In setting up the Liouville equation, we have to take into account the fact that the ith molecule of species will have "beads" which will be indicated by an additional index , and that these beads will be spread out around the center of mass of the molecule. Therefore, the complete phase space will consist of all the bead positions, , and the bead momenta, . We are then concerned with a distribution function that is a function of all the bead positions and momenta; this function will be described by an "equation of continuity" in the complete phase space:
When this equation is combined with Newton's laws of motion for the beads, and , we then get the Liouville equation for the polymer mixture:
where is the Liouville operator.
2b. The general equation of change
When the Liouville equation is multiplied by some property B and then integrated over the entire phase space, we get the general equation of change:
where (2.7; 2.8)