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The subject of this monograph is the appearance of irreversibility in gas dynamics.
At a molecular level, the dynamics is Newtonian. In particular, it is reversible, in
contrast with observations at a macroscopic level.
We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles N goes to infinity and the characteristic length of interaction ε simultaneously goes to 0, in the Boltzmann-Grad scaling Nεd−1 ≡ 1.
The time of validity of the convergence is a fraction of the average time of first collision, due to a limitation of the time on which one can prove uniform estimates for the BBGKY and Boltzmann hierarchies.
Our proof relies on the fundamental ideas of Lanford, and the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina. The main novelty here is the detailed study of pathological trajectories involving recollisions, which proves the term-by-term convergence for the correlation series expansion.
Preface
I Introduction
The low density limit
The Boltzmann equation
Main results
II The case of hard spheres
Microscopic dynamics and BBGKY hierarchy
Uniform a priori estimates for the BBGKY and Boltzmann hierarchies
Statement of the convergence result
Strategy of the proof of convergence
III The case of short-range potentials
Two-particle interactions
Truncated marginals and the BBGKY hierarchy
Cluster estimates and uniform a priori estimates
Convergence result and strategy of proof
IV Termwise convergence
Elimination of recollisions
Truncated collision integrals
Proof of convergence
Concluding remarks