Activated dynamics probability density

Whether the inactive region is a true continuum (e.g., photofragmentation) or a quasi-continuum comprised of an enormous density of rigorously bound eigenstates (polyatomic molecule dynamics, Section 9.4.14) is often of no detectable consequence. The dynamical quantities discussed in Section 9.1.4 (probability density, density matrix, autocorrelation function, survival probability, transfer probability, expectation values of coordinates and conjugate momenta) describe the active space dynamics without any reference to the detailed nature of the inactive space. 

The probability density flow (5.260), or the one of conditioned probability density (5.252), stay on the basis of analytically representation of the no equilibrium dynamic for the electro-reactive chains in a temporary scale which cover, but also it overcome the one of the activated chemical complex. For the complexity of this study, at least from the perspective of the involving the path integrals in the dynamic of the respective equilibrium and no equilibrium, becomes extremely instmctive the solution of Fokker-Planck equation in the conditioned probability density form (5.250), from where, the calculation of the probability density as well as of the associated currents are immediate. 

Abstract. In this paper we show that a well-known model of genetic regulatory networks, namely that of Random Boolean Networks (RBNs), allows one to study in depth the relationship between two important properties of complex systems, i.e. dynamical criticality and power-law distributions. The study is based upon an analysis of the response of a RBN to permanent perturbations, that may lead to avalanches of changes in activation levels, whose statistical properties are determined by the same parameter that characterizes the dynamical state of the network (ordered, critical or disordered). Under suitable approximations, in the case of large sparse random networks an analytical expression for the probability density of avalanches of different sizes is proposed, and it is shown that for not-too-smaU avalanches of critical systems it may be approximated by a power law. In the case of small networks the above-mentioned formula does not maintain its validity, because of the phenomenon of self-interference of avalanches, which is also explored by numerical simulations. 

Conventionally it is considered that it is not possible to derive dynamical information, and hence determine the mechanism of ionic conduction, from Monte Carlo simulations. However, by examining the density distribution along different possible pathways for conduction, one can determine whether such processes are active and make some estimate of their relative probabilities (Fig. 6.8). Clearly in this case the dominant pathway will be via the interstitial... 

The initial geometry for formamide was set to the enol form as in Fig. 7.10. The atoms O, C, and N make a molecular plane, and the bridging water molecule is also placed initially so as to lie in that plane. In what follows, we refer to the molecular orbitals approximately lying on the plane and to those approximately perpendicular to the plane as a and tt orbitals, respectively. Likewise, using only tt orbitals in 7(r,f) and Bab (r.f)) we estimate the tt electron density and tt bond-order, respectively. Similarly, the a electron density and a bond-order are made available. This distinction between the a and tt subspaces is just a matter of convenience, and of course they are not physical observables individually, since Cs symmetry is not imposed on the molecular system. On the contrary, all the vibrational modes are active in the present SET calculations. Since the aim of this study is not to estimate the reaction probability but the mechanism of the electron dynamics associated with proton transfer, we chose somewhat artificial initial conditions of nuclear motion to sample as many paths achieving proton transfer as possible. 

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