Sequence of Random Impulses

In this section we are concerned with a function of time that consists of a sequence of impulses, with random magnitudes, occurring at random times. Specifically, the force produced by molecules colliding with a wall. This function is very similar to "shot noise" (see Papoulis for example).

Analysis

We assume that N collisions occur during a time period T, where N/T approaches a constant rate as T → ∞. We want to compute the DC and AC components of the force within a bandwidth Δf, or equivalently, the mean and variance over a time interval τ=1/Δf. The duration of a single collision is Δt, where Δt/τ → 0. Each collision reverses the x-component of molecular velocity.

The total change in momentum over the time interval is:

Where m is the molecule mass, vn is the x-component of velocity of the nth molecule, and gn equals 1 if the collision occurs within the time interval τ, and zero otherwise. Both vn and gn are real-valued random variables. The collision times are uniformly distributed in the interval 0 ≥ t ≥ T, and statistically independent. The expected value of gn is simply

The velocities are also statistically independent, and independent of gn. The expected value of M is then

The square of M is

For n=q, gngq equals 1 if the nth collision occurs within the time interval τ, and zero otherwise. Therefore

The expected value of M2 is then

Force is the rate of change in momentum, so the expected value and variance of the force are, for τ/T → 0,

Thus given the mass m and time interval τ, the mean and variance of the force are determined by: (1) the molecular collision rate N/T, and (2) the velocity statistics.

Consider the relative power level of the AC and DC components, which is equal to

Nτ/T is the expected value of the number of collisions in the period. The force variance, relative to the square of the force produced by atmospheric pressure, is inversely proportional to that number. The remaining factor, for the case where the collision velocities have a Rayleigh distribution, is

This represents the increase in force variance due to the variation in collision velocity. If the collision velocity is constant, this ratio=1, and the force variance is caused solely by the variation in the number of collisions. Therefore for the Rayleigh case the variation in the number of collisions contributes 78.5% of the force variance, and the variation in the magnitude of the collisions contributes the remaining 21.5%.

Diatomic molecule case

For the case of a diatomic molecule, the first question is: when both ends of the molecule collide with the surface, does this count as two collisions? If both ends always collide within a time interval much smaller than τ, the two impulses are effectively merged into one. For time intervals of interest, this will be the case for an overwhelming fraction of the collisions. Then the force variance component due to collision rate will be the same for the diatomic and monatomic cases. The variance component due the magnitude of the collisions will differ. But the collision magnitudes are nearly equal, and this only effects 21.5% of the variance, so we would not anticipate a great difference between the two molecule types. As discussed in the main section, this difference has been determined by a Monte-Carlo simulation of dumbbell collisions, and it turns out that the diatomic case results in a noise power .9565 times the monatomic noise power.

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