# Maxwell-Boltzmann Equation (avg velocity)

A question asked me to derive a symbolic expression for mean particle speed using the Max-Boltz equation. I know that Max-Boltz equation is a function of velocity (v).
The Max-Boltz equation is ##f=(\frac{m}{2\pi kT})^{3/2}4\pi v^2 exp(\frac{-mv^2}{2kT})##
Apparently the general formula for the average given a statistical function is ##\bar{v}=\int_{0}^{\infty}\frac{fvdv}{n}##
Here is what I don't understand:

1) Where did this formula come from? Does this formula only apply to statistical functions? What is a statistical function?
2) It turns out that division by the number of particles (n) is unnecessary for the Max-Boltz equation. What is reasoning behind this?
3) Why is the integration from zero to infinity? Clearly no particle can have infinite velocity...

Thank you again!

## Answers and Replies

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DrClaude
Mentor
1) Where did this formula come from? Does this formula only apply to statistical functions? What is a statistical function?
Consider a random variable with discrete values (such as a die). The expectation value is given by summing over the probability of each event times its value:
\begin{align} \langle x \rangle &= \sum_i P_i x_i \\ &= \frac{1}{6} 1 + \frac{1}{6} 2 + \frac{1}{6} 3 + \frac{1}{6} 4 + \frac{1}{6} 5 + \frac{1}{6} 6 = 3.5 \end{align}
The first line above is the generic equation, the second line is the specific example of a six-sided die. When the random variable is continuous, the sum becomes an integral:
$$\langle x \rangle = \int f(x) x \, dx$$
where ##f(x)## is the probability density function (pdf), i.e., the probability that the random variable will have a value between ##x## and ##x + dx##.

2) It turns out that division by the number of particles (n) is unnecessary for the Max-Boltz equation. What is reasoning behind this?
I guess it depends on how the pdf is defined. Normally, the MB distribution will give you the speed pdf per particle, so there is no factor 1/n.

3) Why is the integration from zero to infinity? Clearly no particle can have infinite velocity...
Thank you again!
This is a non-relativistic theory. Speed is not bounded, so you have to integrate up to infinity.