Kramers integral equations

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

The relevance of a recorded impedance spectrum is not clear, as is the case for experiments done with instrumental methods. A number of potentially occurring errors can give rise to a distortion (small or large) of the impedance spectrum, with a certain impact on the interpretation of the data and the curves. A method to analyse the obtained impedance spectra makes use of so-called Kramers-Kronig transformations50,51, which are a set of coupled integral equations that describe the relationship between the real and imaginary part of the impedance. For impedance Z ... 

This problem is solved using the Kramers Kronig (KK) integral equations, which connect the dispersive (real part) and dissipative (imaginary part) reaction processes, by using the fundamental principle of causality and... 

In electromagnetic theory it is shown that the real and imaginary part of e (co) are not independent of each other, but are connected by a pair of integral equations, the Kramers-Kronig relation (e.g. Bohren Huffman 1983). Equation (A3.10) satisfies these relations, i.e. using a Lorentz-Drude model fitted to the laboratory data automatically guarantees that the optical data satisfy this basic physical requirement. 

Absorption is one consequence of a (vibrational) transition, the spectral behaviour of the refractive index reflects the same phenomenon. Within the spectral interval of an absorption band the refractive index follows a dispersion curve. Kramers-Kronig integral equations (for applications in optics see Caldwell and Eyring, 1971 Hopfe et al., 1981 ... 

The Kramers-Kronig are integral equations that constrain the real and imaginary components of complex quantities for systems that satisfy conditions of linearity, causality, and stability. These relationships, derived independently by Kronig and Kramers, were initially developed from the constitutive relations associated with the Maxwell equations for description of an electromagnetic field at interior points in matter. 

A second issue is that the integral equations do not account explicitly for the stochastic character of experimental data. This requires solution of the Kramers-Kronig relations in an expectation sense, as discussed in the following section. The methods used to address the incomplete sampled frequency range will be described in a subsequent section. 

Requirements (22.71) and (22.75) place well-defined constraints on the evaluation of the Kramers-Kronig integral equations. 

Qsp were applied for the validation of electrochemical impedance data. Agarwal et al. described an approach that eliminated problems associated with direct integration of the Kramers-Kronig integral equations and accoimted explicitly for stochastic errors in the impedance measurement. 

Since o(co) is causal, the scattering rate and the mass enhancement are not independent, and are connected through the Kramers-Kronig integral equation ... 

The Kramers-Kronig validation [11] is a well-established method to assess consistency and quality of measured impedance spectra. The Kramers-Kronig relations are integral equations, which constrain the real and imaginary components of the impedance for systems that satisfy the conditions of causality, linearity, and... 

The creatmenc of the boundary conditions given here ts a generali2a-tion to multicomponent mixtures of a result originally obtained for a binary mixture by Kramers and Kistecnaker (25].These authors also obtained results equivalent to the binary special case of our equations (4.21) and (4.25), and integrated their equations to calculate the p.ressure drop which accompanies equimolar counterdiffusion in a capillary. Their results, and the important accompanying experimental measurements, will be discussed in Chapter 6 ... 

The Kronig-Kramers relation is of fundamental importance for optics and for physics in general13). Here, these equations do not seem practical because of the integration of the wavelength from 0 to oo. However, these are very useful for calculating the molar ellipticity magnitude from the observed ORD curve 14). 

Thus we have derived the Kramers equation (VIII.7.4) as an approximation for short tc. It becomes exact in the white noise limit (3.12). The coefficient of the fluctuation term is the integrated autocorrelation function of the fluctuating force, in agreement with (IX.3.5) and (IX.3.6).110... 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

The integration of the fractional Klein-Kramers equation (69) over the position coordinate leads in, the force-free limit, to the fractional Rayleigh equation... 

The Kramers-Kronig relations have been applied to electrochemical systems by direct integration of the equations, by experimental observation of stability and linearity, by regression of specific electrical circuit models, and by regression of generalized measurement models. 

Equation (36) agrees with previous results, which have been derived in a more heuristic manner [30-32], The adiabatic electronic potential-energy surfaces (that is, the eigenvalues of H — TVU) are doubly degenerate (Kramers degeneracy). The adiabatic electronic wave functions carry nontrivial geometric phases which depend on the radius of the loop of integration [29-32]. 

Faced with these difficulties, we shall presently illustrate that if a generalization of the Klein-Kramers equation, first proposed by Barkai and Silbey [30], where the fractional derivatives do not act on the Liouville terms, is used, then the desired return to transparency at high frequencies is achieved. Moreover, the Gordon sum mle, Eq. (85), is satisfied. In conclusion of this subsection, we remark that the divergence of the integral absorption is not unusual in models that incorporate inertial effects. For example, in the well-known Van Vleck-Weisskopf model [88], the divergence results from the stosszahlansatz used by them, just as in the present problem. 

The foregoing general conclusions of Kramers were first verified in a specific case by Eyring and Zwolinskj, 19 who took into account the quantized nature of the molecular levels in an elementary way. They compared the results of the exact integration of typical unimolecular kinetic equations to the results based on the assumption that equilibrium was maintained throughout the reaction. By making assumptions which are supposed to cover the extremes likely to be encountered, they found that the nonequilibrium rate may deviate by no more than 20 per cent from its equilibrium rate. 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

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