Dressed-Particle Formulation Of Brownian Motion

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 the current article, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincare nonintegrable systems, through an invertible transformation operator A introduced by Prigogine and collaborators. A 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 Poincare resonances, which breaks time symmetry. The unitarity of U is extended to "star unitarity" for A. We show that A-transformed variables have the same time evolution as stochastic variables obeying Langevin equations and that A-transformed distribution functions satisfy an exact Fokker-Planck equation. The effects of Gaussian white noise are obtained by the nondistributive property of A with respect to products of dynamical variables. Therefore, our method leads to a direct link between dynamics of Poincare nonintegrable systems, probability, and stochasticity. (C) 2004 Wiley Periodicals, Inc.