Non-interacting gas in homogeneous gravitational field

I am not familiar with the Maxwell distribution of speeds myself, because I am not so familiar with the mathematical tools needed (I am a high school senior), but here is my attempt at a partial answer:

Suppose the two layers are separated by a very small height $dh$. Say the average energy of a particle moving from the higher layer to the lower layer is $E$. When they arrive at the lower layer, they must have gained some gravitational potential energy, $nmg, dh$, where $g$ is the field strength. As such, the particles arrive with average kinetic energy $E + mg ,dh$. At the same time, for the same amount of heat (random kinetic energy) to be transferred to the layer above, the particles must also arrive with average kinetic energy $E + mg ,dh$; thus it should follow that they leave the lower layer with an average kinetic energy $E' = E + 2nmg,dh$.

Remembering that the temperature is constant throughout the whole system, the particles must follow the same Maxwell distribution of speeds (and Maxwell-Boltzmann distribution of kinetic energies, if that is something in the toolkit).

Suppose that $E$ is the average kinetic energy of the particles in the upper layer. You could force $E'$ to be the average kinetic energy of the particles with the Maxwell-Boltzmann distribution at temperature T by excluding some proportion of the particles, $p$, $0 < p << 1$ (I suppose this is a mathematical exercise in finding a cut-off area of low-energy particles, at energies above which the mean energy is $E'$ if the mean energy is $E$ over the entire M-B distribution). This implies that, for every particle in the upper layer, there must be $frac{1}{1-p} approx 1 + p$ particles in the lower layer.

The difference in the number of particles (a common ratio of $frac{1}{1-p}$) could be extrapolated to find the difference in density between the two layers, as well as the difference in pressure (from $P,,alpha,,n$, $T, V,constant$).

You could certainly use the full statistical physics approach, if you'd like. From this POV, you would consider adjacent horizontal layers to be separate systems in thermal and chemical equilibrium, and then note that the external potential serves as a chemical potential which varies between those layers.

However, the more down-to-earth treatment would be to basically repeat the standard derivation for the pressure variation in a liquid. One can consider the sum of the forces acting on a small cube of air at height $z$. The forces in question are supplied by gravity and the pressure at the various points in the system. Noting that the average molecular speed is constant, you can relate all of these forces directly to density, which (in the limit as the size of the cube goes to zero) becomes a differential equation for $rho(z)$.