Ensembles in statistical mechanics In statistical mechanics, a macroscopic property of a system S is calculated by averaging that property over a set, or ensemble, of points moving in the phase space of S. The number of ensemble points in any volume of phase space is represented by a density function ρ(q,p,t). If the system is autonomous and in statistical equilibrium, it is required that, even though the ensemble points are moving (in accordance with Hamilton's equations), their density function should remain the same, that is, ρ=ρ(q,p). This places a restriction on possible choices for ρ(q,p). Let R0 be any region of the phase space and suppose that, after time t, the points of R0 occupy the region Rt. Explain why statistical equilibrium requires that
∫R0ρ(q,p)dv=∫Rtρ(q,p)dv
and show that the uniform density function ρ(q,p)=ρ0 satisfies this condition. [It can be proved that the above condition is also satisfied by any density function that is constant along the streamlines of the phase flow.]