"Ergodic" redirects here. For other uses, see Ergodic (disambiguation).
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 same general area, eventually filling the entire space.
Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept than ergodicity is that of mixing, which aims to mathematically describe the common-sense notions of mixing, such as mixing drinks or mixing cooking ingredients.
The proper mathematical formulation of ergodicity is founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis.
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...
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this...
group level data. Ergodic process Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicityErgodicity Loschmidt's paradox...
process that is not ergodic is said to be in non-ergodic regime. A regime implies a time-window of a process whereby ergodicity measure is applied. One...
Ergodicity economics is a research programme aimed at reworking the theoretical foundations of economics in the context of ergodic theory. The project's...
Ergodic literature is a term coined by Espen J. Aarseth in his 1997 book Cybertext—Perspectives on Ergodic Literature to describe literature in which nontrivial...
Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to energy eigenstates of a quantized ergodic Hamiltonian...
contradicting ergodicity. Hence A = B = L∞(R). When all the σt with t ≠ 0 are conservative, the flow is said to be properly ergodic. In this case it...
addresses issues such as theory of price formation, price dynamics, market ergodicity, collective phenomena, market self-action, and market instabilities. Physics...
probability theory, a stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the...
In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties. Let A = { a j } {\displaystyle...
described by critical exponents, universality, fractal behaviour, and ergodicity breaking. Critical phenomena take place in second order phase transitions...
The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that ( X , B , μ ) {\displaystyle (X,{\mathcal {B}}...
Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory. Ergodic Ramsey theory...
exist; then the values of the Lyapunov exponents do not change. Verbally, ergodicity means that time and space averages are equal, formally: lim t → ∞ 1 t...
several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured...
Cybertext as defined by Espen Aarseth in 1997 is a type of ergodic literature where the user traverses the text by doing nontrivial work. Cybertexts are...
Ergodic Theory and Dynamical Systems is a peer-reviewed mathematics journal published by Cambridge University Press. Established in 1981, the journal...
L.; Kryukov, V. I.; Toom, A. L. (1978). Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Manchester University Press. ISBN 9780719022067...
points, the book must be rotated to be read, making it a prime example of ergodic literature. The book is most often described as a horror story, though...
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