Nonequilibrium statistical mechanics: foundations and modern applications Brief description: Nonequilibrium phenomena are ubiquitous in nature, but are much less understood than equilibrium phenomena. The free expansion of a gas in a container, electric currents, the formation of blood vessels, the dynamics of earthquakes, the explosion of supernovas are just a few exaples of nonequilibrium phenomena. The great variety of nonequilibrium systems is associated with the variety of nonequilibrium states in which each given system can be found; for instance nonequilibrium steady steady states and transient states are characterized by quite different behaviours, and they can be realized in very many different guises. This has led to the development of different ad hoc approaches, for the different phenomena, which have not been unified under a general framework, except for the case of near equilibrium systems. Luckily,the macroscopic phenomena which we want to control on the scale of our daily life are most often near-equilibrium phenomena, and we can therefore rely on nonequilibrium thermodynamics for that. However, the modern sciences and technology are ever more concerned with the nanoscales, where the conditions needed for the validity of equilibrium or nonequilibrium thermodynamics are not verified. Understanding the phenomena occurring at the nanoscale provides but one motivation to understand the foundationas of statistical mechanics. Proceeding along this line of research, and motivated by the range of applications which can be envisaged or are already realizable, present day nonequilibrium statistical mechanics has found a possible unifying principle in nonequilibrium fluctuations. This course will illustrate conceptual and technical aspects of current research in nonequlibrium statistical mechanics, cast in the emerging unifying picture, making use of elementary concepts of dynamical systems theory and discrete stochastic processes. Contents: 1. Nonequilibrium thermodynamics: local equilibrium and derivation of balance equations and hydrodynamic equations; Brownian motion and fluctuation-dissipation theorem; Onsager reciprocal relations; Green-Kubo relations; Onsager-Machlup theory. 2. Ensembles and invariant measures: microscopic foundations and the role of chaos; ergodic and chaotic hypotheses; anomalous transport. 3. Far from equilibrium and nanoscale systems: Equivalence and non-equivalence of ensembles; dynamical models; extensions of the linear theory to far from equilibrium systems; trasient fluctuation relations (Evans-Searles, Jarzynski, Crooks, Hatano-Sasa); steady state fluctuation relations; relations with linear response and future developments. ********************************************************** BIBLIOGRAPHY *First block* D. J. Evans, G. P. Morriss, Statistical mechanics of nonequilibrium liquids, Cambridge Univ. Press (2008) H. B. Callen, Thermodynamics, Wiley (1966) M. W. Zemansky, Calore e termodinamica, Zanichelli (1970) E. Fermi, Thermodynamics, Dover (1956) S. R. de Groot, P. Mazur, Non-equilibrium thermodynamics, Dover (1984) J. N. Reddy, Principles of continuum mechanics, Cambridge Univ. Press (2010) J. Bellissard, Coherent and dissipative transport in aperiodic solids: an overview, in Lecture Notes inPhysics, Vol. 597, Springer (2002) N. W. Ashcroft, N. D. 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