  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.
