Relaxation operator

The relaxation operator f carries out rotational broadening as well as vibrational dephasing between 0 and 1 states. 

Fig. 5.8. (a) The weights of the eigenvalues of energy relaxation operator and (b) energy correlation function behaviour at short and long times (in inset) from [215]. 

Let us note that sometimes spectra are so well resolved that even some asymmetry of lines is registered [283]. This means that the wings of individual components of a spectral band become observable. Hence, non-adiabatic secularization becomes too crude an approximation. The experiment [283] was interpreted by Kouzov [280] taking into account the co-dependent diagonal part of the relaxational operator. 

We are interested in the upper left-hand block, whose diagonal and off-diagonal elements govern, respectively, exchange inside and between the O, Q and S branches of the Raman spectrum. Therefore, further on we may consider only 3x3 matrices, meaning that the left-hand upper block does not couple with the right-hand lower one. The quasi-classical expression for the relaxation operator... 

Perhaps the first evidence for the breakdown of the Born-Oppenheimer approximation for adsorbates at metal surfaces arose from the study of infrared reflection-absorption line-widths of adsorbates on metals, a topic that has been reviewed by Hoffmann.17 In the simplest case, one considers the mechanism of vibrational relaxation operative for a diatomic molecule that has absorbed an infrared photon exciting it to its first vibrationally-excited state. Although the interpretation of spectral line-broadening experiments is always fraught with problems associated with distinguishing... 

There is greatly renewed interest in electron solvation, due to improved laser technology. However it is apparent that a simple theoretical description such as implied by Eq. (9.15) would be inadequate. That equation assumes a continuum dielectric with a unique relaxation mechanism, such as molecular dipole rotation. There is evidence that structural effects are important, and there could be different mechanisms of relaxation operating simultaneously (Bagchi, 1989). Despite a great deal of theoretical work, there is as yet no good understanding of the evolution of free-ion yield in polar media. 

The inclusion of the rate of light transitions into the relaxation operator Q in line with the decay rate 1/x, makes the kernel light-dependent, a fact that was completely ignored until now. This rate affects the pair correlation functions that obey two sets of equations that follow from (3.106)... 

Due to strong light pumping, the vector ND has to have a higher rank here than in Eq. (3.258). The light-induced and radiationless transitions D D should be accounted for in the intramolecular relaxation operator... 

Sufficient conditions for optimality of forced unsteady-state operation which provides J > Js, can be determined on the basis of analysis of two limiting types of periodic control [10]. The first limiting type is a so-called quasisteady operation which corresponds to a very long cycle duration compared to the process response time t. In this case the description of the process dynamics is reduced to the equations x(t) = /t(u(t)), where h is defined as a solution of the equation describing a steady-state system 0 = f(/t(u(t),u(t))). The second limiting type of operation, the so-called relaxed operation, corresponds to a very small cycle time compared to the process response time (tc t). The description of the system is changed to ... 

When considering relaxation, a Liouville space representation is typically used in which the Hamiltonian and density matrix are represented as superoperators in addition to the relaxation operator being represented as a superoperator. Once a... 

We have not yet specified if the operator to be handled is Hermitian (real eigenvalues) or whether it is a relaxation operator (eigenvalues either real or in the lower half of the complex plane). Uie moment problem related to a Hermitian operator is addressed as the classical moment problem, while by relaxation moment problem we mean the treatment of relaxation operators. 

Consider a Hermitian or relaxation operator H and a state /g). The moments of H with respect to the state of interest j/g) are defined as the diagonal matrix elements of H ... 

When H is Hermitian, the power moments ju are real quantities, and even moments are positive both properties are lost in the case of a general relaxation operator. 

As a Gnal remark before dosing this section, we emphasize that everything that has been said for Hermitian and relaxation operators also applies to Hermitian or relaxation superoperators (see also Chapters I and IV). Hie formal changes to be performed are trivial the state of interest /q) is to be replaced by the operator of interest. /4o)> operator H by the superoperator (— L) where L = [H,...], and the scalar product by a suitable average on an appropriate equilibrium distribution. The moments now have the form... 

While Eq. (8.7) represents a powerful result to handle SUeltjes operators, no similar result exists for a Hamburger operator or a general relaxation operator. In Fig. 3 we give a schematic representation of the theorem stun-marized by Eq. (8.7). 

The extension of the recursion method to non-Hermitian operators possessing real eigenvalues has been carried out by introducing an appropriate biorthogonal basis set in close analogy with the unsymmetric Lanczos procedure. Non-Hermitian operators with real eigenvalues are encountered, for instance, in the chemical pseudopotential theory. Notice that the two-sided recursion method in formulation (3.18) is also valid for relaxation operators, as previously discussed. 

Another remarkable advantage of the formalism is that the generalization to the non-Hermitian relaxation operator is straightforward. If H is not Hermitian, that is, if H we must go through the entire demonstration of this section we can easily verify the usefulness of introducing the follow-... 

The parameters a and are not necessarily real, and bl can even be negative. The properties of continued fractions in relation to relaxation operators are described in Chapter III. 

We consider a most simplified damping by introducing for a two-level system the Lindblad type relaxation operator... 

The Redfield equation describes the time evolution of the reduced density matrix of a system coupled to an equilibrium bath. The effect of the bath enters via the average coupling V = and the relaxation operator, the last sum on the right of Eq. (10.155). The physical implications of this term will be discussed below. 

The kinetic coefficients R(a>) that appear in the relaxation operator are... 

So far, the spins have been considered as isolated entities, which are completely decoupled from the surrounding environment. This is, in fact a good approximation for short evolution times. For longer times, the coupling to the lattice must be considered. This can be done either by including the lattice in the density matrix or by introducing a relaxation operator T which summarily accounts for the effects of the lattice on each... 

Relaxation is treated quantum mechanically as in ref. 39, the original treatise on the subject. The relaxation operator in the usual rotating frame, Hilbert space representation is... 

The correlation functions in Eq. (5) are then expanded in the usual way2 in terms of spectral densities (see Section 3 for further details). As Eq. (5) shows, the relaxation operator involves products of Hamiltonian matrix elements and thus has the effect of redistributing coherence between the various matrix elements or coherences/populations of the density matrix through its involvement in Eq. (2). Kristensen and Farnan40 use their formalism to calculate the central transition lineshapes for lvO (/= 5/2) for both fully relaxed and partially relaxed conditions under different motional models. Some examples are shown in Fig. 27. 

The time evolution of fhe densify mafrix g of the system under the joint influence of the spin Hamiltonian H comprising also the exchange interaction, of diffusion (operator F), of the chemical reaction (operator K), and of relaxation (operator R) is given by the so-called stochastic Liouville equation, ... 

The application of the relaxative operator theory to predict the emission volume of gas in coal mine... 

In the grey theory, before modeling, we usually take a relaxative operator to the original series according to the quditative analysis. It will result to that the disturbance to the series will be weakened, and also result to a desirable effect. 

We call an operator as the relaxative operator when it meets three axioms below (Si F. et al. 1999). 

We regard the operator meeting the three axioms as the relaxative operator. If the rate or amplitude of the relaxative series is weaker w-hen compared to the original series X, the relaxative operator will be called weakened operator. Otherwise, the relaxative operator will be called strengthened operator. The operator below is a commonly used weakened operator. 

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