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 n n × matrix and a N N × Hermitian random matrix. We show that under certain ‘macroscopicity’ conditions on R, the reduced density matrix of the system Tr S R S R (eq) ρ ρ = ∪ , is given by H exp{ } S c S ( ) ρ β ∼ − , where HS is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using S c ( ) ρ to describe some nano-systems, like biopolymers, in equilibrium with their environment (Seifert 2012 Rep. Prog. Phys. 75 126001). Our results extend those obtained previously in (Lebowitz and Pastur 2004 J. Phys. A: Math. Gen. 37 1517–34); (Lebowitz et al 2007 Contemporary Mathematics (Providence RI: American Mathematical Society) pp 199–218) for a special two-level system.