Star-unitary transformations. From dynamics to irreversibility and stochastic behavior

Abstract

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches Langevin-type equations for {\it 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 {\it dressed} particles, which gives exact Markovian equations. We formulate dressed particles for Poincar\'e nonintegrable systems, through an invertible transformation operator \Lam introduced by Prigogine and collaborators. \Lam 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 Poincar\'e resonances, which breaks time-symmetry. The unitarity of $U$ is extended to ``star-unitarity'' for \Lam. We show that \Lam-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that \Lam-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the non-distributive property of \Lam with respect to products of dynamical variables. Therefore our method leads to a direct link between dynamics of Poincar\'e nonintegrable systems, probability and stochasticity.Comment: 24 pages, no figures. Made more connections with other work. Clarified ideas on irreversibilit