Jeans's theorem

In astrophysics and statistical mechanics, Jeans's theorem, named after James Jeans, states that any steady-state solution of the collisionless Boltzmann equation depends on the phase space coordinates only through integrals of motion in the given potential, and conversely any function of the integrals is a steady-state solution.

Jeans's theorem is most often discussed in the context of potentials characterized by three, global integrals. In such potentials, all of the orbits are regular, i.e. non-chaotic; the Kepler potential is one example. In generic potentials, some orbits respect only one or two integrals and the corresponding motion is chaotic. Jeans's theorem can be generalized to such potentials as follows:^[1]

The phase-space density of a stationary stellar system is constant within every well-connected region.

A well-connected region is one that cannot be decomposed into two finite regions such that all trajectories lie, for all time, in either one or the other. Invariant tori of regular orbits are such regions, but so are the more complex parts of phase space associated with chaotic trajectories. Integrability of the motion is therefore not required for a steady state.

References