On the Equilibrium State of a Small System with Random Matrix Coupling to Its Environment

We consider a random matrix model of interaction between a small $n$-level system, $S$, and its environment, a $N$-level heat reservoir, $R$. The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$, the reduced density matrix of the system $\rho _{S}=\mathrm{Tr}_{R}\rho _{S\cup R}^{(eq)}$, is given by $\rho _{S}^{(c)}\sim \exp {\{-\beta H_{S}\}}$, where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.