Power spectrum classical

Because simulated water is a classical liquid, the computed power spectrum which describes the translational motions, is bound to disagree with that of real water. Figure 37, shows that the power spectrum has peaks at 44 cm-1 and 215 cm-1, whereas for real water they occur at 60 cm-1 and 170 cm-1. A similar discrepancy exists between simulated and real water rotational power spectra (compare the simulated water frequencies 410 cm-1, 450 cm-1 and 800-925 cm-1 with the accepted experimental values 439 cm-1, 538 cm-1 and 717 cm-1). In this model localization of the molecules around their momentary orientations is only marginal. 

Figure 10 The classical power spectrum of a large molecule (RDX, hexahydro-l,3,5-trinitro-1,3,5-triazine) at a very low energy (a) and at its zero-point energy content (78 kcal mol ) (b). Already at the zero point there is extensive classical mode mixing. (Adapted from T. D. Sewell, C. C. Chambers, D. L. Thompson, and R. D. Levine, Chem. Phys. Lett. 208 125 (1993).)...

Figure 12 The classical power spectrum of the OH stretch of OHC1 initially excited to the v = 6 overtone of the OH mode. This provides enough energy to dissociate the weak O—Cl bond but the two modes, which very much differ in their frequency, are not effectively coupled. Two power spectra are shown corresponding to propagating classical trajectories for the first ps after excitation and for the time interval between 1 and 2 ps after excitation, (a) Computation for the isolated molecule, (b) Computations for the molecule in liquid Ar at the (quite high) reduced density of 0.83. Note the solvent induced changes in the spectrum showing that the environment can affect the course of IVR. (Adapted from Y. S. Li, R. W. Whitnell, K. R. Wilson, and R. D. Levine, J. Phys. Chem. 97 3647 (1993).)...

Active vibration-based monitoring method is a classical SHM technique. The main idea behind the method is that structural dynamic characteristics are functions of the physical properties, such as mass, stiffness and damping [22]. Hence, physical property changes due to damages can cause detectable differences in vibration responses. The dynamic characteristic parameters usually used in the technique include frequency, mode shape, power spectrum, mode curvature, frequency response function (FRF), mode flexibility matrix, energy transfer rate (ETR), etc. 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

Summary. We discuss the concept of the Berry phase in a dissipative system. We show that one can identify a Berry phase in a weakly-dissipative system and find the respective correction to this quantity, induced by the environment. This correction is expressed in terms of the symmetrized noise power and is therefore insensitive to the nature of the noise representing the environment, namely whether it is classical or quantum mechanical. It is only the spectrum of the noise which counts. We analyze a model of a spin-half (qubit) anisotropically coupled to its environment and explicitly show the coincidence between the effect of a quantum environment and a classical one. 

On the applied side of quantum chaology we find serious efforts to forge the semiclassical method into a handy tool for easy use in connection with arbitrary classically chaotic systems. Quite frankly, the current status of semiclassical methods is such that they are immensely helpful in the interpretation of quantum spectra and wave functions, but are only of limited power when it comes to accurately predicting the quantum spectrum of a classically chaotic system. In this case numerical methods geared toward a direct numerical solution of the Schrodinger equation are easier to handle, more transparent, more accurate and cheaper than any known semiclassical method. It should be the declared aim of applied semiclassics to provide methods as handy and universal as the currently employed numerical schemes to solve the spectral problem of classically chaotic quantum systems. 

The power spectra of Ar3, shown in Fig. 2, make it clear that there is a distinct change between mean energies of —1.4 x 10-14 erg/atom (equivalent to 18.19 K) and —1.16 x 10 14 erg/atom (equivalent to 28.44 K). From sharp, distinct vibrations, the system has transformed to one with a continuous spectrum of available classical energies. 

As the velocity increases, the energy-loss spectrum of charged particles moving far from the surface into the vacuum is dominated by long-wavelength excitations. In this limit, equation (36) yields the classical stopping power dictated by equation (38). This is illustrated in Fig. 8, where... 

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