Classical moments

This is an analogue of the classical moments-generating functional discussed by Kubo [39]. Upon expanding the exponential as a power series, the operator J f acts to place each term in so-called normal order, in which all creation operators are to the left of all annihilahon operators j/. By virtue of this ordering (and only by virtue of this ordering). 

N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. 

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

Classical moment expressions. Spectral moments expressed in the Heisenberg notation can be immediately interpreted in terms of classical physics. For a discussion of classical moments, we consider the moments Af of the spectral density, J co), which are related to the moments, y , of the absorption coefficient, a(co), according to Eq. 5.8. By combining that equation with Eq. 5.16, we get at once... 

All odd moments vanish, Mi = M3 = = 0 as was mentioned above. The sixth and eighth classical moments have also been given [158]. 

The last line, Table 5.1, reports the purely classical moments. The zeroth classical moment is a little smaller than the zeroth quantum moment, because of the wave mechanical tunneling of the collisional pair into the classically forbidden region which enhances the intensities. All odd moments of classical profiles are, of course, zero. The second and fourth moments are significantly smaller than the quantum moments, because... 

Higher-order classical moments have also been reported. We mention the classical expressions for the translational spectral moments M , with n = 0, 2, 4, and 6, for pairs of linear molecules given in an appendix of [204]. Spectral moments of spherical top molecules have been similarly considered [163, 205], We note that for n > 1, spectral moments show dynamic as well as static quantum correction, which become more important as the order n of the spectral moments is increased. The discussions on pp. 219, and Table 5.1, suggest that, even for the near-classical systems, quantum corrections may be substantial and can rarely be ignored. 

The previous treatment deals with a one-component order parameter (such as for a commensurate Peierls distortion) but does not apply to situations where the order parameter is complex with an amplitude and a phase (superconductivity, incommensurate Peierls, or spin density wave transitions). The latter situation is analogous to classical moments which can rotate freely in an XY plane. The coherence length of the XY model is less strongly divergent at low temperature than for the Ising model,... 

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. 

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. 

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. 

Pq( ) = 1]. The expression of D can be recognized as the standard expression of Hankel determinants, wtiich are known to be essentially positive quantities (in the classical moment problem). 

The Lanczos Method. The use of classical moment theory involves large determinants, thereby implying delicate problems of numerical stability. This instability arises from very severe cancellations in the determinants, which in turn come from the properties of the moments themselves. Whitehead and Watt bypassed this problem by using the Lanczos al-... 

Classical moment expressions are usually sufficient for the massive systems at high enough temperatures, that is, under conditions where many partial waves are required to describe the collisional interactions. For long-range interaction (like DID), quantum corrections are generally less substantial than for short-range (overlap) induction. The second moments require more substantial quantum corrections than zeroth moments under comparable conditions. 

Quantum sum formulas based on exact pair distribution functions (obtained in the isotropic potential approximation) are also known for n = 0,1,2, and 3 [318,319] we mention also unpublished work by J. D. Poll. Levine has given detailed estimates based on classical moments, assessing the bound dimer contributions [302]. 

This naive description of liquid systems actually represents amodel—flie basic model to describe liquids at a local scale. As it has been here formulated, it is a classical model use has been made of physical classical concepts, as energy, colhsions (and, implicitly, classical moments), spatial ordering (i.e., distribution of elements in the space). 

It should be noted that for termination by recombination (Fig. 10.5A), inherently an error is introduced regarding its contribution in the classical moment equations. For the individual living polymer continuity equations, it can be derived that the contribution of termination by recombination is... 

Since in the classical method of moments, no explicit calculation of the individual concentrations is performed, the manipulation introduced previously disappears when writing down the final classical moment equation, explaining the inherent error for this reaction ... 

In their pioneering work. Handy and Bessis showed that by imposing the positivity conditions originating from the classic Moment Problem in mathematics, one could quantize the low lying discrete states of (multidimensional) rational fraction potential problems. 

The evaluation of the integral in Eq. 1 can be computationally difficult some examples are as follows fx is often not well-defined because of the incompleteness of the statistical information available G(X) may have a nonlinear form the computation of the multifold integral can be very difficult if the number of tmcertain parameters is high. Various methods have been proposed for solving the integral form in Eq. 1. These approaches range from the classical moment methods for structural reliability (e.g., first-order second-moment reliability method) to the simulation-based approaches (i.e., Monte Carlo family of methods), and also the PEER approach, which is quite different compared to the other two techniques. In this entry, alternative methods for estimating the probability of failure are described. 

In essence, this "moment method" seeks to construct the quantum distribution based on knowledge of a few classical moments. Justification for this procedure is based on the approximate equality of low-order classical and quantum moments for some systems and on the fact that it is often possible to characterize quantum probability distributions completely in terms of one or two moments. Although moment methods have often been studied for atom-diatom systems,with comparisons of exact quantum and moment method probabilities available, no analogous study of moment methods for polyatomic systems has been done. In addition, it is not particularly clear how best to generalize many of the existing moment methods to the treatment of polyatomic systems. The incorporation of correlations between different degrees of freedom in the final state distributions has not been considered, for example. 

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