H-functions and mixing in violent relaxation

An H-function is any function of the phase space distribution function F(x, υ) which is non-decreasing with time. In collisionless systems Boltzmann's H-function $$-\int F\enspace\text{log}\enspace F\enspace dx \enspace dv$$ is only one of a variety of H-functions of the form $$-\int C(F)\enspace dx \enspace dv$$, where C is any convex function. Every equilibrium stellar system in which the distribution function is a decreasing function of the energy alone is a stationary point of some H-function of this form. During violent relaxation, all such H-functions must increase, and the distribution function is said to become ‘more mixed’. We give a simple criterion for determining whether a given distribution function is more mixed than another; this criterion is used to show that a violently relaxed galaxy resembles observed elliptical galaxies only if the initial state is cold or clumpy.