[플라즈마 물리][Plasma Physics]CH1 - Distribution Function

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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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