Random matrices and chaos in nuclear physics: Nuclear reactions

Abstract

The application of random-matrix theory (RMT) to compound-nucleus (CN) reactions is reviewed. An introduction into the basic concepts of nuclear scattering theory is followed by a survey of phenomenological approaches to CN scattering. The implementation of a random-matrix approach into scattering theory leads to a statistical theory of CN reactions. Since RMT applies generically to chaotic quantum systems, that theory is, at the same time, a generic theory of quantum chaotic scattering. It uses a minimum of input parameters (average S-matrix and mean level spacing of the CN). Predictions of the theory are derived with the help of field-theoretical methods adapted from condensed-matter physics and compared with those of phenomenological approaches. Thorough tests of the theory are reviewed, as are applications in nuclear physics, with special attention given to violation of symmetries (isospin, parity) and time-reversal invariance.PACS numbers: * Electronic address: mitchell@tunl.duke.edu † Electronic address: richter@ikp.tu-darmstadt.de ‡ Electronic address: Hans.Weidenmueller@mpi-hd.mpg.de problem. Motivated by the fundamental interest in CN scattering and by the need in other fields of physics (neutron physics, shielding problems, nuclear astrophysics, etc.) to have a viable theory of CN reactions with predictive power, that work was carried on for a number of years and led to partial insights into the connection between RMT and CN scattering.

Chaotic behavior of the Compound Nucleus, open Quantum Dots and other nanostructures

“…In this section we present the results of a random matrix simulation based on a random Hamiltonian based S-matrix used extensively by Weidenmüller and collaborators [3][4][5],…”

“…[15] In such systems, it is customary to assume that the underlying electronic dynamics is chaotic to statistically describe the electron transport properties using random matrix theory (RMT) [3][4][5] and the Landauer conductance formula [16,17] Within this framework, the conductance fluctuations are universal functions that depend on the quantum dot symmetries, such as time-reversal, and on the number of open modes N connecting the QD to its source and drain reservoirs. In the semiclassical limit of large N, the statistical transmission fluctuations are accurately modeled by Gaussian processes.…”

Current Status of Nuclear Physics Research

“…The observables such as the average cross section and the correlation function are then calculated by performing an average over the ensemble to which H pertains of products of S-matrices of the type given above. For more details we refer the reader to [233,234].…”

“…As a matter fact, recent application of RMT has been mostly in mesoscopic systems, such as electronic conductance in quantum dots and graphene [232]. The test of RMT has been made possible using, among others, microwave resonators [233,234]. The Quantum Chaotic Scattering Theory relies basically on an expression of the S-matrix which exhibits its relation to the random Hamiltonian of the system.…”

Rank One Non-Hermitian Perturbations of Hermitian $$\beta $$ β -Ensembles of Random Matrices

“…The numerous physical applications of such random matrices can be found in the review papers [6,7,10].…”