Aurelio Patelli and Stefano Ruffo
Statistical mechanics and dynamics of long-range interacting systems
Received: 4 February 2013
Accepted: 10 April 2013
Mathematics Subject Classification (2010): 82B05, 82B26, 82C05, 82C22.
Keywords: Long-range interactions, Inequivalence of ensembles, Large deviations, Quasistationary states.
Abstract: This manuscript summarizes the content of a series of five lectures given by one of the authors (SR) at the summer school on Methods and Models of Kinetic Theory in Porto Ercole, june 3-9 2012. The paper is organized in four Sections. In the first, introductory, Section 1 we discuss several physical systems in which long-range interactions appear: self-gravitating systems, mean-field spin systems, Euler's equation in two dimensions, one component electron plasmas, dipolar systems, finite systems, free electron lasers, cold atoms in optical cavities. The following Section 2 is devoted to ensemble inequivalence, which manifests itself with exotic phenomena like negative specific heat, temperature jumps, broken ergodicity. In this Section we present illustrative models for which microcanonical entropy can be obtained by direct counting: the Blume-Capel and the Kardar-Nagel model. In Section 3 we describe how large deviation theory can be used to obtain the microcanonical entropy of several mean-field models, especially those with continuous local variables, for which direct counting is not applicable. Here, we also show how magnetic susceptibility can become negative in the microcanonical ensemble. Finally, Section 4 is fully devoted to describe quasistationary states. These are non equilibrium states that appear generically in the dynamics of system with long-range interactions and for which a beautiful statistical theory has been proposed long ago by Lynden-Bell. The results presented in this set of lectures are mainly contained in Refs. [1, 2, 3].
 A. Campa, T. Dauxois and S. Ruffo,
Statistical mechanics and dynamics of solvable models with long-range interactions,
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 T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, Eds., Dynamics and thermodynamics of systems with long-range interactions: an introduction, Lecture Notes in Phys., 602, Springer, Berlin, 2002 |
 T. Dauxois, S. Ruffo and L. F. Cugliandolo, Eds., Long-range interacting systems, Oxford University Press, Oxford 2010 |