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

TL;DR

This paper introduces a star-unitary transformation framework that connects classical Hamiltonian dynamics with stochastic Langevin and Fokker-Planck equations, revealing how irreversibility and randomness emerge from deterministic systems.

Contribution

It develops an exact Markovian formulation for nonintegrable systems using star-unitary transformations, extending canonical transformations to include divergence removal and time-symmetry breaking.

Findings

Star-unitary transformations produce stochastic-like variables from deterministic dynamics.

Transformed distribution functions satisfy exact Fokker-Planck equations.

The method links Poincaré nonintegrable systems with probability 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.