Random matrices

O. Lumpkin. Diffusion of a reptating polymer interacting with a random matrix. Phys Rev E 45 1910-1915, 1993. 

J. Stockmann, 1999). The main achievement of this field is the establishment of universal statistics of energy levels the typical distribution of the spacing of neighbouring levels is Poisson or Gaussian ensembles for integrable or chaotic quantum systems. This statistics is well described by random-matrix theory (RMT). It was first introduced by... 

Similar to closed chaotic billiards the idea was to adjust RMT to the effective Hamiltonian Heff = H — iT (see the pioneering work (J.J.M. Verbaarschot et.al., 1985) and (H.-J. Sommers et.al., 1999) for references). These matrices correspond to GUE with broken time-reversal symmetry. A next natural step was to assume that in the transition region between GOE and GUE, the eigenfunctions are complex and may be thought as columns of the unitary random matrix (G. Lenz et.al., 1992 E. Kanzieper et.al., 1996) S = S +ieS2, composed of two independent orthogonal matrices. The parameter... 

Another development in the quantum chaos where finite-temperature effects are important is the Quantum field theory. As it is shown by recent studies on the Quantum Chromodynamics (QCD) Dirac operator level statistics (Bittner et.al., 1999), nearest level spacing distribution of this operator is governed by random matrix theory both in confinement and deconfinement phases. In the presence of in-medium effects... 

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... 

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

The very nature of non-Kekule species as reactive intermediates suggests that studies of them under conditions far from those used in conventional investigations of the synthesis and reactions of stable molecules are indispensable. These requirements frequently are met by immobilizing the species in crystalline hosts or randomly oriented matrices, as is described in Chapter 17 by Bally in this book. Although some information available from crystal studies usually must be sacrificed in the random matrix technique, the latter is usually far more convenient, and most smdies of non-Kekule compounds in solids have used it. 

Exercise. Formulate the white noise limit for a random matrix process. 

A famous and only partly solved problem of this type is the linear chain of harmonically bound particles, in which the masses and spring constants are random.5 0 A related problem is the determination of the distribution of eigenvalues of a random matrix. )... 

The preceding considerations are essentially based on the model of random-matrix ensembles proposed by Dyson and others in the 1960s. Recent works, in particular by Casati and co-workers [89], have focused on band random matrices. Such matrices naturally arise in quantum systems with subspaces coupled only to next-neighboring subspaces such as for electronic states in a chain of atoms or in the kicked rotator. In such systems, localized states are observed that present a level statistics interme-... 

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. 

For the above defined matrix A3 4 we can form the matrix product >3X5 = but not F = C4X5 /13X 4 for any 4 by 5 matrix C, such as the random matrix C = randn(4,5) that contains normally distributed random entries. 

On the other hand, we know that some chemical reaction systems, especially when highly excited, exhibit quantum chaotic features [16] that is, statistical properties of eigenenergies and eigenvectors are very similar to those of random matrix systems [17]. We call such systems quantum chaos systems. Researchers have also studied how these quantum chaos systems behave under some external... 

This chapter is organized as follows. In Section II, we show how quantum chaos systems can be controlled under the optimal fields obtained by OCT. The examples are a random matrix system and a quantum kicked rotor. (The former is considered as a strong-chaos-limit case, and the latter is considered as mixed regular-chaotic cases.) In Section III, a coarse-grained Rabi state is introduced to analyze the controlled dynamics in quantum chaos systems. We numerically obtain a smooth transition between time-dependent states, which justifies the use of such a picture. In Section IV, we derive an analytic expression for the optimal field under the assumption of the CG Rabi state, and we numerically show that the field can really steer an initial state to a target state in random matrix systems. Finally, we summarize the chapter and discuss further aspects of controlling quantum chaos. 

In the following subsections, we numerically demonstrate to control multilevel-multilevel transition problems in quantum chaos systems One is a random matrix system, and the other is a quantum kicked rotor. 

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