The Relaxation Moment Problem

It is not possible to extend right away the results of the classical moment problem to the relaxation moment problem. However, our survey of Section V.B has been done in such a way that it is possible to select which relations maintain their validity in the relaxation moment problem and which are to be disregarded. Thus little remains to be said except for a few comments. 

An important property which is preserved in the relaxation case is the Herglotz property of the Green s function Goo( ). It can be expanded in the form 

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 

All the mathematical apparatus of Hankel determinants and continued fractions expansion apply also to Hermitian or relaxation superoperators. 

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The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

Before closing this section, we wish briefly to comment on how to deal with the relaxation moment problem in this case the usual definition of moments,... 

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. 

It is well known that in bulk crystals there are inversions of relative stability between the HCP and the FCC structure as a fxmction of the d band filling which follow from the equality of the first four moments (po - ps) of the total density of states in both structures. A similar behaviour is also expected in the present problem since the total densities of states of two adislands with the same shape and number of atoms, but adsorbed in different geometries, have again the same po, pi, P2/ P3 when the renormalization of atomic levels and the relaxation are neglected. This behaviour is still found when the latter effects are taken into account as shown in Fig. 5 where our results are summarized. 

Below we (1) disregard the chemical reaction running in the c-phase, (2) consider the temperature TB of the surface constant, and (3) consider the relaxation time of processes running in the gas to be quite small compared to the relaxation time (time of variation) of the distribution of heat in the c-phase. Restricting ourselves to consideration of intervals of time which are significant compared to the relaxation time of processes in the gas, we shall consider that the state of the nearest layer to the gas, in which the chemical reaction is concentrated, corresponds at each moment to the heat distribution in the c-phase. After establishing this correspondence the problem reduces to consideration of the comparatively slow variation of the heat distribution in the c-phase. 

However, the NMR properties of solid-phase methane are very complex, due to subtle effects associated with the permutation symmetry of the nuclear spin set and molecular rotational tunnelling.55 Nuclear spin states ltotai = 0 (irred. repr. E), 1 (T) and 2 (A) are observed. The situation is made more complicated since, as the solids are cooled and the individual molecules go from rotation to oscillation, several crystal phases become available, and slow transitions between them take place. Much work has been done in the last century on this problem, including use of deuterated versions of methane for example see Refs. 56-59. Much detail has emerged from NMR lineshape analysis and relaxation time measurements, and kinetic studies. For example, the second moment of the 13C resonance is found to be caused by intermolecular proton-carbon spin-spin interaction.60 Thus proton inequivalence within the methane molecules is created. 

Nuclei with spin 7 1 display another problem arising from the electric quadrupole moment which interacts with the electric field gradient created by the surrounding electrons. This property leads to a decrease of the relaxation times and, accordingly, an increase of the linewidths. This situation theoretically limits the use of NMR to nuclei with highly symmetric environments. The cluster effect , as suggested by Rehder [8], sometimes leads to important exceptions to this rule, as shown below. 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

Measurements of relaxation times fall broadly into two classes, those which monitor the populations of some chosen states, and those which measure in some way the impedance of the system to the propagation of a thermal disturbance many laser experiments fall into the first class, whereas ultrasonic dispersion or shock-tube measurements fall into the second. Although artefacts can occur if unsuitable population v. time profiles are used [76.P3], there is, in general, no real difficulty in using equation (2.14) to obtain the vibrational relaxation rate we need not discuss this point further at the moment. Problems may well arise, though, in the determination of rotational relaxation rates in this way, as I will show. 

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