On the existence of dynamical systems with exponentially decaying collision operators

The paper is concerned with the time-domain collision operator psi Of the B russels school of non-equilibrium statistical mechanics: psi (t) PLQ exp(tQ LQ) QLP, where L is the skew-adjoint Liouville operator of a dynamical syst em, and P and Il are complementary orthogonal projectors. Under the assumpt ion that P is finite rank, we prove that if psi is norm-bounded by a decrea sing exponential, then L; must have a certain spectral property, and that, conversely, this spectral property guarantees the existence of a projector P for which the corresponding psi decays exponentially. We use this charact erization to show that K-systems admit exponentially decaying collision ope rators. We also shaw that this property is enjoyed by the collision operato r of the Pietenpol model for a large class of interactions. This answers in the affirmative a question raised by Coveney and Penrose.