thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which *resembles* a second quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there:
$$H_i=\sum_{q=1}^\infty \gamma_{q,i} \; \lambda_q$$
while $q$ and $i$ positive integers.

What we find is that $\gamma_{q,i}$ is an infinite matrix, and we have all elements of this infinite matrix from observations. $\gamma_{q,i}$ contains only positive integers (and zero), it is a lower triangular matrix with all diagonal elements equal zero except of $\gamma_{11}=\gamma_{22}=1$. The matrix is not random.

I wonder if someone in this community may help me to understand where and how such matrices are dealt with in physics? Is there a specific name/notation for them? Is there a certain intuition behind of such Hamiltonian and such tzpe of matrix?

I want to analyse $\gamma_{q,i}$ further from a physicist's point of view.

Thanks

ensemblefor a multiparticle state. – al-Hwarizmi Jun 26 '13 at 21:24