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Random matrix ensembles from nonextensive entropy.

Fabricio Toscano1, Raúl O Vallejos, Constantino Tsallis

  • 1Universidade Federal do Rio de Janeiro, Instituto de Física, Caixa Postal 68528, 21941-972 Rio de Janeiro, Brazil. toscano@if.ufrj.br

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 13, 2004
PubMed
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Researchers explored generalized random-matrix ensembles using nonextensive entropy (S(q)). For q>1, these ensembles exhibit power-law tails and non-Wigner-Dyson spacing distributions, differing from classical Gaussian ensembles.

Area of Science:

  • Statistical Mechanics
  • Random Matrix Theory
  • Nonextensive Statistical Mechanics

Background:

  • Classical Gaussian ensembles are derived from Boltzmann-Gibbs-Shannon entropy.
  • Generalized entropy, S(q), introduces a nonextensivity parameter, q.
  • The limit q→1 recovers Gaussian ensembles.

Purpose of the Study:

  • To construct and analyze random-matrix ensembles based on generalized entropy S(q).
  • To investigate the impact of the nonextensivity parameter q on ensemble properties.
  • To compare these novel ensembles with classical Gaussian and other non-Gaussian ensembles.

Main Methods:

  • Maximizing generalized entropy S(q) under constraints to define random-matrix ensembles.
  • Analyzing the joint probability distributions P(H) and marginal distributions P(H(ij)).

Related Experiment Videos

  • Performing numerical analyses of nearest-neighbor spacing distributions in the limit of large matrices.

    • Main Results:

    • For q≠1, matrix elements are correlated, unlike Gaussian ensembles.
    • When q<1, fluctuations resemble Gaussian ensembles.
    • For q>1, power-law tails emerge in probability distributions, and spacing distributions are long-tailed, deviating from Wigner-Dyson statistics.

    Conclusions:

    • Generalized entropy leads to non-Gaussian random-matrix ensembles with distinct properties.
    • The parameter q controls the degree of nonextensivity and associated anomalies.
    • These nonextensive ensembles show connections to Lévy and soft confinement ensembles.