Phase-space gaps

Monte Carlo (MC) methods can address the time gap problem of MD. The basis of MC methods is that the deterministic equations of the MD method are replaced by stochastic transitions for the slow processes in the system.3 MC methods are stochastic algorithms for exploring the system phase space although their implementation for equilibrium and non-equilibrium calculations presents some differences. 

First, in order to simplify the description of the dynamics we separate the whole system, locally in the phase space, into two parts based on a gap in characteristic time scales. This is done using the concept of normally hyperbolic invariant manifolds (NHIMs) [4-8]. Here, the characteristic time scales are estimated as the inverses of the absolute values of the local Lyapunov exponents [5,6]. Then, the Fenichel normal form offers a simplified description of the local dynamics near a NHIM [7]. 

One more important ingredient in discussing the transport process in mixed phase space is so-called cantori. They are invariant sets in which the motion has an irrational frequency. They resemble invariant circles, but they have an infinite number of gaps in them. The existence was proposed by Percival [30] and Aubry [31]. They gave an explicit example, and a proof of their existence has been given afterwards [32-34]. 

Further characterization of P(t) and Pg(t) requires details of the dynamics, here formulated in terms of ergodic-theory-based assumptions. However, since ergodic theory11 considers dynamics on a bounded manifold, it is not directly applicable to the unbounded phase space associated with molecular decay. To resolve this problem we first introduce a related auxiliary bounded system upon which conditions of chaos are imposed, and then determine their effect on the molecular decay details of this construction are provided elsewhere.46 What we show is that adopting this condition leads to a new model for decay, the delayed lifetime gap model (DLGM) for P(t) and Pg(t). The simple statistical theory assumption that Pt(t) and P(t) are exponential with rate ks(E) is shown to arise only as a limiting case. 

Figure 3. Illustrating the topological confinement of the orbit in the 4D phase space. The continuous curves T and Y" represent two sets of 2D invariant tori that intersect transversally an energy surface. An orbit with initial datum in the gap between two tori will be eternally trapped in the same region (see text).

According to Nekhoroshev (1977) and to Morbidelli and Giorgilli (1995), the old and crucial question of stability of a dynamical system turns out to be related to the structure and density of invariant tori which foliate the phase space. For instance the puzzle of the 2/1 gap of the asteroidal belt distribution was explained showing that the corresponding region of the phase space is a weak chaotic one (Nesvorny and Ferraz-Mello 1997). 

H. Waalkens, G.S. Ezra, S. Wiggins, Microcanonical rates, gap times, and phase space dividing surfaces, J. Chem. Phys. 130 (2009) 164118. 

Let us address these in turn, without being entirely formal. The sensitive dependence on initial conditions can be taken to mean that if a pair of initial points of phase space is given which are separated by any finite amount, no matter how small, then the gap between these solutions grows rapidly (typically exponentially fast) in time. A problem with this concept is that we often think of molecular systems as having an evolution that is bounded by some sort of domain restriction or a property of the energy function the exponential growth for a finite perturbation can therefore only be valid until the separation approaches the limits of the accessible region of phase space. In order to be able to make sense of the calculation of an exponential rate in the asymptotic t oo) sense, we need to consider infinitesimal perturbations of the initial conditions, and this can be made precise by consideration of the Lyapunov characteristic exponents mentioned at the end of this chapter. 

As a consequence, the gap function il(k) is a matrix function of k only or its corresponding polar and azimuthal angles and (p. We have neglected indices for interband pairing because the available phase space is much smaller than in the case of intraband pairs. 

Individual lines on these diagrams represent the energy gap dependence in individual pseudobinary alloys. Regions contained by four curves are for quartemary alloys, and define the region of phase space that should, if the alloy were completely miscible, be accessible with a given quaternary alloy. 

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