Random matrix prediction

Very accurate results were obtained for the classically chaotic Sinai billiard by Bohigas, Giannoni, and Schmit (see Fig. 2) which led them to the important conclusion (Bohigas, Giannoni and Schmit, 1984) Spectra of time-reversal invariant systems whose classical analogues are K systems show the same fluctuation properties as predicted by the Gaussian orthogonal ensemble (GOE) of random-matrix theory... 

W. H. Miller I would like to ask Prof. Schinke the following question. Regarding the state-specific unimolecular decay rates for HO2 — H + O2, you observe that the average rate (as a function of energy) is well-described by standard statistical theory (as one expects). My question has to do with the distribution of the individual rates about die average since there is no tunneling involved in this reaction, the TST/Random Matrix Model used by Polik, Moore and me predicts this distribution to be x-square, with the number of decay channels being the cumulative reaction probability [the numerator of the TST expression for k(E)] how well does this model fit the results of your calculations ... 

R. Schinke Although the information on the rate for HO2 is rather limited, we performed a statistical analysis and found reasonable agreement with the prediction of random matrix theory. A picture is given in the original publication [A. J. Dobbyn et al., J. Chem. Phys. (15 May 1996)]. 

R. Schinke We extracted the resonance widths from the spectrum . It is clear that resonances are missed, especially the broader ones. Moreover, the widths have some uncertainty, especially at higher eneigies. Therefore, the statistics of rates is not unambiguously defined. The only point which I want to make is that our results are in qualitative accord with the predictions of random matrix theory. 

In Fig. 8, we compare the predicted values, Eqs. (42) and (43), with the numerical results for the random matrix system. Those results agree well each other especially for a large T that is, the CG picture is valid and useful especially for a large target time T. ... 

For moderate-sized molecules with tens of vibrational modes, vibrational energy flow is conveniently described in a vibrational quantum number space. A statistical theory for the vibrational Hamiltonian, called Local Random Matrix Theory (LRMT), exploits the local coupling in the state space. LRMT predicts... 

Here the authors consider the possibility of inferring such statistical characteristics from the spectral features of probe photons or particles that are scattered by the density fluctuations of trapped atoms, notably in optical lattices, in two hitherto unexplored scenarios, (a) The probe is weakly (perturbatively) scattered by the local atomic density corresponding to the random occupancy of different lattice sites, (b) The probe is multiply scattered by an arbitrary (possibly unknown a priori) multi-atom distribution in the lattice. The highlight of the analysis, which is based on this random matrix approach, is the prediction of a semicircular spectral lineshape of the probe scattering in the large-fluctuation limit of trapped atomic ensembles. Thus far, the only known case of quasi-semicircular lineshapes in optical scattering has been predicted [Akulin 1993] and experimentally verified [Ngo 1994] in dielectric microspheres with randomly distributed internal scatterers. 

Ncff is just the inverse of the rate fluctuation. This is the traditional definition of the number of effective decay, or reaction, channels in the random matrix approach to the statistics of decay rates. This approach has been used both in nuclear (16) and chemical (17) physics. Comparing this result with the RRKM prediction, one can see that Mc(t(E) replaces N (E). One can use either a vibrationally adiabatic tunneling model (17) or a model of hopping between two electronic surfaces in the Condon approximation (40) to show that, when a global random matrix model is used for the Hamiltonian, Neff = N in the classical limit. 

In a system described by a local random matrix model there can be corrections to the RRKM prediction for the average reaction rate. Assuming that the total reaction time is the sum of the RRKM contribution and a correction, and using = f dt e Ja ... 

Leitner DM, Wolynes PG. 1997. Vibrahonal mixing and energy flow in polyatomic molecules Quantitative prediction using local random matrix theory. J. Phys. Chem. A 101 541-548. 

We will create yet another set of validation data containing samples that have an additional component that was not present in any of the calibration samples. This will allow us to observe what happens when we try to use a calibration to predict the concentrations of an unknown that contains an unexpected interferent. We will assemble 8 of these samples into a concentration matrix called C5. The concentration value for each of the components in each sample will be chosen randomly from a uniform distribution of random numbers between 0 and I. Figure 9 contains multivariate plots of the first three components of the validation sets. 

In order to test PCR and later PLS, we remove a random selection of 10 test samples from the total data set the 10 test spectra are collected row-wise in the matrix Ys and the corresponding known qualities in a column vector qs,known. The remaining spectra are organised in the same way in the matrix Y of dimensions 70x700. For each one of the samples we also know the protein content we collect these qualities in the vector q with 70 entries. In the following, Y and q serve as the calibration set that is used later to predict the unknown qualities qs for the test set Ys. The predicted qs can then be compared with the known qualities qs,known. 

In this simple form, this expression is a good first approximation to compare the experimental reinforcement achieved upon addition of filler to the matrix, to the theoretical prediction [11]. It provides a measure of how efficiently the properties of the nanofiller are exploited in the composite, but also enables the comparison with the level of reinforcement achieved using other fillers. Note, in addition, that equation (8.2) sets an upper limit between Efl5 = 200 GPa and / = 1000 GPa, depending on whether the nanocarbon is randomly or perfectly oriented (without taking q0 into account). 

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