Ergodicity of the $\Delta_3$ statistic and purity of neutron resonance data

TL;DR

This paper investigates the statistical properties of the $ ext{Delta}_3$ statistic to analyze neutron resonance data, aiming to identify underlying dynamics, missing levels, and sequence mixing, with a focus on ergodicity and variance considerations.

Contribution

It provides a detailed analysis of the $ ext{Delta}_3$ statistic's ergodicity and variance, applying these concepts to neutron resonance data for the first time.

Findings

Results support the use of $ ext{Delta}_3$ for detecting spectral regularity or chaos.

Analysis indicates the neutron resonance data are consistent with maximum likelihood level spacing.

Finite sample size effects are effectively incorporated into variance calculations.

Abstract

The $\Delta_3(L)$ statistic characterizes the fluctuations of the number of levels as a function of the length of the spectral interval. It is studied as a possible tool to indicate the regular or chaotic nature of underlying dynamics, detect missing levels and the mixing of sequences of levels of different symmetry, particularly in neutron resonance data. The relation between the ensemble average and the average over different fragments of a given realization of spectra is considered. A useful expression for the variance of $\Delta_3(L)$ which accounts for finite sample size is discussed. An analysis of neutron resonance data presents the results consistent with a maximum likelihood method applied to the level spacing distribution.