Using the Boltzmann distribution for a small system in contact with a heat reservoir at temperature T, find the Maxwell-Boltzmann distribution.
The Boltzmann distribution states that the probability density of a system in contact with a heat reservoir at temperature T having energy ε is proportional to e-ε/kT where k is the Boltzmann constant.
The Attempt at a Solution
So my book basically does this for me, but I'm having trouble understanding a few things. So say we treat an individual molecule in the gas as our small system, and the reservoir is the rest of the gas at fixed T, then the probability density of having kinetic energy in the x direction of 0.5mvx2 is proportional to
Then if this is the probability density for the KE in the x direction, it must be the same proportionality for velocity in the x direction so the probability density for vx is proportional to
The same logic follows for y and z so we get the same factors for these
e-0.5mvy2/kT and e-0.5mvz2/kT.
Then the probability of having velocity in the interval [vx,vx+dvx]x[vy,vy+dvy]x[vz,vz+dvz] is the product of the individual probability densities so we have
We can get the speed from changing to spherical polars, integrating over theta and phi to get the probability of being in [v,v+dv] and this gives something proportional to
So assuming that is all the correct logic, why couldn't I have started by saying that if our molecule is the small system, and the gas our reservoir, then the probability density of having KE 0.5mv2 is proportional to e-0.5mv2/kT? This disagrees with the actual result but follows the same initial step, just considering the overall KE and not the x,y,z components.