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Absorbing Barrier (finance)
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of Microstate (statistical mechanics), microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. Liouville's theorem (Hamiltonian), Liouville's theorem states that, for Hamiltonian dynamics, Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.e., the Material derivative, convective time derivative is zero). Thus, if the microstates are Uniform distribution (continuous), uniformly distributed in phase space initially, they will remain so at all times. But Liouville's theorem does ''not'' imply that the ergodic hypothesis holds for all Hamiltonian systems. The ergodic hypothesis is often assumed in the statistical a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Hypothesis W Reflecting Rays
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
