Random-matrix theory

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

But even in fields different from atomic and nuclear physics the concept of discrete spectra occurs. Consider, for instance, a microwave resonator. A closed ideal microwave resonator can only support a countable number of discrete electromagnetic field configurations called modes. There 

Since the phenomenon of spectra is so ubiquitous in physics it is important to have a tool to characterize energy or frequency spectra. Such a tool is provided by the theory of (energy) level statistics. Following closely the standard literature (Haake (1991), Mehta (1991)) we will now introduce some elements of this field. 

Assume that we have a set of energy levels Ej, j = 1,2. M, where M can be finite or infinite. The first and most elementary thing we can do with a set of energy levels is to count them. We introduce the counting function 

The function Af E) jumps by one unit whenever an energy level Ej is encountered. An example is shown in Fig. 4.1 it is now obvious why the 

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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... 

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... 

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. 

Wigner was the first to suggest the application of random matrix theory to complicated quantum mechanical systems, such as atoms and nuclei. 

Thus, he is generally credited with creating the field of random matrix theory. 

A surprising result of advanced random matrix theory is that the normalized level spacing distribution Pl x) for M x M matrices in the limit of M —> oo is very close to Pw x), which was obtained on the basis of M = 2. The distribution Pl(x) is shown as the full line in Fig. 4.4. Fig. 4.4 shows that the difference between Pw and Pl is indeed very small. 

In summary, random matrix theory is a very useful tool in the investigation of the spectra of classically chaotic wave systems. 

In cases like D2CO or NO2 comparison with experimental data on a state-specific level are ruled out entirely and one has to retreat to more averaged quantities like the average dissociation rate, (fc), and the distribution of rates, Q(k). If the dynamics is ergodic — the basic assumption of all statistical theories — one can derive a simple expression for Q k), which had been established in nuclear physics in order to describe the neutron emission rates of heavy nuclei [280]. These concepts have since developed into the field of random matrix theory (RMT) and statistical spectroscopy [281-283] and have also found applications in the dissociation of energized molecules [121,284-286]. 

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