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

Used Functions

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

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

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

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

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

CALCULUS TRY 1: Velocity Distribution of Ideal-Gas

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

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

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

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

The Energy Distribution

using:

$${\displaystyle \epsilon ={\frac {1}{2}}~m~v^{2}={\frac {1}{2}}~m~(V_{x}^{2}+V_{y}^{2}+V_{y}^{2})}$$

${\displaystyle k=3}$

${\displaystyle a={\sqrt {\frac {kT}{m}}}}$

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

CALCULUS TRY 2:

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

Rewriting the Energy Distribution as Maxwell-Boltzmann Distribution

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

Equation (x) can be rewritten as

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

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

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

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

Energy Distribution of n-Degrees of Freedom

With the energy of a single degree ${\displaystyle E_{n=1}}$

$${\displaystyle \epsilon =3~E_{n=1}}$$

$${\displaystyle E_{n=1}=\epsilon \times {\frac {1}{3}}~}$$

I believe the following is NOT VALID?

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

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

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