10 - The grand description – Liouville's equation and entropy

Summary

All for one, one for all, that is our device.

Alexandre Dumas, Elder

Langevin's equation, the Fokker–Planck equation, the master equation, and Boltzmann's equation are all just partial descriptions of gravitating systems. Each is based on different assumptions, suited to different conditions. They all arise from physical, rather intuitive, approaches to the problem. But there is also a more general description from which our previous ones emerge as special cases. We know this must be true because Newton's equations of motion provide a complete description of all the orbits. The trouble with Newton's equations is that they are not very compact: N objects generate 6N equations. True, the total angular and linear momenta, and energy, are conserved, at least for isolated systems, but this is not usually a great simplification.

By extending our imagination, we can cope with the problem. We previously imagined a six-dimensional phase space for the collisionless Boltzmann equation. Each point in this phase space represented the three position and three velocity (or momentum) coordinates of a single particle. It was a slight generalization of the twodimensional phase plane whose coordinates are values of a quantity and its first derivative resulting from a second order differential equation for that quantity. The terminology probably arose from the case of the harmonic oscillator where this plane gave the particular stage or phase in the recurring sequence of movement of the oscillator.