Classical Energy Distributions in Terms of Phase Space

 Emphasis seems to be placed on spatial arrangements when developing the Maxwell-Boltzmann distribution and on arrangements over “energy degenerate” cells (i.e. phase space) with a fixed single particle energy ei in quantum mechanics. In the MB case, particles are independent, while in the quantum case, single particle energy levels are independent, immediately “forcing” a grand canonical approach in which the total number of particles is not conserved.

 In this note, we try to argue that the Maxwell-Boltzmann distribution follows directly from the quantum phase space approach in the case that n(ei) is very small compared with the degeneracy (cells) in phase space i.e.  d3p dV / h3.