Derivation of Maxwell Speed Distribution

Introduction

This animation shows how the 1, 2, and 3 dimensional velocity (or speed) and energy probability distributions arise. First I reiterate Maxwell's very elegant derivation of the single dimension velocity distribution which turns out to be a gaussian in speed, v. Then the animation shows graphically how the products of these 1 dimensional speed distributions combine to result in more interesting higher dimensional distributions. The animation also converts these speed distributions into energy distributions in all three dimensions.

Maxwell's Derivation of Velocity Distributions

Maxwell claims that the (vx,vy,vz) components are independent of each other so he can write the three dimensional probability function as the product of three one-dimensional probability functions:

Now, since the F or fs are not expected to be zero or negative, it is acceptable to take the natural log of both sides of this equation:

We can take the derivative of this equation with respect to vx:

Note that the total speed is:

and therefore we can use the chain rule to express the derivative:

Now

so that:

Then using a previous equation we can write:

which can be written:

The next claim is that all of the vx,vy,vz versions of the right hand side of this equation must be equal to the same constant. We will call this constant -b. Then we can write:

or

Obviously

Conversion to Speed Distributions in Various Dimensions

For the speed, v, distribution in two dimensions we must take into account the fact that there are more possible values at large v than at small v. We can think of the product f(v_x)f(v_y) as a weighted distribution of points in the xy plane. Let

be the circle centered on (vx,vy)=(0,0) where the square root of the sum of the squares of (vx,vy) is equal to the constant v.

The number of points in the range v to v+dv is then proportional to:

where

is just the velocity area increment for a ring of width dv and radius v.

For the speed, v, distribution in three dimensions we must take into account the fact that there are many more possible values at large v than at small v. We can think of the product f(v_x)f(v_y)f(v_z) as a distribution of points in the xyz volume. Let

be the spherical surface centered on (vx,vy,vz)=(0,0,0) where the square root of the sum of the squares of (vx,vy,vz) is equal to the constant v.

The number of points in the range v to v+dv is then proportional to:

where

is just the velocity volume increment for a spherical shell of thickness dv and of radius v.

Conversion to Energy Distributions in Various Dimensions

The energy is proportional to v² and the energy increment dE is then proportional to vdv. Then we only need to make the appropriate changes to the speed distributions to convert them to energy distributions.

For 2 dimensions we have the expression

for the number of points in the energy range E to E+dE. This results in a simple exponential in E.

For 3 dimensions we have the expression

for the number of points in the energy range E to E+dE.

For 1 dimension, where there is no power of v multiplying the exponential we have