 The volume of phase space formula_1, occupied by a system of formula_2 degrees of freedom is the product of the configuration volume formula_3 and the momentum space volume.Since the energy is a quadratic form of the momenta for a non-relativistic system, the radius of momentum space will be formula_4 so that the volume of a hypersphere will vary as formula_5 giving a phase volume of
where formula_7 is a constant depending upon the specific properties of the system and formula_8 is the Gamma function.In the case that this hypersphere has a very high dimensionality, formula_9, which is the usual case in thermodynamics, essentially all the volume will lie near to the surface
where we used the recursion formula formula_11.