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Temperature
The one-dimensional Maxwellian distribution is given by
\[\begin{aligned} f(v)=A e^{-\frac{mv^2}{2K_BT}}\end{aligned}\]
Unlike normal distribution gaussian equation can have a form of
\[\begin{aligned} \label{eq_4} f(x) = \frac{n}{\sqrt{2\pi}\sigma}e^{-\frac{(v-v_{\mu})^2}{2\sigma^2}}\end{aligned}\]
Where \(n\) is the number density. \(fdv\) is the number of particles per [\(m^3\)] with velocity between \(v\) and \(v+dv\), \(\frac{1}{2}mv^2\) is the kinetic energy, and \(K_B\) is the Boltzmann’s constant. The density \(n\), or number of particles per [\(m^3\)], is given by
\[\begin{aligned} n=\int_{-\infty}^{\infty}f(v)dv\end{aligned}\]
so that the constant \(A\) is found to be
\[\begin{aligned} A=n\sqrt{\frac{m}{2\pi K_BT}}\end{aligned}\]
Where \[\begin{aligned}
\int_{-\infty}^{\infty}e^{-ax^2}dx = \sqrt{\frac{\pi}{a}} \end{aligned}\] is used.
\(\bullet\) meaning of T = Distribution of the particles
\[\begin{aligned} \label{eq_5} E_{av}=\frac{\int_{-\infty}^{\infty}\frac{1}{2}mu^2 f(u) du}{\int_{-\infty}^{\infty} f(u) du}\end{aligned}\]
Defining \(v_{th}=\sqrt{\frac{2K_BT}{m}}\) and \(y=\frac{u}{v_{th}}\), 1-D Maxwellian distribution can be written as
\[\begin{aligned} f(u)=Ae^{-\frac{u^2}{v_{th}^2}}\end{aligned}\]
By substitution average kinetic energy becomes
\[\begin{aligned} E_{av}&=\frac{\frac{1}{2}mAv_{th}^3 \int_{-\infty}^{\infty} e^{-y^2}y^2 dy}{Av_{th}\int_{-\infty}^{\infty} e^{-y^2}dy}\\ &=\frac{\frac{1}{2}mAv_{th}^3 \frac{1}{2}}{A v_{th}}=\frac{1}{4}mv_{th}^2=\frac{1}{2}K_BT\end{aligned}\]
Thus the average kinetic energy is \(\frac{1}{2}K_BT\).
In three dimensions,
\[\begin{aligned} f(u,v,w)=n\left( \frac{m}{2\pi K_BT } \right)^{\frac{3}{2}}e^{-\frac{\frac{1}{2}m\left( u^2 + v^2 + w^2 \right)}{K_BT}}\end{aligned}\]
Using similar calculation we get
\[\begin{aligned} E_{av}=\frac{3}{2}KT\end{aligned}\]
The general result is that \(E_{av}\) equals \(\frac{1}{2}K_BT\) per degree of freedom.
\(\bullet\) Since \(T\) and \(E_{av}\) are so closely related, it is customary in plasma physics to give temperatures in units of energy.
\(\bullet\) To avoid confusion, it is not \(E_{av}\) but the energy corresponding to \(KT\) that is used to denote the temperature.
\[\begin{aligned} T=\frac{1.6 \times 10^{-19}}{1.38 \times 10^{-23}}=11600\end{aligned}\]
Thus the conversion factor is
\[\begin{aligned} 1eV = 11,600 ^{\circ}K \end{aligned}\]
\(\bullet\) By a \(2eV\) plasma we mean that \(KT=2eV\), or \(E_{av}=3eV\) in three dimensions.
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