Distribution functions, asymptotic expansion

For the calculation of the distribution at large values of X, the first term of the asymptotic expansion of the Bessel function, fi(X), can be used. The following distribution is obtained ... 

Of course, the calculated distribution function cannot be an arbitrary function of and must satisfy the mathematical conditions of non-negativity and of normalization physical conditions too have to be satisfied for instance, yPc ) must be identically null for < 0 and must be temperature independent. There is no choice of m and m for which function (35) is temperature-independent however, the asymptotic expansion (36) shows that y>c ) varies with T only in a region centred on of width Oik T), while the calculated distribution function is approximately constant in the whole interval (finj iw) provided that the extremes and 5m i nd therefore the difference 5m — in, are constant with T. 

Equations (11-71), (11-72), and (11-73) will prove very useful in the determination of the first few terms of the asymptotic expansion of the distribution function. 

It is well known that Zq , the kinetic part of the partition function, is irrelevant in the determination of the equation of state and in the case of equilibrium between two states. In this work we will be interested only in the configurational part of the distribution function Wn/zq ). Asymptotic expansions of the rotational part of for linear rotators and S3mimetric tops can be found elsewhere. We are now in a position to determine the general form of an expansion of Wn/Zq in powers of the quantum parameter A. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The choice of the basis functions depends crucially on the behavior of the self-similar distribution (see footnote 5). For example, suppose that the self-similar distribution z6 (z) has the asymptotic behavior (ju < 1) in the region of z close to zero. Then it is possible to show (see Appendix of Sathyagal et al, 1995) that the function g u) is approximated by for u close to zero. In other words, g(u) is of order 0(m ). Consequently, g u) is not analytic at w = 0, and a very large number of basis functions in the expansion (6.1.9) are required to describe adequately the behavior near the origin. This problem can be overcome by choosing basis functions that have the same dependence on u near w = 0, as g u) does. Incorporating as much known analytical information as possible about the nature of the solution is an important aspect of the solution of inverse problems. Let us see how... 

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