Many-Particle Dynamics and Kinetic Equations - C. Cercignani

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Résumé :
This book is devoted to the evolution of infinite systems interacting via a short range potential. The Hamilton dynamics is defined through its evolution semigroup and the corresponding Bogolubov-Born-Green-Kirkwood-Yvo n (BBGKY) hierarchy is constructed. The existence of global in time solutions of the BBGKY hierarchy for hard spheres interacting via a short range potential is proved in the Boltzmann-Grad limit and by Bogolubov's and Cohen's methods. br/ emAudience:/em This volume will be of interest to graduate students and researchers whose work involves mathematical and theoretical physics, functional analysis and probability theory.

Sommaire:
I: The Bbgky Hierarchy.- 1.1. Introduction.- 1.2. The Hamilton dynamics of a system of particles with hard core.- 1.3. The evolution operator for a system with finite number of particles.- 1.4. The derivation of the BBGKY hierarchy.- 1.5. The steady BBGKY hierarchy.- Appendix I. The infinitesimal operator of the groupSN (t).- Appendix II. The formal derivation of the infinitesimal operator [BB] of the group SN(t).- II: The Initial Value Problem For The Bbgky Hierarchy of a System of a Finite Number of Particles.- 2.1. Introduction.- 2.2. The evolution operator of the BBGKY hierarchy.- 2.4. Existence of solutions for a one-dimensional BBGKY hierarchy.- 2.5. The iteration series.- III: The Initial Value Problem For L? Data: Thermodynamic Limit.- 3.1. Introduction.- 3.2. A local existence theorem for the BBGKY hierarchy of hard spheres.- 3.3. Global existence theorems.- 3.4. Method of the interaction region.- IV: The Derivation of The Boltzmann Equation.- 4.1. Introduction: On the Boltzman-Grad limit.- 4.2. The Boltzmann-Grad limit for equilibrium states.- 4.3. The Boltzmann hierarchy and the Boltzmann equation.- 4.4. The Boltzmann-Grad limit for solutions of initial value problem for the BBGKY hierarchy.- 4.5. The Boltzmann-Grad limit for equilibrium states of systems of hard spheres in the framework of the canonical ensemble.- V: On the Derivation of Kinetic Equations From the Bbgky Hierarchy.- 5.1. Introduction: kinetic equations.- 5.2. Bogolubov's method of constructing kinetic equations.- 5.3. The non-equilibrium cluster expansions method.- 5.4. Justification of the generalized kinetic equation.- References.