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” (), evolving through the Liouville equation for the non-equilibrium classical distribution , with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution , as weight function to generate orthogonal polynomials (’s) in momenta. The moments of , implied by the ’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 are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions 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.
Highlights
The initial condition is Wc,eq ([0])−1/2 Wc,in ([0]). Both the exact hierarchy for the g([n])’s and the closed approximate one for them after the long-term approximation are genuinely different from the non-equilibrium classical BBGKY hierarchy [7,8]
In the latter, in the equation for the distribution function for n particles, one leaves unintegrated their position vectors and momenta, while those for the remaining N − n particles are integrated over. Such an equation depends on the distribution function for n + 1 particles but not on that for n − 1 ones, a feature which, beyond the approximate framework of the standard Boltzmann equation [2,7,8], does not seem to shed much light on the long-term approach to thermal equilibrium for larger n
In the equation for g([n]) in the actual non-equilibrium hierarchy based upon Wc,eq, the contributions from g([n + 1])’s are neatly different from those coming from g([n − 1])’s, at least in the long-term approximation [25]
Summary
Research leading from equilibrium statistical mechanics [1,2,3,4,5,6] to non-equilibrium statistical mechanics [2,7,8,9] in both classical and quantum regimes faces many open fundamental difficulties. The present author has carried out several detailed analyses of the classical Liouville and quantum Wigner equation, through infinite hierarchies for suitable non-equilibrium moments, in order to analyze approaches to thermal equilibrium in long-term approximations [25,26,27,28,29,30,31,32]. The corresponding equilibrium distributions will be used to generate an infinite family of orthogonal polynomials The latter will allow construction of non-equilibrium moments which, through either the Liouville equation or the Wigner one, will imply infinite linear hierarchies. These hierarchies, under suitable assumptions and approximations, will yield an approach to thermal equilibrium for the long term. Successive results are explicitly indicated, for compactness, within the main text
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