Spectral density limit

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

In optical domain, preamplifier is no more an utopia and is in actual use in fiber communication. However quantum physics prohibits the noiseless cloning of photons an amplifier must have a spectral density of noise greater than 1 photon/spatial mode (a "spatial mode" corresponds to a geometrical extent of A /4). Most likely, an optical heterodyne detector will be limited by the photon noise of the local oscillator and optical preamplifier will not increase the detectivity of the system. 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

Dealing with the restrictive situation of equal dampings, where yQ = y8 = y, the spectral density (107) may be written as the limit of /sf (oj) ex [Eq. (88)] when neglecting the anharmonic coupling—that is, when aG = 0 ... 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

An analytical theory of the outer-sphere PRE for slowly rotating systems with an arbitrary electron spin quantum number S, appropriate at the limit of low field, has been proposed by Kruk et al. (144). The theory deals with the case of axial as well as rhombic static ZFS. In analogy to the inner sphere case (95), the PRE for the low field limit could be expressed in terms of the electron spin spectral densities s ... 

Models for the outer-sphere PRE, allowing for faster rotational motion, have been developed, in analogy with the inner sphere approaches discussed in the Section V.C. The outer-sphere counterpart of the work by Kruk et al. 123) was discussed in the same paper. In the limit of very low magnetic field, the expressions for the outer-sphere PRE for slowly rotating systems 96,144) were found to remain valid for an arbitrary rotational correlation time Tr. New, closed-form expressions were developed for outer-sphere relaxation in the high-field limit. The Redfield description of the electron spin relaxation in terms of spectral densities incorporated into that approach, was valid as long as the conditions A t j 1 and 1 were fulfilled. The validity... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

The most remarkable feature of these average properties is that they are determined to within rigorous error bounds just by the knowledge of the general properties of the spectral densities discussed in Section II. That is, if we calculate a certain finite number of moments of a spectral density, then averages such as Eqs. (14) (18) must lie between certain calculable limits, no matter what (positive) functional form 1(a) actually has, as long as 1(a) has the specified moments. 

Often one expects on physical grounds that the spectral density for a problem should be a smooth, or at least continuous, function of frequency. However, the special spectral densities constructed in the previous section were very singular functions, sums of 5 functions (i.e., sharp lines). These special spectral densities were constructed uniquely to minimize or maximize various functionals of the spectral densities, subject to the constraints of having given values of a finite number of moments. One would like to be able to construct also some of the smoother spectral densities, which also have the correct moments, and thus will necessarily have functionals interpolating somewhere between the error limits. Of course such functions will not be unique unless other constraints are added. 

Another approach to estimating spectral densities, which has the advantage of guaranteeing that the approximate functions are positive, can be based on the error bounds constructed in Section III-A for the spectral density broadened by a Lorentzian slit function. If we had a sufficient number of moments to make the error bounds very precise, then we could reduce the broadening as much as we like, so that the broadened distribution of spectral density becomes as close as we like to the true distribution. In order to estimate these higher moments, we should need to take advantage of some special feature of the distribution. For example, in the case of the harmonic vibrations of a crystalline solid, the distribution of frequencies lies between limits — co,nax and +comax, and is zero outside... 

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

S. Mukamel In order to represent situations in which nuclear and electronic dynamics take place on the same time scale, one needs to incorporate nuclear degrees of freedom into the description. A frequency-dependent Redfield superoperator can capture some effects, but in general is very limited and may even yield negative probabilities. A method for decomposing a given spectral density into a few collective coordinates and identifying these coordinates was presented in Ref. 1. 

If we again consider Fig. 21.4, we can see that the cross section vanishes at v2 = 0.32 and that the profile does not match the spectral density, A2, of the autoionizing state. The Beutler-Fano profile of Fig. 21.4 is periodic in v2 with period 1, so the spectrum from the ground state consists of a series of Beutler-Fano profiles. At higher values of v2 the profiles become compressed in energy since dW/dv2 = l/vf. Fig. 19.2 shows two regular series of Beutler-Fano profiles between the Ba+ 6p1/2 and 6p3/2 limits. In this case the absorption never vanishes because there is more than one continuum. 

In contrast to the diffusion processes, the fluctuating force 3F t) can no longer be derived from the Wiener process, and its spectral density defined as 2kBT 0 times the Fourier transform of the memory function C(f) is frequency limited and has no more the white noise characteristics. 

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