A perturbation expansion for correlation functions via the Wigner distribution

Abstract

We compute the coordinate correlation 1 2 < q ( t) q ( o) + q ( o) q ( t)> the Wigner phase space distribution, for a system with Hamiltonian H = H 0 + λ H 1, where the canonical ensemble Wigner distribution corresponding to H 0 is known exactly. Perturbation expansions in powers of λ, for the correlation function and the Wigner distribution, are developed. By avoiding an expansion in powers of ħ, we obtain results whose validity is not restricted to the near-classical regime, in contrast to the Wigner-Kirkwood approach. We illustrate our results by application to the one-dimensional anharmonic oscillator, and the relation to other perturbation methods (Lindstedt-Poincaré, Green's Function) is explored. By virtue of the fluctuation-dissipation theorem, it is anticipated that such results will also be useful for the investigation of transport problems.