The solution of the Liouville equation in the infinite limit

Abstract

Methods have been developed for performing the Cauchy integral identified in a previous paper. The singularities in the complex plane of this integral determine the time dependence of the solutions of the Liouville equation in the infinite limit for systems describable in angle-action variables. The singularities do not have the form assumed previously in analogy with the work of Prigogine's group. Solutions in the infinite limit have the same form as for finite systems. Taking the infinite limit, with continuous spectrum of the Liouville operator, is equivalent to expressing the solution to the Liouville equation as a Fourier integral in the angle variables instead of a Fourier series as for a finite system with periodic boundary conditions. The solution in this case is no longer multiply periodic in time. The phase space on which ensemble avarages are computed must be extended from the range zero to one for the angle variables for finite systems to the range of the entire real axis for systems in the infinite limit. The Liouville operator is Hermitian on this space. For systems for which angle-action variables exist, the solutions of the Liouville equation cannot become ime-independent even in the infinite limit.