Microcanonical ergodicity

Definition 5.1 A Hamiltonian system is said to be ergodic on an energy surface X e (microcanonically ergodic) if, for almost all trajectories F emanating from initial conditions on He, and for any observable g, the following holds... 

There is a key difference between the concept of microcanonical ergodicity and canonical ergodicity. For deterministic dynamics, the distribution solves the Liouville equation... 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

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

A) The microcanonical distribution p is everywhere zero except between the two energy surfaces E=Eo and E=E0+BE0) where BE0 is very small. Within this shell p has a constant value. This distribution of volume density p(q, p) for BE0=0 is obviously equivalent to the ergodic surface distribution (Eq. 31) of Section 10b. 

According to (XII) it is first of all plausible that in general the average (Eq. 62) over the canonical ensemble will be very nearly identical with the average value taken over a microcanonical or even ergodic ensemble with E=E0> In fact, in that case also Eq. (57), for example, coincides with a relationship derived by Boltzmann (1871) for ergodic ensembles.182 Furthermore, the micro-canonical ensemble is very nearly equivalent to an ensemble that is distributed (cf. Section 12c) with constant density over the shell in T-space belonging to... 

L. Omstein2 discusses related questions. One should emphar size (1) his critical attitude toward the ergodic hypothesis and, as a result, his introduction of the microcanonical ensemble not as the only possible one, but only as the simplest stationary ensemble (2) some remarks on the properties of ensembles of systems which are distributed around the surface of most probable energy following a different law, e.g., with density proportional to... 

The microcanonical ensemble may be depleted in the vicinity of the transition state by the absence of trajectories in the reverse direction. This assumption is often referred to as the ergodic approximation, that the microcanonical ensemble is rapidly randomized behind the reaction bottleneck faster that reactive loss can perturb the distribution. 

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

These phenomena lead us to a rather complicated situation. The first phenomenon reminds us of ergodicity, the realization of microcanonical distribution in systems with many degrees of freedom and the validity of statistical mechanics. We know that KAM tori cannot divide the phase space (or energy surface) for systems with many degrees of freedom, and the first phenomenon tells us that two neighborhoods in different parts of the phase space are connected not only topologically but also dynamically. In this sense the phenomenon can be considered as an elementary process of relaxation in systems with many degrees of freedom. 

The irregular trajectories in Fig. 15.6 display the type of motion expected by RRKM theory. These trajectories moves chaotically throughout the coordinate space, not restricted to any particular type of motion. RRKM theory requires this type of irregular motion for all of the trajectories so that the intramolecular dynamics is ergodic [1]. In addition, for RRKM behavior the rate of intramolecular relaxation associated with the ergodicity must be sufficiently rapid so that a microcanonical ensemble is maintained as the molecule decomposes [1]. This assures the RRKM rate constant k E) for each time interval f —> f + df. If the ergodic intramolecular relaxation is slower than l/k(E), the unimolecular dynamics will be intrinsically non-RRKM. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

PT has also been used successfully to perform ergodic simulations with Lennard-Jones clusters in the canonical and microcanonical ensembles [53,54]. Using simulated tempering as well as PT, Irback and Sandelin studied the phase behavior of single homopolymers in a simple hydro-phobic/hydrophilic off-lattice model [55]. Yan and de Pablo [56] used multidimensional PT in the context of an expanded grand canonical ensemble to simulate polymer solutions and blends on a cubic lattice. They indicated that the new algorithm, which results from the combination of a biased, open ensemble and PT, performs more efficiently than previously available teehniques. In the context of atomistic simulations PT has been employed in a recent study by Bedrov and Smith [57] who report parallel... 

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