Aurelio Patelli and Stefano Ruffo
Statistical mechanics and dynamics of longrange interacting systems
Pages: 345396
Received: 4 February 2013
Accepted: 10 April 2013
Mathematics Subject Classification (2010): 82B05, 82B26, 82C05, 82C22.
Keywords: Longrange 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 39 2012. The paper is organized in four Sections. In the first, introductory, Section 1 we discuss several physical systems in which longrange interactions appear: selfgravitating systems, meanfield 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 BlumeCapel and the KardarNagel model. In Section 3 we describe how large deviation theory can be used to obtain the microcanonical entropy of several meanfield 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 longrange interactions and for which a beautiful statistical theory has been proposed long ago by LyndenBell. The results presented in this set of lectures are mainly contained in Refs. [1, 2, 3].
References
[1] A. Campa, T. Dauxois and S. Ruffo,
Statistical mechanics and dynamics of solvable models with longrange interactions,
Physics Reports 480 (2009), 57159

[2] T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, Eds.,
Dynamics and thermodynamics of systems with longrange interactions: an introduction,
Lecture Notes in Phys., 602, Springer, Berlin, 2002

[3] T. Dauxois, S. Ruffo and L. F. Cugliandolo, Eds.,
Longrange interacting systems, Oxford University Press, Oxford 2010
