Star-unitary transformations: from dynamics to irreversibility and stochastic behavior.

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches, Langevin-type equations for bare particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time-reversal invariant. In contrast, the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time-evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincaré nonintegrable systems, through an invertible transformation operator Lambda introduced by Prigogine and co-workers. Lambda is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincaré resonances, which breaks time symmetry. The unitarity of U is extended to "star unitarity" for Lambda. We show that Lambda-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Lambda-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the nondistributive property of Lambda with respect to products of dynamical variables.