由$k$ -体相互作用产生的多粒子随机矩阵系综跃迁强度分布的二元$q$ -正态分布
Bivariate $q$-normal distribution for transition strengths distribution from many-particle random matrix ensembles generated by $k$-body interactions
摘要：
最近，通过低阶矩建立了单变量q正态分布，其是$q$ -Hermite多项式的权重函数，描述了由$k$ -体相互作用产生的多粒子随机矩阵系综的系综平均特征值密度[Manan Vyas和V.K.B. Kota，J. Stat. Mech. { \bf 2019}，103103（2019）]。这些系综统称为$k$ -身体相互作用的嵌入系综[EE（$k$）]，其GOE和GUE版本分别称为EGOE（$k$）和EGUE（$k$）。在此基础上，利用哈密顿量的EGOE（$k$）[或EGUE（$k$）]和作为$t$ -体的跃迁算符${\cal O}$的独立EGOE（$t$）生成的跃迁强度密度的低阶二元约化矩，建立了系综平均的二元跃迁密度服从二元$q$ -正态分布.还提出了双变量相关系数$ρ$和$q$值作为粒子数$m$、粒子所占据的单粒子状态数$N$和身体等级$k$和$t$的函数的公式，分别为$H$和${\cal O}$。最后，利用二元$q$规范形，给出了能量为$E$的态的跃迁强度的混沌测度主成分数的计算公式。
英文摘要：
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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