Random Matrix Results Research Articles

The autocorrelation function for spectral determinants of quantum graphs

The paper considers the spectral determinant of quantum graph families with chaotic classical limit and no symmetries. The secular coefficients of the spectral determinant are found to follow distributions with zero mean and variance approaching a constant in the limit of large network size. This constant is in general different from the random matrix result and depends on the classical limit. A closed expression for this system dependent constant is given here explicitly in terms of the spectrum of an underlying Markov process.

Quantum spectra of triangular billiards on the sphere

We study the quantal energy spectrum of triangular billiards on a spherical surface. Group theory yields analytical results for tiling billiards while the generic case is treated numerically. We find that the statistical properties of the spectra do not follow the standard random matrix results and their peculiar behaviour can be related to the corresponding classical phase space structure.

Semiclassical cross section correlations

We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth classical limit the autocorrelation function of such matrix elements has two contributions with relative weights determined by classical dynamics. We show how the random matrix result can be obtained if the operator approaches a projector onto a single initial state. The expressions are verified in calculations for the kicked rotor.

Dissipation in finite Fermi systems

We present a systematic theory of dissipation in finite Fermi systems. This theory is based on the application of periodic-orbit theory to linear response of many-body systems. We concentrate only on the mesoscopic aspect of the phenomena wherein a many-body system can be reduced to a single-body in an effective mean-field. We obtain semiclassical periodic-orbit corrections on top of the two-time correlation function for the rate of energy dissipation. We show that this energy dissipation is irreversible on an observational time-scale. To do so, we derive a generalised Smoluchowski equation for the energy distribution in the quantal domain. Employing the Weyl–Wigner expansion, we also obtain an equation governing the evolution of energy distribution in the combination of semiclassical and adiabatic approximations. Further, we show how the periodic orbit corrections are related to geometric phase acquired by a single-particle wavefunction as it evolves with the slow, time-varying mean-field. We present our results for the important case of mixed dynamics also. We obtain random-matrix results for response functions. We incorpoate chaos in the underlying classical system by writing an ansatz for a generic wavefunction which leads to various central results of equilibrium statistical mechanics. This new formalism is extended here to include dissipation. Finally, we present an expression for the viscosity tensor encountered in nuclear fission in terms of periodic orbits of single particle in an adiabatically deforming nucleus.

Spectral statistics for unitary transfer matrices of binary graphs

Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalise this approach by considering arbitrary, directed graphs with unitary transfer matrices. An exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes. A special class of graphs, so-called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low--dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs.

Conductivity of a small metallic particle

AbstractThe size‐dependent ac‐conductivity of a small metallic particle is studied for electrons enclosed in a box with random impurities. The exact wave functions and energies are calculated numerically up to 500 electrons, then the Kubo‐formula is evaluated, and an ensemble average is performed. A strong size‐ and frequency dependence of the conductivity is found which is in good agreement with random matrix results. The size dependence of the dc‐conductivity agrees qualitatively with microwave absorption measurements on submicron particles.

26Al(n,p)26Mg reaction: Comparison between the Hauser-Feshbach formula and the exact random-matrix result for the cross section.

We have calculated the $^{26}\mathrm{Al}$(n,p${)}^{26}$Mg reaction rate using an exact expression for the compound-nucleus cross section derived recently by Verbaarschot et al. This reaction is astrophysically important, and it happens to fall in a kinematic regime where the exact expression is expected to yield large deviations from the standard Hauser-Feshbach formula. We find that the exact statistical cross section is 20% lower at energies greater than 300 keV, and drops to 40% lower at 0 energy but still does not describe the available data. These deviations are quite similar to those predicted by the formula of Tepel et al.