Exchangeability, chaos and dissipation in large systems of particles
The sixth problem of Hilbert is concerned with the mathematical treatment of the axioms of mechanics and, more precisely, ``addresses the question of describing the transition between atomistic and continuous models for gas dynamics by rigorous mathematical convergence results'' [\textit{T. Bodineau} et al., C. R., Math., Acad. Sci. Paris 353, No. 7, 623--627 (2015; Zbl 1405.35125)]. The author presents some approaches towards the verification of Boltzmann's assumption of microscopic chaos, which is a key assumption to macroscopic theories of gas dynamics. This also constitutes a first step of a programme to solving Hilbert's sixth problem. In particular, the hard sphere system of a finite number of balls in a multidimensional torus is considered. The balls have the same mass and diameter and the evolution of the system is governed by the effects of transport and elastic collisions. When the ball diameter is small, it can be predicted that the system is strongly unstable in the sense that the change of the position of a particle may result in the change of the trajectory of the system. The created disorder by such phenomena is also called chaos, but is caused by two different mechanisms (ergodicity and uncorrelation, see in the article, page~19). Exchangeability, in the article's title, is with respect to momentum and kinetic energy upon brief collisions. To understand certain assumptions of uncorrelation in perfect gases at equilibrium, that the balls are independent besides exchangeability, the author presents the rigorous results on the ergodicity of the process, the invariant measures that are anticipated to depend on the energy only, entropy and chaos. Furthermore, the Kac model is presented, which is a stochastic model for the propagation of chaos and the ``underlying scheme in Monte Carlo simulations of the Boltzmann equation.'' Finally, a deterministic result on the propagation of chaos is presented. The hard sphere system is a deterministic system (Hamiltonian, singular and non-dissipating) and randomness is introduced as the ball diameter becomes small. In this part, the approach of \textit{O. E. Lanford III} [in: Math. Aspects statist. Mech., Proc. Sympos. appl. Math. Am. Math. Soc. Soc. industr. appl. Math. New York City 1971, SIAM-AMS Proc. 5, 65--75 (1972; Zbl 0273.70015)] on the asymptotic propagation of chaos in the low density limit is presented.
scientific article; zbMATH DE number 6669058