Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times

Abstract

We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” (hb), evolving through the Liouville equation for the non-equilibrium classical distribution Wc, with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution Wc,eq, as weight function to generate orthogonal polynomials (Hn’s) in momenta. The moments of Wc, implied by the Hn’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a hb are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions Weq generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.