Discrete & Continuous Dynamical Systems - A, 2012
The evolutions of states is described corresponding to the Glauber dynamics of an infinite system... more The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro-and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.
Journal of Dynamics and Differential Equations, 2013
The dynamics of an infinite system of point particles in R d , which hop and interact with each o... more The dynamics of an infinite system of point particles in R d , 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 μ 0 → μ t 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 k 0 → k t , t ∈ [0, T), in a scale of Banach spaces; (b) proving that each k t is a correlation function for a unique measure μ t. 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 t , t ∈ [0, +∞).
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Papers by Jerzy Kozicki