The Boltzmann-Grad limit of the periodic Lorentz gas

Abstract

We study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit).A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two. ContentsJENS MARKLOF and ANDREAS STR ÖMBERGSSON 4.4.Proof of Theorem 1.1 270 4.5.Proof of Theorem 1.3 273 5. Convergence to the stochastic process Ξ(t) 275 5.1.Four lemmas 276 5.2.Proof of Theorem 5.1 279 5.3.A counterexample 281 5.4.Macroscopic initial conditions 282 6.A continuous-time Markov process 282 6.1.Transcription of Theorem 1.4 283 6.2.A semigroup of propagators 285 6.3.The Fokker-Planck-Kolmogorov equation 288

Locations

• arXiv (Cornell University) - View - PDF
• Annals of Mathematics - View - PDF