Ergodic behavior

Figure 8.16 from Frantz, D D, D L Freeman and J D DoU 1990. Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking applications to atomic clusters. The Journal of Chemical Physics 93 2769-2784. 

DD Frantz, DL Freeman, ID Doll. Reducing quasi-ergodic behavior m Monte Carlo simulations by J-waUcmg Applications to atomic clusters. I Chem Phys 93 2769-2784, 1990. 

Frantz, D.D. Freeman, D.L. Doll, J.D., Reducing quasi-ergodic behavior in Monte Carlo simulation by J-walking Applications to atomic clusters, J. Chem. Phys. 1990, 93, 2769... 

Gutzwiller, M.C. (1980). Classical quantization of a Hamiltonian with ergodic behavior, Phys. Rev. Lett. 45, 150-153. 

Depke, G. Lifshitz, C. Schwarz, H. Tzidony, E. Non-ergodic behavior of excited radical cations in the gas phase, Angew. Chem. Int. Ed. Engl. 1981,20, 792-793. 

Thus we have seen some ways to explore the local aspects of chaotic and ergodic behavior. However, at this stage of progress, a significant challenge loomed, which was only resolved when one of the organizers of the 2003 Kyoto Conference appeared, some five years after the work here was carried out. The... 

Figure 16 provides67 a comparison between the classical [Eq. (4.6)], and quantum [Eq. (4.1)] spatial correlation functions for four chaotic stadium eigenfunctions. The classical and quantum correlation functions are seen to agree very well for all distances other than those comparable to the size of the stadium s linear dimension. Thus, the rapid decay of the quantum spatial autocorrelation is a measure relating directly to classical ergodic behavior. 

There is one caveat to note in relation to these reference systems Molecular dynamics simulations of harmonic and nearly harmonic models are known to exhibit very poor ergodicity behavior, leading to very poor sampling of phase space. Special measures should be taken if applying molecular dynamics simulation in a free-energy calculation involving such a model reference system [71]. If MC simulation is employed, there is no intrinsic problem with ergodicity and the issue is of no concern. 

Numerical simulations have shown that the presence of the perturbation V in Eq. (4.52) is not sufficient for ergodic behavior. It was discovered that a system of N weakly coupled harmonic oscillators will not freely exchange energy as long as there is no collection of integers nj for which... 

Static susceptibility and other Nonergodic behavior at Ergodic behavior at all... 

Quasi-ergodic Behavior in Monte Carlo Simulations by J-walking Applications to Atomic Clusters. 

What are the implications of these studies on the calculation of macroscopic dynamical properties of ionic liquids At the very least, they suggest that one should be careful when applying standard computational techniques used for simple liquids to ionic liquids. Most of these techniques assume ergodic behavior, but the work described above shows this may not always be the case. Due to the sluggish dynamics of ionic liquid systems, one should carry out very long simulations to ensure adequate sampling. 

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