of ergodicity in statistical physics are hard to come by; most high-dimensional many-body systems are assumed to be ergodic, without mathematical proof. Exceptions include the dynamical billiards, which model billiard ball-type collisions of atoms in an ideal gas or plasma. The first hard-sphere ergodicity theorem was for Sinai's billiards, which considers two balls, one of them taken as being stationary, at the origin. As the second ball collides, it moves away; applying periodic boundary conditions, it then returns to collide again. By appeal to homogeneity, this return of the "second" ball can instead be taken to be "just some other atom" that has come into range, and is moving to collide with the atom at the origin (which can be taken to be just "any other atom".) This is one of the few formal proofs that exist; there are no equivalent statements e.g. for atoms in a liquid, interacting via van der Waals forces, even if it would be common sense to believe that such systems are ergodic (and mixing). More precise physical arguments can be made, though. In geometry Ergodicity is a wide-spread phenomenon in the study of Riemannian manifolds. A quick sequence of examples, from simple to complicated, illustrates this point. The geodesic flow of a flat torus following any irrational direction is ergodic; informally this means that when drawing a straight line in a square starting at any point, and with an irrational angle with respect to the sides, if every time one meets a side one starts over on the opposite side with the same angle, the line will eventually meet every subset of positive measure. More generally on any flat surface there are many ergodic directions for the geodesic flow. There are similar results for negatively curved compact Riemann surfaces; note that in this case the definition of geodesic flow is much more involved since there is no notion of constant direction on a non-flat surface. More generally the geodesic flow on a negatively curved compact Riemannian manifolds is ergodic, in fact it satisfies the stronger property of being an Anosov flow. In finance Models used in finance and investment assume ergodicity, explicitly or implicitly. The ergodic assumption is prevalent in modern portfolio theory, discounted cash flow (DCF) models, and aggregate indicator models that infuse macroeconomics, among others. The situations modeled by these theories can be useful. But often they are only useful during much,