Distribution function
\(\bullet\) The most detailed description of a plasma gives the location and velocity of each plasma particle as a function of time.
\(\bullet\) It is impossible to obtain such a description of a real plasma. Rather than require an exact knowledge of a system with many particles, the behavior of such a particle system can be studied statistically.
\(\bullet\) It is customary to use the distribution function to describe a plasma. The distribution function is the number of particles per unit volume in phase space. \[f(\vec{r},\vec{v},t)d\vec{r}d\vec{v}\] represents the expected number of particles at time \(t\) in \((\vec{r},\vec{v})\) (6D phase) space with coordinates \(\vec{r}\) and \(\vec{r}+d\vec{r}\) and velocity \(\vec{v}\) and \(\vec{v}+d\vec{v}\)
\(\bullet\) A gas in thermal equilibrium has particles of all velocities, and the most probable distribution of these velocities is known as the Maxwellian distribution.
Maxwellian distribution
\(\bullet\) Maxwellian distribution is nothing but a Gaussian distribution. Recall, 1D Gaussian equation is given as
\[\begin{aligned} \label{eq_2} f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(v-v_{\mu})^2}{2\sigma^2}}\end{aligned}\]
where \(v_{\mu}\) is mean or expectation of the distribution (and also its median and mode), \(\sigma\) is the standard deviation, and \(\sigma^2\) is the variance.
\(\bullet\) Let us consider the mean velocity is \(0\); \(v_{\mu}=0\), and thermal speed(standard deviation) is \(\sigma=\sqrt{\frac{K_BT}{m}}\). Then, by simple substitution, we get Maxwellian distribution for particles.
\[\begin{aligned} \label{eq_3} f(u)= \sqrt{\frac{m}{2\pi K_BT}} e^{-\frac{\frac{1}{2}mv^2}{K_BT}}\end{aligned}\]
What is f(u)?
\(f\) would be a fractional distribution. Suppose a teacher gave a 100-point quiz to a large number N of students. \(n_i\) of students got \(s_i\) scores. The fractional distribution would be
\[\begin{aligned} f_i = \frac{n_i}{N} \end{aligned}\]
Notice that \(\sum_i f_i =1\)
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