Systems ergodic

As t becomes large, this should approach the equilibrium value (fi), which for an ergodic system is... 

For ergodic systems, the probability of visiting the neighborhood of each point in phase space converges to a unique limiting value as T —> oo, such that the time average of / is equal to its ensemble average... 

For an ergodic system, if the simulation length n —> oo, then du(n) —> 0. By analogy with molecular dynamics, for large n we expect the form of the convergence to be [5]... 

RMT). K systems are most strongly mixing classical systems with a positive Kolmogorov entropy. The conjecture turned out valid also for less chaotic (ergodic) systems without time-reversal invariance leading to the Gaussian unitary ensemble (GUE). 

Domains having different degrees of fluidity may appear as a consequence of density fluctuations at intermediate length scales. Note that in an ergodic system the probability of fluctuation of the wavenumber-dependent density, pv is given by the following well-known expression ... 

The single, undisturbed motion of the system, if pm-sued without limit in time, will finally traverse every phase point which is compatible with its given total energy. A mechanical system satisfying this condition is called by Boltzmann an ergodic system. "... 

For an ergodic system all motions with the same total energy take place on the same (7-path.M... 

It is on account of this last property that the definition of ergodic systems and the assumption that the gas models are ergodic appear in Boltzmann s investigations (ef. Section 11). 

When Maxwell and Boltzmann first introduced these special density distributions, they justified them by referring explicitly to the hypothesis that the gas models are ergodic systems.102... 

The fundamental assumption underlying this investigation is the hypothesis that the gas models are ergodic systems (cf. Section 10). With the help of this hypothesis Boltzmann computed the time average of, for instance, the kinetic energy of each atom (the same value is obtained for all atoms ).108 Likewise he calculated the time average of other functions (q, p) which characterize the average distribution of state. 

One should be careful to distinguish between the following two concepts (a) an ergodic system, (b) ergodic density distribution in the T-space. For the relationship of (a) to (b), see note 101. 

If, for the time being, we call systems satisfying requirement (I) of note 98 "quasi-ergodic, then, instead of statements 1), 2), and 3) in the text, we must Bay that for a "quasi-ergodic system on each surface E(q, p) =E there will be a continuum of oo different (7-paths with different values of the constants ct, , ... 

Molecular dynamics is an appropriate tool for the study of the pick-up procedure. It is imperative to perform dynamical (and not thermodynamical) simulation because (i) we do not know in forward the final temperature of the system and (ii) for non-ergodic systems the final distribution can represent just a local minimum on the free energy surface of the system, i.e., a metastable state with a high ki-... 

F. Tal and E. Vanden-Eijnden (2006) Transition state theory and dynamical corrections in ergodic systems. Nonlinearity 19, p. 501 31. E. Vanden-Eijnden and F. Tal (2005) Transition state theory Variational formulation, dynamical corrections, and error estimates. J. Chew,. Phys. 123, 184103 T. S. van Erp and P. G. Bolhuis (2005) Elaborating transition interface sampling method. J. Cow,p. Phys. 205, p. 157... 

Ergodic Systems An ergodic system displays the equality of space and time averages almost everywhere, that is,... 

Eigendistributions for ergodic systems may also be readily obtained, although they prove less useful than those obtained for the regular case. Specifically, stationary eigenfunctions that are L2 on the energy hypersurface for this case are given by... 

In general Ai may depend on the initial condition xo- But for ergodic systems it has the same value for almost all initial positions (Eck-mann and Ruelle, 1985), i.e. everywhere except in a set of measure zero. The characteristic signature of chaotic advection is that at least one of the Lyapunov exponents is positive, representing exponential growth of the distance separating the two particles. 

For an ergodic system the long-time average is equal to the ensemble average - the average with respect to the configuration of the systems (average over all possible positions and shapes). 

Birkhoff proved this for an ergodic system that is, a system that is metrically indecomposable (Uhlenbeck and Ford, 1963). 

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