Physics of long-range interacting systems

## Informations pratiques

Discipline : |
Physique |

Niveau : |
Master 2 |

Semestre : |
S4a |

Crédits ECTS : |
4 |

Volume Horaire : |
21h Cours |

Responsable : |

Thierry Dauxois |

CNRS & École Normale Supérieure de Lyon, Laboratoire de Physique |

Intervenants : |

Thierry Dauxois |

## Language of instruction

The course will be given in english if one student is non french-speaking (or more).

## Objective

This lecture tries to gradually acquaint with the subject. In particular we will provide a satisfactory understanding of properties generally considered as oddities only a couple of decades ago: ensemble inequivalence, negative specific heat, negative susceptibility, ergodicity breaking, out-of-equilibrium quasi-stationary-states, anomalous diffusion, etc.

We will describe the theoretical and computational instruments needed for addressing the study of both equilibrium and dynamical properties of systems subject to long-range forces. We will discuss also the applications of such techniques to the most relevant examples of long-range systems.

## Syllabus

*1. Equilibrium Statistical Mechanics of Long-Range Interactions*

Definition of long-range interactions. Extensivity vs Additivity. The large deviations method. Few solvable simple toy models: the BEG, the HMF, the three states Potts models.

*2. Dynamical Properties*

Kinetic theory. Boltzmann equation. The H-theorem and irreversibility. Klimontovich equation. Vlasov equation. Out-of-equilibrium dynamics and slow relaxation. Quasi-stationary states.

*3. Gravitational systems*

Equilibrium statistical mechanics of self-gravitating systems. Lynden-bell’s entropy.

*4. Coulomb systems and plasma*

Main parameters. Temperature, Debye Shielding. The Vlasov-Maxwell equations. Waves Particle interactions. Free Electron Laser (FEL).

*5. Two-dimensional and geophysical Fluid Mechanics*

Elements of fluid mechanics. The statistical approach of the Onsager point vortex model. The Robert-Sommeria-Miller theory for the 2D Euler equation. The quasi-geostrophic model for fluid geophysical fluid dynamics.

## Prerequisite

A basic course in statistical mechanics.

## Keywords

Statistical mechanics; mean-field models; ensemble inequivalence; negative specific heat; Out-of-equilibrium dynamics; kinetic theory; gravitation; Coulomb; plasma; wave-particles interaction; dipolar; two-dimensional hydrodynamics.

## Exam

Written