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Topic: Ergodic hypothesis

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ergodic theory

ergodic hypothesis

Ergodic literature

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individual ergodic theorem

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quasi ergodic set

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	Ergodic theory - Wikipedia, the free encyclopedia
		The ergodicity of the geodesic flow on manifolds of constant negative curvature was discovered by E.
		Ergodicity of geodesic flow in symmetric spaces was given by F.
		A simple criterion for the ergodicity of a homogeneous flow on a homogeneous space of a semisimple Lie group was given by C.
	en.wikipedia.org /wiki/Ergodic_theory (629 words)

	Ergodic hypothesis - Wikipedia, the free encyclopedia
		In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equally probable over a long period of time.
		Ergodic theory is a branch of mathematics which deals with dynamical systems which satisfy a version of this hypothesis, phrased in the language of measure theory.
		The fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking.
	en.wikipedia.org /wiki/Ergodic_hypothesis (322 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Ergodic theory justifies the replacement of the infinite time averages with the phase averages relying on the geometric properties of the phase space (invariance), on the properties of transformations defined therein (measuring-preserving transformation) and on a particular hypothesis (MT).
		Ergodic theory vs. Khinchin’s approach: a comparison Ergodic theory attempts to resolve the ergodic problem by using structural characteristics of the phase space of Hamiltonian systems and the hypothesis of the MT. From this point of view, the probabilistic aspect of the problem is clearly subordinate to questions of general dynamics.
		The hypothesis of annihilation of dynamic correlations, the large number of degrees of freedom and the derivation of approximate formulas for the calculation of phase averages are all typical instruments of non-equilibrium theories.
	philsci-archive.pitt.edu /archive/00002277/01/The_Foundational_Role_of_Ergodic_Theory.doc (6760 words)

	Capturing chaos : Nature
		Ergodicity: a fundamental assumption of statistical physics —; anything that can happen will happen — was thrown into question 50 years ago.
		Fermi and his colleagues' experiment cast doubt on a fundamental assumption of physics —; the so-called 'hypothesis of ergodicity' — and thereby on the foundations of the physics of solids, liquids and other forms of matter.
		As it turns out, ergodicity seems to come into play when the energy given to the chain is about ten times greater than that applied by Fermi and colleagues in their original study.
	www.nature.com /nature/journal/v435/n7040/full/435281a.html (944 words)

	Ergodic theory group
		Ergodic hypothesis was first formulated for systems of particles, subject to known interactions and having known masses, evolving in a domain of the 3 dimensional space.
		Ergodic hypothesis is equivalent in stating that most of the trajectories are uniformly distributed on surfaces of constant energy of the phase space; moreover, asymptotically, it allows the replacement of time averages by spatial averages.
		Beyond the strict probabilistic framework, methods of ergodic theory are applied to problems of geometric or arithmetic nature or to models stemming from mathematical physics.
	www.math.univ-rennes1.fr /theoergo/index.html.en.html (406 words)

	Boltzmann's ergodic hypothesis. (Site not responding. Last check: 2007-10-20)
		Boltzmann's ergodic hypothesis is usually understood as the assumption that the trajectory of an isolated mechanical system runs through all states compatible with the total energy of the system.
		Ergodicity was formulated as the condition that only one integral of motion, the total energy, is preserved in time.
		Boltzmann found it difficult to ascribe ergodic behavior to a single system where the theoretical dependence on initial conditions, though never observed, has to be admitted as possible.
	www.helsinki.fi /~vonplato/boltzsum.html (157 words)

	Search Results for hypothesis
		For example he mentions the hypothesis that the sun is at the centre of the planets as proposed by Hipparchus but rejects it immediately since it contradicted the views of a Chaldean whom he says that it is unlawful not to believe.
		The result of the hypothesis of a stationary ether is shown to be incorrect, and the necessary conclusion follows that the hypothesis is erroneous.
		A hypothesis is conceived and defined with all necessary exactitude; its logical consequences are ascertained by a deductive argument; these consequences are compared with the available observations; if these are completely in accord with the deductions, the hypothesis is justified at least until fresh and more stringent observations are available.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=hypothesis&CONTEXT=1 (6420 words)

	Boltzmann's Work in Statistical Physics
		Particularly notorious are the role of the ergodic hypothesis and the status of the so-called H-theorem.
		It seems that Boltzmann regarded the ergodic hypothesis as a special dynamical assumption that may or may not be true, depending on the nature of the system, and perhaps also on its initial state.
		It is indeed evident that if the ergodic hypothesis holds, a state will spend time in the various regions of phase space in proportion to their volume.
	plato.stanford.edu /entries/statphys-Boltzmann (13000 words)

	Domokos Szász (Technical University of Budapest)
		The Ergodic Hypothesis for Hard Balls says that the systems of N elastic hard balls moving on the d-torus is ergodic modulo the trivial invariants of motion.
		In 1999, Simányi and Szász gave a partial solution of the Boltzmann-Sinai Ergodic Hypothesis by establishing that typical hard ball systems are hyperbolic, i.
		Several experts, however, were surprised that their methods relied quite heavily on the fact that in the isomorphic billiard system the boundaries of the obstacles are quadratic algebraic manifolds (indeed, they are cylinders with spherical bases).
	www.math.psu.edu /dynsys/DW2001/abstracts/node23.html (257 words)

	Education activities of karma
		The roots of ergodic theory go back to Boltzmann's ergodic hypothesis concerning the equality of the time mean and the space mean of molecules in a gas, i.e., the long term time average along a single trajectory should equal the average over all trajectories.
		The hypothesis was quickly shown to be incorrect, and the concept of ergodicity (`weak average independence') was introduced to give necessary and sufficient conditions for the equality of these averages.
		Nowadays, ergodic theory is known as the probabilistic (or measurable) study of the average behavior of ergodic systems, i.e., systems evolving in time that are in equilibrium and ergodic.
	www.math.uu.nl /people/dajani/eduM.html (459 words)

	Ergodic - TheBestLinks.com - Mathematics, Thermodynamics, Ergodic hypothesis, Ergodic theory, ...
		Ergodic - TheBestLinks.com - Mathematics, Thermodynamics, Ergodic hypothesis, Ergodic theory,...
		The ergodic hypothesis is a postulate of thermodynamics.
		Ergodic literature is literature that requires effort to read.
	www.thebestlinks.com /Ergodic.html (107 words)

	Ergodic theory and experimental visualization of chaos in 3D flows (Site not responding. Last check: 2007-10-20)
		In his motivation for the ergodic hypothesis Gibbs invoked an analogy with fluid mixing: “Yet no fact is more familiar to us than that stirring tends to bring a liquid to a state of uniform mixture, or uniform densities of its components”.
		Although proof of the ergodic hypothesis is possible only for the simplest of systems using methods from ergodic theory, the use of the hypothesis has led to many accurate predictions in statistical mechanics.
		We also show that ergodic theory methods provide a rigorous theoretical justification for this approach whose main objective is to reveal the non-ergodic regions of the flow.
	flux.aps.org /meetings/YR00/DFD00/abs/S860003.html (255 words)

	[No title]
		BOLTZMANN enunciated the ergodic hypothesis, the original form of which posits that the trajectory of any given every initial condition passes through every point of the surface in phase space having the same total energy as the initial condition.
		If the ergodic hypothesis were true, BOLTZMANN felt, then the time averages of physical observables would tend, as t ->[[infinity]], to a spatial average, the average of the observable over the energy shell.
		Recurrence and ergodicity (the ability to equate time-averages and phase averages) depend on the existence of invariant measures.
	www.mathphysics.com /dynam/ch7c.html (5045 words)

	ergodic (Site not responding. Last check: 2007-10-20)
		If the system wanders about ergodically, the relative sizes of the volumes are a direct measure of their probability.
		This is the so-called ergodic hypothesis -- that the system's trajectory will wander aimlessly, or ergodically, in phase space.
		An ergodic system may also have an attractor whose form remains the same but whose size is unlimited and can cover all of phase space (see Prigogine and Stengers p.
	www.christianhubert.com /hypertext/ergodic.html (436 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The shortcoming of their model, however, is that, on one hand, they restrict the types of all feasible ball-to-ball collisions, on the other hand, they introduce some additional scattering effect with the collisions at the strictly concave wall of the container.
		The annoying shortcoming of that result is that the larger the number of balls $N$ is, larger and larger dimension $\nu$ of the ambient container is required by the method of the proof.
		Pesin's theory [P(1977)] on the ergodic properties of non-uniformly hyperbolic, smooth dynamical systems has been generalized substantially to dynamical systems with singularities (and with a relatively mild behavior near the singularities) by A. Katok and J-M. Strelcyn [K-S(1986)].
	www.ma.utexas.edu /mp_arc/html/papers/03-84 (6178 words)

	Model Reduction
		This latter issue is usually swept under the rug, the usual justification being that the system sufficiently mixes or satisfies the ergodic hypothesis.
		To Boltzmann, the ergodic hypothesis essentially means that all infinite time averages are equal to ensemble averages (with an appropriate Boltzmann weight).
		To Gibbs, this criterion is weakened to more vague notion of "sufficient mixing" where trajectories in phase space mix enough such that the time average of important quantities (observables) are equal to their ensemble averages.
	www.physics.ucsb.edu /~complex/research/reduction.htm (371 words)

	SE377 (Site not responding. Last check: 2007-10-20)
		Convergence of random variables- in probability, almost sure, in weak topology, in norm, complete convergence and r-quick convergence, rate of convergence.
		Ergodic hypothesis, Measure preserving transformations, Poincare's recurrence theorem, ergodic theorem, mixing conditions.
		Walters (2000), An Introduction to ergodic Theory, Springer, available in paperback or 1982 first edition by Springer.
	www.iitk.ac.in /scienceelectives/SE377.HTML (167 words)

	UML Math Department Authors
		The quasi-ergodic hypothesis states that for dynamical systems admitting a finite invariant measure, such as Hamiltonian systems, the average amount of time spent in a certain region of phase space is proportional to the volume of the region.
		In 1931 G. Birkhoff proved that the quasi-ergodic hypothesis is equivalent to the fact that every invariant set of the dynamical system has zero measure or full measure.
		In 1941 Oxtoby and Ulam proved that ergodicity is generic, or typical, for measure-preserving homeomorphisms of compact manifolds, and in particular showed the existence of ergodic homeomorphisms.
	www.uml.edu /Dept/Math/umlauthors/umlauthors.html (1304 words)

	DAM: Seminars for Oct 26 - Oct 30, 1998
		In the limit of infinite steepness of the potential the particle travels with a constant speed in a region, undergoing elastic collisions at the region's boundary - namely it performs a billiard motion.When the boundary is concave, causing neighboring trajectories to diverge upon reflection, the billiard is called scattering.
		We prove the existence and derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in physically natural smooth Hamiltonian approximations of such scattering billiards.
		This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect.
	www.cfm.brown.edu /cgi-bin/dam/view_seminars.cgi?Oct-26-Oct-30-1998 (1075 words)

	Ergodic hypothesis: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-20)
		Ergodic hypothesis: Facts and details from Encyclopedia Topic
		(the ergodic hypothesis says that, EHandler: no quick summary.
		Ergodic theory[For more facts and a topic of this subject, click this link] is a branch of mathematics mathematics quick summary:
	www.absoluteastronomy.com /encyclopedia/e/er/ergodic_hypothesis.htm (646 words)

	BOOK REVIEW
		We recall that Gibbs assumed the ergodic hypothesis to justify his use of ensemble averages in place of time averages.
		century mathematicians realized that this hypothesis had to be false, on the basis of measure-theoretical arguments, but it was replaced with the "quasi-ergodic" hypothesis, that the point comes arbitrarily close to every point.
		Ergodic theory has become a topic studied by pure mathematicians.
	www.pzweifel.com /music/boltzmann_secondo.htm (1549 words)

	[No title]
		In class today, I mentioned the ergodic hypothesis to connect Q with q.
		At the extreme, I could have selected one particle from each of N different collections, the ergodic hypothesis says that it does not matter to the probability expression.
		That’s the point I was trying to make Bottom line: The ergodic hypothesis says that the likelihood of any of the many collections of particles in substantially different than any other collection is vanishingly small and therefore all of the collections behave essentially the same.
	faculty.une.edu /cas/cnash/CH327/ErgodicHypothesis.doc (350 words)

	List of Participants
		The progress obtained in quantum chemistry, non-linear mechanics and in the correspondence of classical to quantum mechanics [1] in the last forty years has unveiled the complexity of the interactions among the nuclei in a molecule and their internal motions.
		The simple statistical theory named RRKM [2], which is based on the ergodic hypothesis, is inadequate to explain detailed experimental results now available from single molecule spectroscopy and molecular beams [3].
		Numerous studies show that the polyatomic molecules seen as a set of non-linear coupled oscillators is not an ergodic system, but instead, they form a mixed phase space with regular and chaotic regions [4,5].
	tccc.iesl.forth.gr /cecam2004/specr_compch.html (1402 words)

	[No title]
		As Tolman (1938) noted long ago in criticizing the role of ergodic theory, the important point is to relate ensemble averages to experimental values, which are acquired over relatively short time intervals.
		In addition, since time averages are intrinsic to ergodicity, there is very little the concept has to say about nonequilbrium phenomena, and so it can hardly be considered a foundational tool.
		While the ergodic concept and its generalizations seem to have little bearing on the behavior of macroscopic physical systems and their thermodynamic description, there is relevance to the study of general nonlinear dynamical systems, usually with very few degrees of freedom.
	w3.uwyo.edu /~wtg/Issues/Issues7.html (3071 words)

	Limits to classification and regression estimation from ergodic processes, Andrew B. Nobel
		It is shown that no measurable procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in the unit interval.
		The results of the paper are derived via reduction arguments and are based in part on recent work concerning density estimaton from ergodic processes.
		Delecroix, M. and Rosa, A. Nonparametric estimation of a regression function and its derivatives under an ergodic hypothesis.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aos/1018031110 (509 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Morphodynamics is a general theory of stationary complex systems, such as living systems, or mental and social systems; it is based on the thermodynamics of physical systems and built on the same lines.
		By means of the ergodic hypothesis, thermodynamics is known to connect the particle dynamics to the emergence of order parameters in the equations of state.
		In the same way, morphodynamics connects order parameters to the emergence of higher level variables; through recurrent applications of the ergodic hypothesis, a hierarchy of equations of state is established which describes a series of successive levels of organization.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=443603 (224 words)

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