Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions

Two very different problems that can be studied by renormalization group methods are discussed with the aim of showing the conceptual unity that renormalization group has introduced in some areas of theoretical Physics. The two problems are: the ground state theory of a one dimensional quantum Fermi liquid and the existence of quasi periodic motions in classical mechanical systems close to integrable ones. I summarize here the main ideas and show that the two treatments, although completely independent of each other, are strikingly similar.Comment: Plain TeX, 20 pages 7 figures (included in ps format

Rigorous Theory Of The Boltzmann Equation In The Lorentz Gas

: The Boltzmann limit conjecture of Grad is discussed in general and proved for the Lorentz gas case (where the Boltzmann equation is linear). This is a reprint of an unpublished preprint of 1972, with one footnote added, one postscript (to quote the Lanford theorem), and improved with language editing. I reprint it in this form to make it accessible, as it has been quoted by other authors in later papers. The original preprint was commissioned for a book that eventually was not published. 1. --- Introduction The Boltzmann equation is an approximation to the "true" evolution equation: this is due to the fact that in its derivation the following assumptions are made [1,2]: 1) only binary collisions are considered 2) "Molecular chaos" is assumed at all times: i.e. the high order correlation functions can be expressed in terms of the one--particle distribution as: f(r 1 v 1 ; r 2 v 2 ; : : : ; r n v n ; t) = n Y i=1 f(r 1 v 1 ; t) (1:1) 3) in the computation of the collision term one ..