From dynamics to irreversibility and stochastic behavior

We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a classical harmonic oscillator (Brownian particle) coupled to a field (heat bath). In traditional approaches Langevin-type equations have been derived from Hamiltonian dynamics, but these equations contain memory terms and are both time-reversible and deterministic. In contrast the original Langevin equation has no memory terms (it is a Markovian equation). In recent years, Prigogine and collaborators have introduced an invertible transformation operator Λ that brings us to a new “kinetic” representation. In this representation dynamics is decomposed into independent Markovian components, including Brownian motion. Λ is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions. While U can be constructed for integrable systems in the sense of Poincaré, for nonintegrable systems there appear divergences in the perturbation expansion, due to resonances. The removal of divergences leads to the Λ transformation. This transformation is “star-unitary.” Starunitarity for nonintegrable systems is an extension of unitarity for integrable systems. We show that Λ-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Λ-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the non-distributive property of Λ with respect to products of dynamical variables.