I don't know how to add a caret within the text).
At the moment the collision occurs, a line drawn between the centers of the two spheres is assumed to make an angle θ with respect to the x-axis. Assuming conservation of momentum and energy, the velocities after the collision are found to be
The first observation is that if the same constant vo is added to the pre-collision x velocity component of both particle "a" and particle "b", the result is that the same constant is added to the post-collision x velocity component of both particles.
If one solves the collision problem in a coordinate system moving with a constant velocity, and then transforms back to a stationary system, it is seen that this is true in general, for any component, and for collisions in 3 dimensions between particles of any shape.
Therefore if a group of particles has a non-zero mean velocity, we can subtract it from all of the velocities, solve (in principle) for all the collisions, and then add the mean velocity back in at the very end.
We now assume that θ is a random variable uniformly distributed over 0-2π. The expected values of the post-collision velocities are then all of the form
So in addition to momentum being conserved, after many collisions momentum will tend to be distributed equally among particles within a group.
Finally we assume that the velocity means are all subtracted out, and that all of the velocity components are statistically independent. The variances are then all of the form
The expectation is of course that energy is conserved. The expected value of energy due to the x component of velocity is the same for both particles. Finally, there is a probable transfer of energy between the x and y components. The overall result is that energy tends to be distributed equally among all velocity components of both particles, which is of course equipartition of energy.
This result is based on spheres colliding in two dimensions, but the principle is actually quite general (see Feynman for example). For a diatomic molecule that can spin with respect to two axis, the energy tends to be distributed equally between the five "degrees of freedom", two spinning motions and three translational velocity components. The equivalent of equations (3) and (4) are different in 3-dimensions, and for diatomic molecules, but the eventual equal distribution still holds.
Collision Frequency and Mean-Free Path
An approximate analysis of collision frequency can be based on a volume swept out by the particles per unit time. We assume all particles are spherical with diameter D. If the center of one particle gets within a distance D of the center of another particle, a collision occurs. A particle traveling with velocity v then sweeps out a volume equal to πD2vdt in time dt, meaning that if another particle center falls within this volume a collision will occur. N particles sweep out N times this volume. For a given particle center, the probability that it falls within the total volume swept out by the other particle is then approximately NπD2vdt/V, where V is the total volume, N>>1, and v is now the average particle speed. The probability of no collision for a given particle is then,
The probability of no collision after m time intervals dt is
where t=mdt. The probability of a collision is 1.0 minus equation (6), and the expected value of time for a collision is found to be
and the probability density of t can be written
The probability distribution of the distance traveled between collisions is also exponential. A Monte-Carlo simulation described next is generally consistent with this prediction, except that the mean values differ a bit from the expected value.
The first simulation is of a group of spherical particles inside a cube. The initial particle velocities are random, zero-mean Gaussian, and the initial positions random, uniform within the volume. The simulation moves the particles in time, and when two particles collide a 3-D version of equation (2) is solved to obtain the post-collision velocities. So the initial conditions are random, but the subsequent behavior is deterministic. Wall collisions assume flat walls, and all collisions are perfectly elastic. With this level of detail the number of particles is limited to about 1000 with the 600 Mhz computer I had at the time. With a density of 2.7x1025 particles per cubic meter, velocity sigma of 463 meters per second, and diameter of 3.3x10-10 meters, the mean time between collisions is 1.8x10-10 seconds, and the mean travel between collisions is 8.6x10-8 meters. This is close to handbook values for air. The probability distributions closely match an exponential distribution as expected. The time between collisions is about 25% less than the estimated value.
A second Monte-Carlo simulation simply collides particles with random orientations and velocities. In this case 200,000 particles were employed. This simulation provides an additional important result: the probability distributions of the velocity components rapidly approach equal Gaussian distributions, regardless of the initial distributions. The following (rather extreme) case was run: the initial distributions of the three Cartesian velocity components were selected to be three uniform distributions of different widths. In the figure seen here [45kb] the nearly rectangular green, black, and blue curves are histograms of the initial sample velocities for the x, y, and z components respectively. There are 200,000 particles in the sample. After each particle has had merely 10 collisions, the three velocity histograms almost perfectly overlay the Gaussian distribution shown in red. One consequence of this result is that the expected value of the cube of each velocity component is equal to the value for a Gaussian distribution. This fact is used in the derivation of the Kinetic energy flux given in Table 3. The mean value of the post -collision momentum was 2/3 of the particle's original momentum plus 1/3 of the momentum of the particle it collided with, rather than the 1/2 and 1/2 of equation (3) for the 2-dimensional case.
A third Monte-Carlo simulation was run long after first two. In this case the initial x-velocities were set equal to a perturbed Gaussian distribution, where an asymmetry was introduced to mimic the effect discussed in the section on wave attenuation. The y and z-velocities were Gaussian. All three distributions have zero mean and equal variances. The asymmetry results in a non-zero expected value of the cube of the x-velocity - the 3rd moment of the probability distribution. The collisions reduced the 3rd moment by an exponential rate, exp(-number of collisions/two).
If momentum and energy are not distributed equally among the particles initially, collisions will quickly make the distributions approach equality. Energy also tends to be distributed equally among all degrees of freedom of motion, and the probability distribution approaches a Gaussian distribution.
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