# How is the Maxwell-Boltzmann Distribution a Chi-square Distribution?

This Wikipedia states that the MB Distribution in terms of energy is a Chi-square Distribution with 3 degrees of freedom. I know that the probability density formula of a Chi-square Distribution with $$n=3$$ would then be: $$f_{n=3}(x)=\frac{1}{2^{\frac{3}{2}}\Gamma(\frac{3}{2})}\cdot x^{\frac{3}{2}-1}\cdot e^{-\frac{x}{2}}$$ Where $$\Gamma(\frac{3}{2})=\int_0^{\infty}x^{\frac{3}{2}-1}\cdot e^x\cdot dx=\frac{\sqrt\pi}{2}$$.

However, the MB Distribution in terms of energy $$x$$ is: $$f(x)=\frac{2}{\sqrt\pi} \cdot \bigg(\frac{1}{k_B T}\bigg)^{\frac{3}{2}}\cdot \sqrt x\cdot e^{\frac{-x}{k_BT}}$$

The factor $$\frac{1}{2^{1.5}\cdot \frac{\sqrt\pi}{2}}$$ as well as the $$e^{\frac{-1}{2}}$$ are missing in the MB Distribution.

Why are the formats of the formulas different?

• Does this answer your question? How to explain the Maxwell Boltzmann distribution graph (physically)? – Semoi Mar 25 at 17:45
• @Semoi I don't see how that is a duplicate – BioPhysicist Mar 25 at 17:47
• $\Gamma(\frac32)=\sqrt\pi/2$ – BioPhysicist Mar 25 at 17:53
• @AaronStevens: Well, there it is explicitly derived that it is a $\chi^2$ distribution. – Semoi Mar 25 at 17:54
• @Semoi Ah ok, I just looked at the question – BioPhysicist Mar 25 at 17:55

Distributions are typically characterized by their "shape". i.e. in your case the dependence on $$x$$. As you can see, both distributions depend on $$x$$ by the form $$A\sqrt x\cdot e^{-x/a}$$ for some constants $$A$$ and $$a$$, so we can consider them to be of the same distribution type.
You can also see this by taking your Chi-squared distribution, making the change of variables $$x\to2x/k_bT$$, and then renormalizing. This will give you the form of the MB distribution you have posted. So, you can view the two distributions as the same thing, just expressed over a different "$$x$$-scale" (?)
• Ah, your edit in your answer helped me a bit. I thought the stated Chi distribution in the link is considering the same $x$-scale as any other distribution of that nature. – JohnnyGui Mar 25 at 18:18