Collisional systems relax to Maxwell-Boltzmann equilibria. Collisionless self-gravitating systems and plasmas, despite the lack of frequent interparticle collisions, nevertheless exhibit quasi-equilibrium nonthermal states with a degree of universality. I will discuss a framework that combines Lynden-Bell's statistical mechanics with the gradual time evolution of the system's infinite set of Casimir invariants, to explain equilibria arising from “violent relaxation” in both gravitational systems and plasmas. The main conclusion is that (closed) collisionless systems tend simultaneously toward states of minimum energy and maximum entropy at long times. This process is driven by a turbulent cascade of a particular kind: in phase space, but reminiscent of the Batchelor scalar turbulence in fluid dynamics. This framework may be able to explain how nonthermal distributions arise in a variety of contexts, from flat-top and power-law distributions in the solar wind to the equilibria of galaxies and dark-matter halos.