skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory

Abstract

We consider the statistics of time delay in a chaotic cavity having M open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix Q = − iħS{sup †}dS/dE, where S is the scattering matrix. Our results do not assume M to be large. In a companion paper, we develop a semiclassical approximation to S-matrix correlation functions, from which the statistics of Q can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches.

Authors:
 [1]
  1. Instituto de Física, Universidade Federal de Uberlândia, Ave. João Naves de Ávila, 2121, Uberlândia, MG 38408-100 (Brazil)
Publication Date:
OSTI Identifier:
22479692
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 6; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; CORRELATION FUNCTIONS; POLYNOMIALS; RANDOMNESS; S MATRIX; SCATTERING; SEMICLASSICAL APPROXIMATION; STATISTICS; T INVARIANCE; TIME DELAY

Citation Formats

Novaes, Marcel. Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory. United States: N. p., 2015. Web. doi:10.1063/1.4922746.
Novaes, Marcel. Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory. United States. doi:10.1063/1.4922746.
Novaes, Marcel. 2015. "Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory". United States. doi:10.1063/1.4922746.
@article{osti_22479692,
title = {Statistics of time delay and scattering correlation functions in chaotic systems. I. Random matrix theory},
author = {Novaes, Marcel},
abstractNote = {We consider the statistics of time delay in a chaotic cavity having M open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix Q = − iħS{sup †}dS/dE, where S is the scattering matrix. Our results do not assume M to be large. In a companion paper, we develop a semiclassical approximation to S-matrix correlation functions, from which the statistics of Q can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches.},
doi = {10.1063/1.4922746},
journal = {Journal of Mathematical Physics},
number = 6,
volume = 56,
place = {United States},
year = 2015,
month = 6
}
  • We consider S-matrix correlation functions for a chaotic cavity having M open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over E of the quantities Tr[S{sup †}(E − ϵ) S(E + ϵ)]{sup n}, for general positive integer n. Our result is an infinite series in ϵ, whose coefficients are rational functions of M. From this, we extract moments of the time delay matrix Q = − iħS{sup †}dS/dE and check that the first 8 of them agree with the random matrix theory prediction from our previous paper [M. Novaes, J. Math. Phys.more » 56, 062110 (2015)].« less
  • Assuming the validity of random matrices for describing the statistics of a {ital closed} chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its {ital open} counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via Mopen channels; a=1,2,{hor_ellipsis},M. A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derivemore » an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the {chi}{sup 2} distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer{endash}Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases {theta}{sub a} of the S-matrix (taken modulo 2{pi}). We use it to find the distribution of derivatives {tau}{sub a}={partial_derivative}{theta}{sub a}/{partial_derivative}E of these eigenphases with respect to the energy (``partial delay times``) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner{endash}Smith time delay {tau}{sub w}(E)=(1/M){summation}{sub a}{partial_derivative}{theta}{sub a}/{partial_derivative}E at two different energies E{minus}{Omega}/2 and E+{Omega}/2 as well as at two different values of the external parameter. (Abstract Truncated)« less
  • We calculate the probability distribution of the matrix Q=-i{h_bar}S{sup -1}{partial_derivative}S/{partial_derivative}E for a chaotic system with scattering matrix S at energy E . The eigenvalues {tau}{sub j} of Q are the so-called proper delay times, introduced by Wigner and Smith to describe the time dependence of a scattering process. The distribution of the inverse delay times turns out to be given by the Laguerre ensemble from random-matrix theory. {copyright} {ital 1997} {ital The American Physical Society}
  • Using the random matrix description of open quantum chaotic systems we calculate in closed form the universal autocorrelation function and the probability distribution of the total photodissociation cross section in the regime of quantum chaos. {copyright} {ital 1998} {ital The American Physical Society}