On the validity of the Brussels formalism in statistical mechanics

Abstract

The so-called 'Brussels' approach to the derivation of kinetic equations usually proceeds by representing the Liouville operator L in the form L0+ lambda L1 where lambda is a perturbation parameter. It can be formulated in terms of an operator P (or a set of such operators) commuting with L0 and projecting from a Hilbert space H, spanned by all square-integrable phase-space densities or density matrices rho , into a subspace H1 in which the reduced or kinetic description is to apply. For the case where H1 is finite-dimensional, the authors prove two main results: (i) if the time-domain collision operator, defined by psi (t)=PL1Q exp(QLQt)QL1P where Q=1-P, is bounded above in norm by a decreasing exponential function of mod t mod and satisfies the condition that the Hermitian part of integral 0infinity eiyt psi (t)dt be invertible for all real y, then for sufficiently small positive lambda the long-time asymptotic approach to equilibrium in the subspace H1 is an exponential decay or exponentially decaying oscillation and is correctly given by the Brussels perturbation method. (ii) If the norm of the collision operator decays to zero at t to infinity more slowly than any exponential then, regardless of the value of lambda , the asymptotic behaviour does not have the exponential form predicted by the Brussels method.