Ergodic dynamics

This model is clearly incomplete, since it does not account for vague tori [355] and the complex Arnold web [357, 358] structure of a multidimensional phase space with both chaotic and quasi-periodic trajectories. However, Eq. (74) does properly describe that, with non-ergodic dynamics, the lifetime distribution will have an initial component that decays faster than the RRKM prediction as found in the simulations by Bunker [323,324] and the more recent study of HCO dissociation [51]. Additionally, there will be a component to the classical rate, which is slower than /srrkm, for example, in the dissociations of NO2 and O3 this component cannot be described by an expression as simple as the one in Eq. (74). 

A complete model for the non-ergodic classical dynamics of a polyatomic molecule will need to represent the complete Arnold web structure of the phase space. There may be multiple bottlenecks for IVR and vague tori may exist in the vicinity of invariant tori. These complex phase space structures, leading to non-ergodic dynamics, are the origins of the... 

One of the most important features of a numerical method for ergodic dynamics (such as Langevin dynamics) is its preservation of the theoretical global phase space... 

A classical microcanonical ensemble for an intrinsie non-RRKM molecule consists of chaotic, vague tori and quasiperiodic trajeetories. Such a complex non-ergodie phase spaee strueture leads to a non-exponential P i). As an applieation of the KAM theorem, Oxtoby and Riee have shown that the intrinsic non-RRKM dynamics that Bunker found for model triatomic Hamiltonians results from insufficient internal resonances to yield ergodic dynamics. 

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

Polik W F, Guyer D R, Miller W H and Moore C B 1990 Eigenstate-resolved unimolecular reaction dynamics ergodic character of Sq formaldehyde at the dissociation threshold J. Chem. Phys. 92 3471-84... 

Assuming the MPC dynamics is ergodic, the stationary distribution is microcanonical and is given by... 

Specialized Methods for Improving Ergodic Sampling Using Molecular Dynamics and Monte Carlo Simulations... 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

Here, w(xfc) is the weighting factor for any property at a given position on the fcth step xfc. For example, for a constant-temperature molecular dynamics or a Metropolis MC run, the weighting factor is unity. However, we wish to leave some flexibility in case we want to use non-Boltzmann distributions then, the weighting factor will be given by a more complicated function of the coordinates. The ergodic measure is then defined as a sum over N particles... 

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]... 

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