| Abstract: |
| A key problem for kinetic theorists is to rigorously derive Boltzmann`s equation starting from a system of $N$ particles (e.g., hard spheres), obeying Newton`s laws, in a natural scaling limit where $N$ tends to infinity. Lanford showed that a hard sphere gas in the so-called Boltzmann-Grad limit, subject to a molecular chaos assumption, will evolve according to Boltzmann`s equation for a short time. One step in Lanford`s proof involves comparing solutions of the Liouville equation against solutions of an infinite system of equations known as the Boltzmann hierarchy; this hierarchy is directly analogous to the Gross-Pitaevskii hierarchy, which appears in several well-known approaches to deriving certain dispersive equations such as cubic NLS. We will write down some toy Boltzmann hierarchies, not directly related to any known $N$-particle system, and discuss techniques for solving these hierarchies (existence and uniqueness) on a short time interval. The results employ Hilbert-Schmidt type norms (unusual in the literature for Boltzmann hierarchies), and the proof makes use of the (inverse) Wigner transform, spacetime estimates \`{a} la Klainerman-Machedon, and combinatorial arguments (in boardgame form) originally due to Erd\{o}s-Schlein-Yau. |
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