On the existence of dynamical systems with exponentially decaying collision operators

The paper is concerned with the time-domain collision operator ψ of the Brussels school of non-equilibrium statistical mechanics: ψ(t)=PLQexp (tQLQ)QLP, where L is the skew-adjoint Liouville operator of a dynamical system, and P and Q are complementary orthogonal projectors. Under the assumption that P is finite rank, we prove that if ψ is norm-bounded by a decreasing 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 ψ decays exponentially. We use this characterization to show that K-systems admit exponentially decaying collision operators. We also show that this property is enjoyed by the collision operator of the Pietenpol model for a large class of interactions. This answers in the affirmative a question raised by Coveney and Penrose (Coveney P.V. and Penrose O. 1992 J. Phys. A: Math. Gen. 25 4947-66). (author)