Low Density Limit and the Quantum Langevin Equation for the Heat Bath

Abstract

We consider a repeated quantum interaction model describing a small system [Formula: see text] in interaction with each of the identical copies of the chain ⊗ℕ* ℂn+1, modeling a heat bath, one after another during the same short time intervals [0, h]. We suppose that the repeated quantum interaction Hamiltonian is split into two parts: a free part and an interaction part with time scale of order h. After giving the GNS representation, we establish the connection between the time scale h and the classical low density limit. We introduce a chemical potential µ related to the time h as follows: h2 = eβµ. We further prove that the solution of the associated discrete evolution equation converges strongly, when h tends to 0, to the unitary solution of a quantum Langevin equation directed by the Poisson processes.