Star-unitary transformations: From dynamics to irreversibility 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 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 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 co-workers. 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 exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the nondistributive property of Λ with respect to products of dynamical variables.
- Publication:
-
Physical Review E
- Pub Date:
- May 2003
- DOI:
- 10.1103/PhysRevE.67.056117
- arXiv:
- arXiv:cond-mat/0206382
- Bibcode:
- 2003PhRvE..67e6117K
- Keywords:
-
- 02.50.Fz;
- 05.40.-a;
- 05.70.Ln;
- Stochastic analysis;
- Fluctuation phenomena random processes noise and Brownian motion;
- Nonequilibrium and irreversible thermodynamics;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 24 pages, no figures. Made more connections with other work. Clarified ideas on irreversibility