Derivation of the Liouville Equation

As each of the N molecules moves through space according to Eq. (2.2), the phase point traces out a unique trajectory in phase space. 

Next consider a collection, or ensemble of phase points. Each phase point of the ensemble represents the same group of molecules, with certain identical groups or macroscopic properties (such as number density and total energy), differing only in their individual positions and momenta. In short, there are many different values or realizations of 

As each phase point traces out a trajectory in phase space, the collection of points, at any time, resembles a cloud as shown in Fig. 2.2.1 We can describe this cloud of points by a density function that represents the number of phase points in a specific region of phase space. In particular. 

Since pm is a density function, it obeys a conservation equation similar in development to the equation of continuity, or mass conservation equation, in fluid mechanics. The Liouville equation simply represents a conservation equation for the phase points through a fixed differential volume of phase space, called a hypercube. The rate of phase points into the hypercube through the faces at X is the flux pnX times the 

Dividing through by the volume of the hypercube, Axi... Appiz), and taking limits as Ax through Aptfz 0 leads to the Liouville equation 

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Here we describe an alternative derivation of the Liouville equation (1.104) for the time evolution of the phase space distribution function f (r, p t). The derivation below is based on two observations First, a change inf reflects only the change in positions and momenta of particles in the system, that is, of motion of phase points in phase space, and second, that phase points are conserved, neither created nor destroyed. 

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