Approximate Eigenfunctions of the Liouville Operator in Classical Many-Body Systems

A variational criterion is used to find approximate eigenfunctions and eigenvalues of the Liouville operator in classical many-body systems. The trial functions are taken to be sums over molecules of functions depending on the position and momentum of a single molecule. In a harmonic lattice, this approach leads to exact eigenfunctions and eigenvalues. In a fluid, the eigenvalue spectrum is continuous, and the eigenfunctions are related to those found by Van Kampen in his study of the linearized Vlasov equation for a plasma. The time dependence of the fluid current density is found by means of these eigenfunctions and eigenvalues. The results show persistent free-particle propagation and damped sound-wave propagation, with relative importance depending on the magnitude of the sound velocity.