Heat and Thermodynamics

13.6 Liquyille's Theorem

We shall now study the variation of density with time. The density of distribution ρ of representative points in phase space can describe the condition of an ensemble at any instant. We consider now the motion of points in phase space according to the principle of mechanics and study the changes of density with time. Let us consider any point q₁, q₂, … q_m, p_l, p₂ … p_m in phase space, the differential element of extension that may be defined at that point by δq₁, … δq_m, δp₁, … δp_m. The number of representative points within the element at any instant

 

n = ρδq₁ … δq_m δp₁… δp_m     (13.15)

 

This number will change with time as the number of representative points entering the volume in phase space through any face will, ...