Thermalization in many-fermion quantum systems with one- plus random $k$-body interactions

We study the mechanism of thermalization in finite many-fermion systems with random $k$-body interactions in presence of a mean-field. The system Hamiltonian $H$, for $m$ fermions in $N$ single particle states with $k$-body interactions, is modeled by mean field one-body $h(1)$ and a random $k$-body interaction $V(k)$ with strength $\lambda$. Following the recent application of $q$-Hermite polynomials to these ensembles, a complete analytical description of parameter $q$, which describes the change in the shape of state density from Gaussian for $q=1$ to semi-circle for $q=0$ and intermediate for $0<q<1$, and variance of the strength function are obtained in terms of model parameters. The latter gives the thermalization marker $\lambda_t$ defining the thermodynamic region. For $\lambda \ge \lambda_t$, the smooth part of the strength functions is very well represented by conditional $q$-normal distribution ($f_{CN}$), which describes the transition in strength functions from Gaussian to semi-circle as the $k$-body interaction changes from $k = 2$ to $m$ in $H$. In the thermodynamic region, ensemble averaged results for the first four moments of the strength functions and inverse participation ratio (IPR) are found to be in good agreement with the corresponding smooth forms. For higher body rank of interaction $k$, system thermalizes faster.