Dressed‐particle formulation of Brownian motion

Abstract

AbstractWe 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 Poincaré nonintegrable systems, through an invertible transformation operator Λ introduced by Prigogine and collaborators. Λ 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 Λ. We show that Λ‐transformed variables have the same time evolution as stochastic variables obeying Langevin equations and that Λ‐transformed distribution functions satisfy an exact Fokker–Planck equation. The effects of Gaussian white noise are obtained by the nondistributive property of Λ with respect to products of dynamical variables. Therefore, our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability, and stochasticity. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004