Topics in chaotic dynamics

Various kinematical quantities associated with the statistical properties of dynamical systems are examined: statistics of the motion, dynamical bases and Lyapunov exponents. Markov partitons for chaotic systems, without any attempt at describing ``optimal results''. The Ruelle principle is illustrated via its relation with the theory of gases. An example of an application predicts the results of an experiment along the lines of Evans, Cohen, Morriss' work on viscosity fluctuations. A sequence of mathematically oriented problems discusses the details of the main abstract ergodic theorems guiding to a proof of Oseledec's theorem for the Lyapunov exponents and products of random matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos, nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov exponents, random matrices, gaussian thermostats, ergodic theory, billiards, conductivity, gas.

ℏ\hbar corrections in semi-classical formula for smooth chaotic dynamics

The validity of semiclassical expansions in the power of $\hbar$ for the quantum Green's function have been extensively tested for billiards systems, but in the case of chaotic dynamics with smooth potential, even if formula are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), based on the classical Green's functions, we present an efficient method allowing the calculation of $\hbar$ corrections for the propagator, the quantum Green's function, and their traces. Especially, we show that the previously published expressions for $\hbar$ corrections to the traces are incomplete.Comment: 19 pages, 4 figures, to be published in Phys. Rev. E shortened version, corrected typo

ERGODICITY, ENSEMBLES, IRREVERSIBILITY IN BOLTZMANN AND BEYOND

The contents of a not too well known paper by Boltzmann are critically examined. The etymology of the word "ergodic" and its implications are discussed. The connection with the modern theory of Ruelle is attempted