2006/08/02 by Dmitri Finkelshtein, Dmitri L. Finkelshtein, Finkelshtein, Dmitri L. +5
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60J75 #60J80 #60K35 #82C21 #82C22 #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #math-ph #math.MP #math.PR #msc:60J75 #msc:60J80 #msc:60K35 #msc:82C21 #msc:82C22
paper · pdf · doi:10.48550/arxiv.math/0608051
arxiv created 2006/08/02 · openalex publication_date 2006/08/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in ℝd which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure mu as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of ``infinite length'' will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in ℝd). We prove that, in the low activity-high temperature regime, the generators of the Kawasaki dynamics converge to the generator of a Glauber dynamics. The convergence is on the set of exponential functions, in the L2(μ)-norm. Furthermore, additionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-dimensional distributions of the processes.