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Consider an unmagnetized, uniform, 1dimensional plasma consisting of electrons and unitcharged
ions.
Now, ions are much more massive than electrons.
Hence, on short timescales, we can treat the ions as a static
neutralizing background, and only consider the motion of the electrons. Let be the
coordinate of the th electron. The equations of motion of the
th electron are written:
where is the magnitude of the electron charge, the electron mass, and the component
of the electric fieldstrength at position . Now, the electric fieldstrength can be
expressed in terms of an electric
potential:

(291) 
Furthermore, from the PoissonMaxwell equation, we have

(292) 
where is the permittivity of freespace, the electron number
density (i.e., is the number of electrons in the interval to ),
and the uniform ion number density. Of course, the average value of is
equal to , since there are equal numbers of ions and electrons.
Let us consider an initial electron distribution function consisting of two counterpropagating
Maxwellian beams of mean speed and thermal spread : i.e.,

(293) 
Here,
is the number of electrons between and with velocities
in the range to . Of course,
.
The beam temperature is related to the thermal velocity via
,
where is the Boltzmann constant. It is wellknown that if is significantly
larger than then the above distribution is unstable to a plasma instability
called the twostream instability.^{38} Let us investigate this instability numerically.
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Richard Fitzpatrick
20060329