Plasma

Lecture 1:
Basic gas Properties Fluid Dynamics

As gas atoms/molecules (or particles) move through out a rule book they collide and randomly distribute their vigour. A basic live of statistics is that random processes result in Normally distributed results. This is know as the underlying limit theorem, see for example Box hunting watch and Hunter, Statistics for experimenters (Yu Lei has the book right now so I dont go for the title quite right.) The universal distribution is

[pic].
hither [pic] is the population variance, [pic] is the population standard deviation, and [pic] is the central value. The distribution looks like

[pic]

Concept of Temperature

Maxwell and Boltzman proposed that this same distribution endure be used to model the velocity distribution of particles in thermal equilibrium. (This is known as the Maxwell Distribution, Boltzman Distribution and the Maxwell-Boltzman Distribution.) use this assumption we have that the velocity distribution is

[pic],

while the faculty distribution is

[pic].

Here k is Boltzmans constant and T is the temperature. Notice that the temperature is simply a measure of the random energy of the system. NOTE that often kT is simply written as T and hence T is given units of energy.

Problem 1
Show that if in one dimension
[pic]
then
[pic].




Also show that if in three dimensions
[pic]
then
[pic].

Here n is the particle density.

Typically, the velocity distributions are well behaved and we have distributions as such:

The Maxwellian Distribution

[pic]

Where
[pic].
This is what happens if there are enough collisions to every bit distribute the energy. In other cases, however, we will be works in with a system in which the rate of collisions is not large enough to evenly distribute the energy. Some classifiable examples are:

The bi-Maxwellian Distribution

[pic]

Where
[pic].

The drifting Maxwellian Distribution

[pic]

Where
[pic].

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