2011/09/22 by Christoph Berns, Yuri kondratiev, Berns, Christoph +6
Mathematics · Physics and Astronomy · #60J75 #60K35 #92D40 #Cold Atom Physics and Bose-Einstein Condensates #Cosmology and Gravitation Theories #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #math-ph #math.DS #math.MP #math.PR #msc:60J75 #msc:60K35 #msc:92D40
paper · pdf · doi:10.48550/arxiv.1109.4754
revised version
openalex publication_date 2011/09/22 · arxiv created 2012/08/19 · arxiv updated 2012/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The dynamics of an infinite system of point particles in ℝd, which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval [0,T), the evolution of states μ0 ↦ μt is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution k0 ↦ kt, t∈ [0,T), in a scale of Banach spaces; (b) proving that each kt is a correlation function for a unique measure μt. The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles' density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution \varrhot, t∈ [0,+∞).