Energy Distribution of an Ideal-Gas with n-degrees of Freedom

Used Functions

$\Gamma (x)$ and $\gamma (x)$ , are the upper and lower incomplete gamma function.

With the value $\Gamma ({\frac {3}{2}})={\frac {\sqrt {\pi }}{2}}$

The PDF and the CDF of the Chi-squared distribution are:

$\chi _{PDF}^{2}(x)={\frac {1}{{2}^{\frac {k}{2}}~\Gamma ({\frac {k}{2}})}}~{x}^{{\frac {k}{2}}-1}~\exp(-{\frac {x}{2}})$ $\chi _{CDF}^{2}(x)={\frac {1}{\Gamma ({\frac {k}{2}})}}~\gamma ({\frac {k}{2}},x)$

CALCULUS TRY 1: Velocity Distribution of Ideal-Gas

The distribution for a velocity in a single direction (Degrees of Freedom n=1)

$f(V_{i})={\sqrt {\frac {m}{2~\pi ~kT}}}~\exp \left[-{\frac {m~V_{i}^{2}}{2~kT}}\right]$

The distribution for absolute velocity (Degrees of Freedom n=3)

$f(v)={({\frac {m}{2~\pi ~kT}})}^{\frac {3}{2}}~4~\pi ~v^{2}\exp \left[-{\frac {m~v^{2}}{2~kT}}\right]$

The Energy Distribution

using: $\epsilon ={\frac {1}{2}}~m~v^{2}={\frac {1}{2}}~m~(V_{x}^{2}+V_{y}^{2}+V_{y}^{2})$ $k=3$ and a scale parameter $a={\sqrt {\frac {kT}{m}}}$

$f(v)={({\frac {1}{2~\pi }})}^{\frac {1}{2}}~2~{\frac {v^{2}}{a^{3}}}~\exp \left[-{\frac {1}{2}}~{({\frac {v}{a}})}^{2}\right]$

CALCULUS TRY 2:

$f_{\epsilon }({\frac {\epsilon }{kT}})={\frac {2}{\sqrt {\pi }}}~{\sqrt {\frac {\epsilon }{kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]$

Rewriting the Energy Distribution as Maxwell-Boltzmann Distribution

Using $k=3$ and $x=2~{\frac {\epsilon }{kT}}$ .

Equation (x) can be rewritten as

$f_{\epsilon }({\frac {x}{2}})={\frac {1}{\Gamma {\frac {k}{2}}}}~{x}^{{\frac {k}{2}}-1}~\exp({\frac {x}{2}})=\chi _{PDF}^{2}(x)$

So the Energy Cumulative Distribution Function (CDF) of three degree of freedom $n=3$ is:

$F_{\epsilon }({\frac {x}{2}})=\chi _{CDF}^{2}(x)={\frac {1}{\Gamma {\frac {k}{2}}}}~\gamma ({\frac {k}{2}},x)$

$F_{\epsilon }({\frac {\epsilon }{kT}})={\frac {1}{\Gamma {\frac {3}{2}}}}~\gamma ({\frac {3}{2}},2~{\frac {\epsilon }{kT}})$

Energy Distribution of n-Degrees of Freedom

With the energy of a single degree $E_{n=1}$ $\epsilon =3~E_{n=1}$ We have $E_{n=1}=\epsilon \times {\frac {1}{3}}~$

I believe the following is NOT VALID?

the energy in a diatomic gas with n=5 is $E_{n=5}=5\times E_{n=1}={\frac {5}{3}}\times \epsilon$

$\epsilon =E_{n=5}\times {\frac {3}{5}}$

$F_{\epsilon }({\frac {\epsilon }{kT}})={\frac {1}{\Gamma {\frac {3}{2}}}}\gamma ({\frac {3}{2}},2{\frac {3}{5}}~{\frac {E_{n=5}}{kT}})$

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