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Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shan-PACS numbers: Key words:This review is dedicated to our masters and to our friends; in particular to Andrei N. Kolmogorov (1903Kolmogorov ( -1987 and Giovanni Paladin whose influence on our work runs deeper than we can know.All the simple systems are simple in the same way, each complex system has its own complexity (freely inspired by Anna Karenina by Lev N. Tolstoy)

We describe a new, biased Monte Carlo scheme to determine the crystal structures of zeolites from powder diffraction data. We test the method on all publicly known zeolite materials, with success in all cases. We show that the method of parallel tempering is a powerful supplement to the biased Monte Carlo.

While statistical mechanics describe the equilibrium state of systems with many degrees of freedom, and dynamical systems explain the irregular evolution of systems with few degrees of freedom, new tools are needed to study the evolution of systems with many degrees of freedom. This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.