# 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)? Commented Mar 25, 2020 at 17:45
• @Semoi I don't see how that is a duplicate Commented Mar 25, 2020 at 17:47
• $\Gamma(\frac32)=\sqrt\pi/2$ Commented Mar 25, 2020 at 17:53
• @AaronStevens: Well, there it is explicitly derived that it is a $\chi^2$ distribution. Commented Mar 25, 2020 at 17:54
• @Semoi Ah ok, I just looked at the question Commented Mar 25, 2020 at 17:55

Any constants multiplying the distribution aren't really important for your question, as they are just in charge of normalizing the distribution.

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" (?)

• But the wiki en.wikipedia.org/wiki/… states that the format for a Chi-square Distribution must have these factors if it's written as a probability density, even if they are constants.
– Phy
Commented Mar 25, 2020 at 17:40
• @JohnnyGui Can you quote exactly where that is said? It says something like "If the constants are not these exact values then it is not a Chi-squared distribution"? Commented Mar 25, 2020 at 17:42
• It shows the formula right under Probability density function
– Phy
Commented Mar 25, 2020 at 17:46
• @JohnnyGui Right. That doesn't mean if those constants are not exactly the same that it doesn't classify as a Chi-squared distribution. Commented Mar 25, 2020 at 17:46
• 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.
– Phy
Commented Mar 25, 2020 at 18:18