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We consider a simple model of a classical harmonic oscillator coupled to afield. In standard approaches Langevin-type equations for {\it bare} particlesare derived from Hamiltonian dynamics. These equations contain memory terms andare time-reversal invariant. In contrast the phenomenological Langevinequations have no memory terms (they are Markovian equations) and give a timeevolution split in two branches (semigroups), each of which breaks timesymmetry. A standard approach to bridge dynamics with phenomenology is toconsider the Markovian approximation of the former. In this paper we present aformulation in terms of {\it dressed} particles, which gives exact Markovianequations. We formulate dressed particles for Poincar\'e nonintegrable systems,through an invertible transformation operator $\Lam$ introduced by Prigogineand collaborators. $\Lam$ is obtained by an extension of the canonical(unitary) transformation operator $U$ that eliminates interactions forintegrable systems. Our extension is based on the removal of divergences due toPoincar\'e resonances, which breaks time-symmetry. The unitarity of $U$ isextended to ``star-unitarity'' for $\Lam$. We show that $\Lam$-transformedvariables have the same time evolution as stochastic variables obeying Langevinequations, and that $\Lam$-transformed distribution functions satisfy exactFokker-Planck equations. The effects of Gaussian white noise are obtained bythe non-distributive property of $\Lam$ with respect to products of dynamicalvariables. Therefore our method leads to a direct link between dynamics ofPoincar\'e nonintegrable systems, probability and stochasticity.Comment: 24 pages, no figures. Made more connections with other work.  Clarified ideas on irreversibilit

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