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arXiv · 2003.09191

Bivariate $q$-normal distribution for transition strengths distribution from many-particle random matrix ensembles generated by $k$-body interactions

V. K. B. Kota·Manan Vyas
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Abstract

Recently it is established, via lower order moments, that the univariate q-normal distribution, which is the weight function for $q$-Hermite polynomials, describes the ensemble averaged eigenvalue density from many-particle random matrix ensembles generated by $k$-body interactions [Manan Vyas and V.K.B. Kota, J. Stat. Mech. {\bf 2019}, 103103 (2019)]. These ensembles are generically called embedded ensembles of $k$-body interactions [EE($k$)] and their GOE and GUE versions are called EGOE($k$) and EGUE($k$) respectively. Going beyond this work, the lower order bivariate reduced moments of the transition strength densities, generated by EGOE($k$) [or EGUE($k$)] for the Hamiltonian and an independent EGOE($t$) for the transition operator ${\cal O}$ that is $t$-body, are used to establish that the ensemble averaged bivariate transition densities follow the bivariate $q$-normal distribution. Presented are also formulas for the bivariate correlation coefficient $ρ$ and the $q$ values as a function of the particle number $m$, number of single particle states $N$ that the particles are occupying and the body ranks $k$ and $t$ of $H$ and ${\cal O}$ respectively. Finally, using the bivariate $q$ normal form a formula for the chaos measure number of principal components (NPC) in the transition strengths from a state with energy $E$ is presented.

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BibTeXRIS

V. K. B. Kota, Manan Vyas. 2020-03-20. Bivariate $q$-normal distribution for transition strengths distribution from many-particle random matrix ensembles generated by $k$-body interactions. https://doi.org/10.1088/1742-5468%2Fababfc

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