Porter-Thomas

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

The coimection between the Porter-Thomas P(lc) distribution and RRKM theory is made tln-ough the parameters j -and v. Waite and Miller [99] have studied the relationship between the average of the statistical... 

Figure 2. Statistics of current for the transmission through the Sinai billiard for T 0. The upper left panel shows the computed distribution for p = 2 together with the Porter-Thomas distribution P(p) (solid curve). In the inset in the same panel the computed wave function statistics f(p) for the real part of ip is compared with a random Gaussian distribution (solid curve). 

The irregularity of the spectrum has consequences on the properties of the matrix elements of observables like the electric dipole moment and, thus, on the radiative transition probabilities. For radiative transitions, a single channel is open and the statistics of the intensities follow a Porter-Thomas or x2 distribution with parameter v = 1, as observed in NO2 [5, 6]. 

Let us emphasize that the Porter-Thomas distribution is here applied to the resonances of the molecular Hamiltonian in the absence of a radiation held. In the case of NO2 mentioned in Section III, the same distribution with v = 1 was applied, by contrast, to the radiative linewidths of the molecular Hamiltonian [5, 6]. 

In order to obtain the frequency function of the amplitudes for a fixed value W one now has to integrate over W the product of the secular behaviour of jj and the Porter-Thomas distribution. The distribution function of the amplitudes for arbitrary values of initial and final energies then follows after a second integration [VER79]... 

The fluctuations in neutron peak intensities arise from the Porter-Thomas distributed beta decay widths to levels in the NE nuclide. In the simplest case only a single state in the GC nuclide can be fed and only one neutron partial wave is significant. The observed levels will be a subset of levels in the NE nuclide and will be distributed in energy following a Wigner distribution. In a typical GC nuclide, however, there will be a number of accessible final states and the delayed neutron spectrum will be a superposition of transitions from several parts of the NE nuclide level structure. 

From the distributions of the decay widths Ti and r2, which for a chaotic dot are given by the Porter-Thomas distribution [Jalabert 1992 Prigodin 1993]... 

The intensity distribution given by (2.17) has been derived12 for real transition amplitudes. It is referred to as the Porter-Thomas distribution,17 and has been extensively studied in nuclear physics. It is the distribution of transition... 

The expression for N t, E) in equation (A3.12.67) has been used to study [103.104] how the Porter-Thomas P k) affects the collision-averaged monoenergetic unimolecular rate constant k (Si, E) [105] and the Lindemann-Hinshelwood unimolecular rate constant T) [47]. The Porter-Thomas P k) makes k, E) pressure... 

P k) only affects yjjj( , T) in the intermediate pressure regime [40.104]. and has no affect on the high- and low-pressure limits. This t5q)e of analysis has been applied to HO2 H + O2 resonance states [106]. which decay in accord with the Porter-Thomas P k). Deviations between the mjj( , T) predicted by the Porter-Thomas and exponential P k) are more pronounced for the model in which the rotational quantum number K is treated as adiabatic than the one with K active. 

Figure 8.4 Porter-Thomas distribution of state specific rate constants, Eq. (8.15), for v 1, 2,4, 8, and In these plots x = k and (x) = k (Polik et al, 1990b).

Thus, for state-specific decay and the most statistical (or nonseparable) case, a micro-canonical ensemble does not decay exponentially as predicted by RRKM theory. It is worthwhile noting that when v/2 becomes very large, the right-hand side of Eq. (8.24) approaches exp -kt) (Miller, 1988), since lim (1 + xln) " = exp (-x), when n-> °o. Other distributions for P(k), besides the Porter-Thomas distribution, have been considered and all give M(f, E) expressions which are nonexponential (Lu and Hase, 1989b). 

The connection between the Porter-Thomas nonexponential N(r, E) distribution and RRKM theory is made through the parameters k and v. The average of the statistical state-specific rate constants k is expected to be similar to the RRKM rate constant k(E). This can be illustrated (Waite and Miller, 1980) by considering a separable (uncoupled) two-dimensional Hamilton H = + Hy whose decomposition path is... 

Three-dimensional quantum mechanical calculations have been performed to determine the unimolecular rate constants for the resonances in HOj H + O2 dissociation (Dobbyn et al., 1995). The resonances are not assignable and the fluctuations in the resonance rate constants can be represented by the Porter-Thomas distribution. Equation 8.17. Thus, the unimolecular dissociation of HO2 is apparently statistical... 

The Porter-Thomas distribution for N(t, ), Eq. (8.24), can be inserted into Eq. (8.35) to obtain the Porter-Thomas lifetime distribution ... 

Figure 8.6 Microcanonical pressure-dependent rate constant (normalized to the average microcanonical rate /c(f), as a function of the reduced collision frequency =

For the high-energy unresolved resonances, statistical methods have to be applied. The radiation width can be assumed to be constant, but the neutron widths show a Porter-Thomas [22] distribution. The probability for Tn to be in an interval dFn is... 

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