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 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.
Invariant tori of regular orbits are such regions, but so are the more complex parts of phase space associated with chaotic trajectories.
's, Then this function is the solution of the collisionless Boltzmann equation, as can be verified by substituting this function into the collisionless Boltzmann equation to find[2][3] This proves the theorem.
A trivial set of integration constants are the initial location
In this case, any function is a solution of the collisionless Boltzmann equation.