Moment dynamics

In this part we will explain a method that we extensively used to describe stochastic dynamical systems. It is based on the dynamics of the moments of a distribution. We applied it successfully to a variety of globally coupled systems. Advantages of the method are simple applicability and quick numerical investigations. Let us consider a globally coupled stochastic system that is described by the following set of Langevin equations  

(x) = Xi is the mean value of the system. Special focus will be on systems where the function / has the form 

We are especially interested in the mean of the ensemble x). We therefore average over Eq. 1.25 and make a Taylor s expansion of the right hand side around x). We obtain  

25 is not the most general form we can treat with the method of the moment dynamics. Especially models with more than one dynamical variable, like the FHN system, are important to us. In this case (Let us call the second variable y) we have to introduce the mixed central moments gn,m = x — x)) y — (y)) ) (and equivalently for more variables). 

If we look closely at Eqs. 1.28 and 1.29 we notice that in general they incorporate infinite sums. It is only for polynomials / and g that the sums break off at some final value. Even if we deal with polynomials and the sums break off we notice that the dynamics of the n-th central moment generally depends on other, higher moments. The system of equations 1.29 forms an infinite set of coupled ordinary differential equations. It is only for linear functions / and g that the system decouples. 

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Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

As a practical matter, the tank crew s overall success depends on the speed and accuracy of the tank commander. As long as the commander sees the enemy tank or fighting vehicle before the enemy sees him, he and his crew can usually destroy the enemy. This is termed staying inside the enemy s decision loop. If the enemy sees him first then the outcome is likely to be catastrophic for him and his crew. Speed is of the essence. Also, crucial is the accuracy of the friend/foe determination to prevent friendly-fire casualties. Identification of friend or foe is highly contextual and depends on what the Army calls situational awareness. Situational awareness is highly complex and depends on an accurate understanding of the moment-to-moment dynamics of the fire and maneuver of the particular battle. 

In order to have a closer look at the observed phenomena we derive the moment dynamics for the system 1.44 in Gaussian approximation ... 

The moment dynamics method allows us to quickly discover the bifurcation diagram. It is given in Fig. 1.22. We see that the transition from stable fixed point of the mean to spiking and back to a stable fixed point is exceptionally complicated. We see a period-doubling cascade followed by a Canard explosion of the chaotic attractor, then a reverse period adding sequence towards a state of uninterrupted spiking. At higher noise inten-... 

These results are supported by simulations of the Langevin equations 1.44. Here the fluctuations in the order parameter are too large to discover the transition in all its detail. Thus, the moment dynamics has has proved a convenient way to increase the resolution of our investigations. 

Figure 12. Plots of the binding energy ) . anharmonic vibrational fiequency static dipole moment dynamic dipole moment /to, Hitshfeld charge atom Pt clusters. Reprinted with permission from S. A. Wasileski eial.,J. Chem. Phys., 115, (2001) 8193. Copyright 2001, American Institute of Physics...

The reconstruction of the bandshape of the imidazole crystal was also performed using Car-Parrinello molecular dynamics (CPMD) simulation [73] of the unit cell of the crystal the results reproduce both the frequencies and intensities of the experimental IR spectrum of bands reasonably well, which we attribute to the application of dipole moment dynamics. The results are presented in Fig. 8 [70]. These and other recent CPMD calculations, on 2-hydroxy-5-nitrobenzamide crystal [71], oxalic acid dihydrate [72], and other systems [64-69], show that the CPMD method is adequate for spectroscopic investigations of complex systems with hydrogen bonds since it takes into account most of mechanisms determining the hydrogen bond dynamics (anharmonicity, couplings between vibrational modes, and intermolecular interactions in crystals). 

The polarization dependence of the photon absorbance in metal surface systems also brings about the so-called surface selection rule, which states that only vibrational modes with dynamic moments having components perpendicular to the surface plane can be detected by RAIRS [22, 23 and 24]. This rule may in some instances limit the usefidness of the reflection tecluiique for adsorbate identification because of the reduction in the number of modes visible in the IR spectra, but more often becomes an advantage thanks to the simplification of the data. Furthenuore, the relative intensities of different vibrational modes can be used to estimate the orientation of the surface moieties. This has been particularly useful in the study of self-... 

The anisotropy of the product rotational state distribution, or the polarization of the rotational angular momentum, is most conveniently parametrized tluough multipole moments of the distribution [45]. Odd multipoles, such as the dipole, describe the orientation of the angidar momentum /, i.e. which way the tips of the / vectors preferentially point. Even multipoles, such as the quadnipole, describe the aligmnent of /, i.e. the spatial distribution of the / vectors, regarded as a collection of double-headed arrows. Orr-Ewing and Zare [47] have discussed in detail the measurement of orientation and aligmnent in products of chemical reactions and what can be learned about the reaction dynamics from these measurements. 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

Two different types of dynamic test have been devised to exploit this possibility. The first and more easily interpretable, used by Gibilaro et al [62] and by Dogu and Smith [63], employs a cell geometrically similar to the Wicke-Kallenbach apparatus, with a flow of carrier gas past each face of the porous septum. A sharp pulse of tracer is injected into the carrier stream on one side, and the response of the gas stream composition on the other side is then monitored as a function of time. Interpretation is based on the first two moments of the measured response curve, and Gibilaro et al refer explicitly to a model of the medium with a blmodal pore... 

TIk experimentally determined dipole moment of a water molecule in the gas phase is 1.85 D. The dipole moment of an individual water molecule calculated with any of thv se simple models is significantly higher for example, the SPC dipole moment is 2.27 D and that for TIP4P is 2.18 D. These values are much closer to the effective dipole moment of liquid water, which is approximately 2.6 D. These models are thus all effective pairwise models. The simple water models are usually parametrised by calculating various pmperties using molecular dynamics or Monte Carlo simulations and then modifying the... 

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

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