Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

Let denote the space of all locally nite subsets (congurations) in R d . A stochastic dynamics of binary jumps in continuum is a Markov process on in which pairs of particles simultaneously hop over R d . We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a nite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density.

[1]  Yuri Kozitsky,et al.  Kawasaki Dynamics in Continuum: Micro- and Mesoscopic Descriptions , 2011, 1109.4754.

[2]  Dmitri Finkelshtein,et al.  Operator approach to Vlasov scaling for some models of spatial ecology , 2011 .

[3]  Yuri Kozitsky,et al.  Glauber Dynamics in Continuum: A Constructive Approach to Evolution of States , 2011 .

[4]  Yuri G. Kondratiev,et al.  Binary jumps in continuum. I. Equilibrium processes and their scaling limits , 2011, 1101.4765.

[5]  E. Lytvynov,et al.  A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes , 2010, 1005.4537.

[6]  Dmitri Finkelshtein,et al.  Vlasov scaling for the Glauber dynamics in continuum , 2010 .

[7]  Dmitri Finkelshtein,et al.  Vlasov Scaling for Stochastic Dynamics of Continuous Systems , 2010 .

[8]  Yuri G. Kondratiev,et al.  Markov evolutions and hierarchical equations in the continuum. I: one-component systems , 2007, 0707.0619.

[9]  E. Lytvynov,et al.  A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes , 2007, math/0702338.

[10]  Y. Kondratiev,et al.  Diffusion approximation for equilibrium Kawasaki dynamics in continuum , 2007, math/0702178.

[11]  M. Rockner,et al.  Non-equilibrium stochastic dynamics in continuum: The free case , 2007, math/0701736.

[12]  Yuri G. Kondratiev,et al.  Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics , 2006 .

[13]  Oleksandr Kutoviy,et al.  On the metrical properties of the configuration space , 2006 .

[14]  V. Kolokoltsov.  Kinetic equations for the pure jump models of k-nary interacting particle systems , 2006 .

[15]  M. Rockner,et al.  Equilibrium Kawasaki dynamics of continuous particle systems , 2005, math/0503042.

[16]  V. Belavkin,et al.  On a general kinetic equation for many–particle systems with interaction, fragmentation and coagulation , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Tobias Kuna,et al.  HARMONIC ANALYSIS ON CONFIGURATION SPACE I: GENERAL THEORY , 2002 .

[18]  J. Carstensen.  One Component Systems , 2000 .

[19]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[20]  Sergio Albeverio,et al.  Analysis and Geometry on Configuration Spaces , 1998 .

[21]  Time reversible and Gibbsian point processes, II. Markovian particle jump processes on a general phase space , 1982 .

[22]  V. Belavkin,et al.  Asymptotic dynamics of a system of a large number of particles described by the Kolmogorov-Feller equations , 1981 .

[23]  A. Lenard,et al.  States of classical statistical mechanical systems of infinitely many particles. I , 1975 .

[24]  A. Lenard,et al.  States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures , 1975 .

[25]  A. Lenard,et al.  Correlation functions and the uniqueness of the state in classical statistical mechanics , 1973 .

[26]  David Ruelle,et al.  Superstable interactions in classical statistical mechanics , 1970 .

[27]  R. Phillips,et al.  Dissipative operators in a Banach space , 1961 .

[28]  J. Doob.  Stochastic processes , 1953 .