concentration, velocity, and temperature gradients. The most satisfactory of these begin with an equation for a distribution function in the phase-space for a gas composed of N molecules. That is, we imagine a hyperspace of 6N dimensions, with one axis each for the x, y, z coordinates of the N molecules and one axis each for the x, y, z components of the momenta. Then one point in this phase space describes the current dynamical state of the system. The equations of motion then describe all future states of the system.

1a. The Liouville equation

Next we imagine an ensemble of systems: a very large number of identical containers of gas, each one represented by a point in the phase space. These points move around and appear just like a flowing fluid. Since no systems, and hence system points, are lost, there will be an "equation of continuity" that describes the motion:

(1.1)

where f is the distribution of points in the phase space and thus a function of all . When Newton's laws of motion, and , for the ith molecule of species with mass , are substituted into Eq. 1.1, we then get the Liouville equation for the gas mixture:

(1.2)