Harmonic perturbations

A. Error Bounds for the Response to a Damped Harmonic Perturbation. 85... 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

It is of interest here to compare one of these methods with Mori s continued fraction representation for spectral densities.40 Consider the case of Lorentzian broadening of the spectral density (which describes the response to a damped harmonic perturbation, Section III-A). If we set... 

Note that the summation over n in this equation has been omitted because we are considering only one spherical harmonic perturbation at a time. 

A classical resonance-absorption theory [66, 67] was aimed to obtain the formulas applicable for calculation of the complex permittivity and absorption recorded in polar gases. In the latter theory a spurious similarity is used between, (i) an almost harmonic perturbed law of motion of a charge affected by a parabolic potential (ii) and the law of motion of a free rotor, this law being expressed in terms of the projection of a dipole moment onto the direction of an a.c. electric field. 

To get excitation properties of the plasma-embedded atomic system, usually a time dependent perturbation approach within linear response theory is used. The response of the systems is studied using an external harmonic perturbation... 

TDDFRT considers a linear response of the ground-state density to the time-dependent perturbation of the external potential dvext(r1>t) which is, usually, a harmonic perturbation <5vexl(r,f) switched on at time t0... 

It will be noted that a function analogous to ipz which arises in the shear viscosity, does not appear in the thermal conductivity. Its absence, of course, is due to the lack of second order surface harmonic perturbation of the radial distribution function g0m in the case of heat conduction. It may be anticipated that this difference in the form of the number density perturbation might lead to the thermal conductivity coefficient leaving a functional dependence on the temperature which is quite different from that of the shear viscosity coefficient. However, the exact temperature dependence of the two coefficients (Eqs. 42 and 47) has not yet been explored. 

The success of the moments of inertia as a useful structural parameter is at first surprising, since the mathematical expressions for the moments of inertia of idealised clusters are complex, do not relate to any recognisable feature, and vary from structure to structure. In this analysis, the moments of inertia are used as a parameter to describe only the gross deviation from sphericality. As was illustrated in Sect. 2.2, the shape of the ellipsoid enveloping the atoms is not sufficient to describe clusters as oblate or prolate, information about atomic positions is also required. The splitting of the orbitals, which is defined by the spherical harmonic perturbation, is dependent on the same factors which determine the moments of inertia, the cluster shape and distribution of atoms about the surface. The moments of inertia therefore provide a generalised shape parameter, without referring to specific features of the individual structures. 

Note that Eq. (9.14) obtained for the discrete modulation (9.15) practically coincides with induced drift equations known for a continuous periodic modulation [14-16]. The only difference is that the constant

displacement direction induced by a sequence of d-perturbations, but not by a harmonic perturbation I t) = Acos u)t). In both cases this angle, which determines the drift direction for 0 = 4>mod = 0 is a characteristic parameter that depends on the properties of the excitable medium and on the applied modulation method. 

There are transient and harmonic perturbations of the interfacial area. As it was shown by Loglio et al. [144] the theoretical basis is the same and therefore transient relaxations correspond also to a certain characteristic frequency. For harmonic relaxation processes there is a phase difference between the generation of the oscillation and the response function which is a measure of the exchange of matter. 

This technique has been used for relation experiments in the transient as well as harmonic perturbation mode (196-199) and is suitable also for liquid/liquid interfaces (200,... 

Many time-resolved methods do not record the transient response as outlined in the earlier example. In the case of linear systems, all information on the dynamics may be obtained by using sinusoidally varying perturbations x(t) (harmonic modulation techniques) [27], a method far less sensitive to noise. In this section, the complex representation of sinusoidally varying signals is used, that is, A (r) = Re[X( ) exp(I r)]> where i = The quantity X ( ) contains the amplitude and the phase information of the sinusoidal signal, whereas the complex exponential exp(I )f) expresses the time dependence. A harmonically perturbed linear system has a response that is - after a certain transition time - also harmonic, differing from the perturbation only by its amplitude and phase (i.e. y t) = Re[T( ) exp(i > )]). In this case, all the information on the dynamics of the system is contained in its transfer function which is a complex function of the angular frequency, defined as [27, 28]... 

In the case of a nonlinear system, a similar approach using harmonic perturbations is possible if a small-signal perturbation x t) = Re[AX( ))exp(/time-independent bias perturbation, is applied to the system. If the signal level of the perturbation is sufficiently small, a linear dependence of the response on the perturbation can be achieved (i.e. y t) = Re[AT(transfer function defined in Eq. (3a) becomes a differential quantity ... 

Theoretical models have reached a state that allows a quantitative description of the equilibrium state by thermodynamic models, the adsorption kinetics of surfactants at fluid interfaces, the transfer across interfaces and the response to transient or harmonic perturbations. As result adsorption mechanisms, exchange of matter mechanisms and the dilational rheology are obtained. For some selected surfactant systems, the characteristic parameters obtained on the various levels coincide very well so that a comprehensive understanding was reached. 

Before the first light pulse is applied, the system is in the lower level 1>, which means ai = 1 and a2 = 0. The harmonic perturbation (Vol. 1, (2.46))... 

The solutions to the anisotropic diffusion equation can be written as a series expansion, each term of which can be associated with a particular relaxation time. For a harmonic perturbation of the rotational distribution function, as occurs in a dielectric relaxation experiment with an ac electric field, it was found that a single relaxation time was sufficient to describe the relaxation of p, and this could be expressed in terms of the relaxation time Xq) for in the absence of a nematic potential by ... 

Figure 2.1 Shape of the perturbation (in full line) used in chemical relaxation methods and response of the system (dashed line) (A) step perturbation as in T-jump (B) rectangular perturbation as in shock-tube (0 = duration of tiie perturbation) (C) harmonic perturbation as in ultrasonic absorption relaxation.

Rheological studies show some similarities with chemical relaxation studies. For instance, a rectangular shear rate is applied and the relaxation of the stress is monitored. This directly yields the stress relaxation time(s). One can also apply a sinusoidal deformation or strain of angular frequency 0). The response of the system is a two-component sinusoidal shear stress. The first component is in phase with the strain and corresponds to the elastic (storage) properties of the system. The second component is out of phase with the strain with a phase angle 5, and corresponds to the viscous loss in the system. These quantities give access to the storage (elastic) modulus G (co) and to the loss (viscous) modulus G"(o)), with G" ((o)/G (co) = tg5. As in the case of chemical relaxation methods with harmonic perturbation, the variations of G (w) and G" (co) with co yield the relaxation time(s) of the system. 

Por a molecule in a radiation field, the definitions of the frequency-dependent properties are related to the induced (in general also frequency-dependent) multipole moments. Let us consider a small harmonic perturbation of angular frequency w from a plane-wave radiation field. We then have (Barron 2004) ... 

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