The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated. Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external heat bath in the canonical ensemble. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the hard-sphere system of Yakov Sinai. The requirements for isolated systems to ensure ergodicity —and, thus equipartition—have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that is usually a good assumption.