Why is the Maxwell velocity distribution Gaussian while the speed distribution is not Gaussian

Question

Why is the Maxwell velocity distribution Gaussian while the corresponding speed distribution is not Gaussian?

Answer 1

Distribution of speeds is: $$ w(v_x, v_y, v_z)=\left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}}e^{-\frac{m(v_x^2+v_y^2+v_z^2)}{2k_BT}} $$ Transforming this to polar coordinates we have $$ w(v, \phi,\theta)=\left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2k_BT}}v^2\sin\theta $$ integrating out the angles gives $$ w(v)=\left(\frac{m}{2\pi k_B T}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2k_BT}}v^24\pi $$

Answer 2

When I was young, this question bothered me for quite some time. Here I am presenting a heuristic picture without any mathematics.

You may think as follows.

Suppose you are observing the movement of gas atoms about a point. Since the velocity distribution is an average of many particles, you will average the velocity of particles passing through that point at regular intervals to generate velocity distribution. Now maximum chance will be that the average of this velocity is zero because the average velocity of two particles (having the same speed), one going from left to right and the other from right to left, is zero. Hence there is a peak in the velocity distribution at zero velocity. Finding average velocities other than zero will become less and less probable with the increase in velocity. Hence velocity distribution is Gaussian.

On the other hand, if you average the speed of particles (without bothering if they are going from left to right or right to left), you will find that there is almost zero chance that the speed of a particle is zero. The peak of the distribution will depend on the temperature of the gas. Now finding a high-speed particle than the most probable speed will be less likely but never zero. That is why you will see a long tail, and hence the speed distribution is not Gaussian.

Answer 3

Speed can only be non-negative while gaussian distributions extends to $-\infty$ and $\infty$. In general, it is impossible for any non-negative quantity to become normally distributed. What you can do is a change of variables to transform this quantity to normally distributed one.

Lets see the relation between velocity and speed distribution. Consider a thin spherical shell of thickness $dv$ in velocity space at velocity magnitude $v$. The Volume of the shell is $4\pi v^2 dv$. So, the particles with $v$ in that range is proportional to that volume weighted by gaussian distribution. Clearly, this is not gaussian. At small speeds, this volume is very less while at high speeds, gaussian weight become very small. Both of these lead to small total number of particles. In between, both are significant enough and their product becomes maximum.