Random-matrix physics: spectrum and strength fluctuations

TLDR

The deviations from uniformity in nuclear spectra are universally similar across the spectrum and the periodic table, and can be described by standard Hamiltonian ensembles derived from information‑theory concepts, though departures arise from symmetry and collective effects requiring more complex ensembles. This review aims to provide a self‑contained account of the theory using stochastic‑process‑compatible methods and to establish a solid foundation for ensemble theory, clarifying the origins of observable fluctuation simplicities and deriving additional general fluctuation results. The authors employ stochastic‑process‑based methods to analyze nuclear‑energy‑level data and discuss observable effects in nuclear transitions, reactions, and the low‑temperature thermodynamics of small metallic particle aggregates. Their analysis confirms the theory’s applicability to experimental data and highlights observable effects in nuclear transitions, reactions, and low‑temperature thermodynamics of small metallic particle aggregates.

Abstract

It now appears that the general nature of the deviations from uniformity in the spectrum of a complicated nucleus is essentially the same in all regions of the spectrum and over the entire Periodic Table. This behavior, moreover, is describable in terms of standard Hamiltonian ensembles which could be generated on the basis of simple information-theory concepts, and which give also a good account of fluctuation phenomena of other kinds and, apparently, in other many-body systems besides nuclei. The main departures from simple behavior are ascribable to the moderation of the level repulsion by effects due to symmetries and collectivities, for the description of which more complicated ensembles are called for. One purpose of this review is to give a self-contained account of the theory, using methods---sometimes approximate---which are consonant with the usual theory of stochastic processes. Another purpose is to give a proper foundation for the use of ensemble theory, to make clear the origin of the simplicities in the observable fluctuations, and to derive other general fluctuation results. In comparing theory and experiment, the authors give an analysis of much of the nuclear-energy-level data, as well as an extended discussion of observable effects in nuclear transitions and reactions and in the low-temperature thermodynamics of aggregates of small metallic particles.