Stochasticity and Time Symmetry Breaking in Hamiltonian Dynamics

In recent years, we have introduced a new type of transformation operator Λ, leading to irreversible kinetic equations from dynamics, both classical and quantum. In our approach we have no loss of information, since the Λ transformation is invertible. In this paper we consider classical mechanics. Our transformation is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions. While U can be constructed for integrable systems in the sense of Poincaré, for nonintegrable systems there appear divergences in the perturbation expansion, due resonances. The removal of divergences leads to the Λ transformation. This transformation is "star-unitary". Star-unitarity for non-integrable systems is an extension of unitarity for integrable systems. In addition, Λ is non-distributive with respect to products of dynamical variables. This gives fluctuations usually associated with noise in phenomenological equations such as the Langevin equation.