On Global-in-time Chaotic Weak Solutions of the Liouville Equation for Hard Spheres

Abstract

We outline a new method of construction of global-in-time weak solutions of the Liouville equation - and also of the associated BBGKY hierarchy - corresponding to the hard sphere singular Hamiltonian. Our method makes use only of geometric reflection arguments on phase space. As a consequence of our method, in the case of $N=2$ hard spheres, we show for any chaotic initial data, the unique global-in-time weak solution $F$ of the Liouville equation is realised as $F=\mathsf{R}(f\otimes f)$ in the sense of tempered distributions on $T{\mathbb{R}}^6\times (-\infty ,\infty )$, where $\mathsf{R}$ is a 'reflection-type' operator on Schwartz space, and $f$ is a global-in-time classical solution of the 1-particle free transport Liouville equation on $T{\mathbb{R}}^3$.

arXiv comment31 pages

Cited by in corpus (1)

• A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions