  The baker’s transformation provides a simple model of deterministic macro-randomness—a system which has a \(\mu\)-measure-preserving temporal evolution, and produces a sequence of coarse grained states that have the Bernoulli property. It is a question of considerable interest whether there are physically more realistic systems which exhibit the same features. We may conceive of an \(n\)-particle classical (Newtonian) system as having, at each moment, a state that is characterised by a single point in a \(6n\)-dimensional state space (each point a \(6n\)-tuple characterising the position and momentum of each particle). The evolution of the system over time is characterised by its Hamiltonian, a representation of the energetic and other properties of the system. The evolution under the Hamiltonian is \(\mu\)-measure-preserving (by Liouville’s theorem), so it might be hoped that at least some systems could be shown to be Bernoulli too. Yet for closed systems, in which energy is conserved over time, this is not generally possible. Indeed, for closed systems it is not generally possible to satisfy even a very weak randomness property, ergodicity. A system is ergodic just in case, in the limit, with probability one, the amount of time a system spends in a given state is equal to the (standard) measure of state space that corresponds to that state (Earman 1986: 159–61; Sklar 1993: 60–3; Albert 2000: 59–60). While a Bernoulli system is ergodic, the converse entailment does not hold; if the system moves only slowly from state to state, it may be ergodic while the state at one time is strongly dependent on past history (Sklar 1993: 166–7). While ergodicity has been shown to hold of at least one physically interesting system (Yakov Sinai has shown that the motion of hard spheres in a box is ergodic, a result of great significance for the statistical mechanics of ideal gases), a great many physically interesting systems cannot be ergodic. This is the upshot of the so-called KAM theorem, which says that for almost all closed systems in which there are interactions between the constituent particles, there will be stable subregions of the state space—regions of positive measure such that if a system is started in such a region, it will always stay in such a region (Sklar 1993: 169–94). Such systems obviously cannot be ergodic.