Ergodicity
This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map.In recreational mathematics, it underpins the period-doubling fractals; in analysis, it appears in a vast variety of theorems.The proof of equivalence is very abstract; understanding the result is not: by adding one at each time step, every possible state of the odometer is visited, until it rolls over, and starts again.Systems that generate (infinite) sequences of N letters are studied by means of symbolic dynamics.Important special cases include subshifts of finite type and sofic systems.In order to do this, it was necessary to state what exactly it means for gases to mix well together, so that thermodynamic equilibrium could be defined with mathematical rigor.In all cases, the notion of ergodicity is exactly the same as that for dynamical systems; there is no difference, except for outlook, notation, style of thinking and the journals where results are published.Glasses present a challenge to the ergodic hypothesis; time scales are assumed to be in the millions of years, but results are contentious.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).The formal study of ergodicity can be approached by examining fairly simple dynamical systems.Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process.One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolza surface, topologically equivalent to a donut with two holes.Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this.Such flows commonly occur in classical mechanics, which is the study in physics of finite-dimensional moving machinery, e.g. the double pendulum and so-forth.The flows on such systems can be deconstructed into stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results.In this sense, chaotic behavior with ergodic orbits is a more-or-less generic phenomenon in large tracts of geometry.This rather neatly ties these systems back into the definition of ergodicity given for a stochastic process, in the previous section.The anti-classification results state that there are more than a countably infinite number of inequivalent ergodic measure-preserving dynamical systems.All of the previous sections considered ergodicty either from the point of view of a measurable dynamical system, or from the dual notion of tracking the motion of individual particle trajectories.There, the resonant interaction allows for the mixing of normal modes, often (but not always) leading to the eventual thermalization of the system.One of the earliest systems to be rigorously studied in this context is the Fermi–Pasta–Ulam–Tsingou problem, a string of weakly coupled oscillators.Resonant interactions between waves helps provide insight into the distinction between high-dimensional chaos (that is, turbulence) and thermalization.[7] However, there is a quantum ergodicity theorem stating that the expectation value of an operator converges to the corresponding microcanonical classical average in the semiclassical limitThe definition given above admits the following immediate reformulations: Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only: Letone obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus.The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.In the examples considered above, irrational rotations of the circle are uniquely ergodic;[24] shift maps are not.The simplest case is that of an independent and identically distributed process which corresponds to the shift map described above.A similar interpretation holds for continuous-time stochastic processes though the construction of the measurable structure of the action is more complicated.