Kawasaki Dynamics in Continuum: Micro- and Mesoscopic Descriptions

The dynamics of an infinite system of point particles in $$\mathbb {R}^d$$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 $$[0,T)$$[0,T), the evolution of states $$\mu _0 \mapsto \mu _t$$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 $$k_0 \mapsto k_t$$k0↦kt, $$t\in [0,T)$$t∈[0,T), in a scale of Banach spaces; (b) proving that each $$k_t$$kt is a correlation function for a unique measure $$\mu _t$$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 $$\varrho _t$$ϱt, $$t\in [0,+\infty )$$t∈[0,+∞).

[1]  Stefan Grosskinsky,et al.  Interacting particle systems MA 4 H 3 , .

[2]  Dmitri Finkelshtein,et al.  Semigroup approach to birth-and-death stochastic dynamics in continuum , 2011, 1109.5094.

[4]  O. Lanford.  The classical mechanics of one-dimensional systems of infinitely many particles , 1968 .

[5]  R. Dobrushin,et al.  Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction , 1977 .

[6]  T. Liggett.  Interacting Particle Systems , 1985 .

[7]  N. N. Bogolyubov,et al.  Problems of a Dynamical Theory in Statistical Physics , 1959 .

[8]  Dmitri Finkelshtein,et al.  Individual Based Model with Competition in Spatial Ecology , 2008, SIAM J. Math. Anal..

[9]  D. Ruelle.  Statistical Mechanics: Rigorous Results , 1999 .

[10]  E. Lytvynov,et al.  Equilibrium Kawasaki dynamics of continuous particle systems , 2005 .

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

[12]  Claudia Neuhauser,et al.  Mathematical Challenges in Spatial Ecology , 2001 .

[13]  尾畑 伸明.  Configuration Space and Unitary Representations of the Group of Diffeomorphisms , 1987 .

[14]  E. Caglioti,et al.  Non-Equilibrium Dynamics of Three-Dimensional Infinite Particle Systems , 2000 .

[15]  Yu. M. Sukhov,et al.  Dynamical Systems of Statistical Mechanics , 1989 .

[16]  Errico Presutti,et al.  Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics , 2008 .

[17]  K. Kawasaki.  Diffusion Constants near the Critical Point for Time-Dependent Ising Models. I , 1966 .

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

[19]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[20]  Yuri Kondratiev,et al.  On non-equilibrium stochastic dynamics for interacting particle systems in continuum , 2008 .

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

[22]  Dmitri Finkelshtein,et al.  An approximative approach for construction of the Glauber dynamics in continuum , 2009 .

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

[24]  François Trèves.  Ovcyannikov theorem and hyperdifferential operators , 1968 .

[25]  K. Deimling.  Ordinary differential equations in Banach spaces , 1977 .

[26]  Nancy L. Garcia,et al.  Spatial birth and death processes as solutions of stochastic equations , 2006 .

[27]  E. Olivieri,et al.  Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics , 2009 .

[28]  Thomas M Liggett,et al.  Stochastic models for large interacting systems and related correlation inequalities , 2010, Proceedings of the National Academy of Sciences.

[29]  J. T. Cox,et al.  Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$ , 1989 .

[30]  D. Finkelshtein,et al.  Markov Dynamics in a Spatial Ecological Model with Dispersion and Competition , 2011 .

[31]  Yuri Kondratiev,et al.  Ergodicity of non-equilibrium Glauber dynamics in continuum , 2010 .

[32]  H. Thieme.  Stochastic semigroups : their construction by perturbation and approximation , 2006 .

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

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

[35]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[36]  Non-equilibrium stochastic dynamics in continuum: The free case , 2007, math/0701736.

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

[38]  Pavel I. Naumkin,et al.  Nonlinear Nonlocal Equations in the Theory of Waves , 1994 .

[39]  T M Li Ge Te.  Interacting Particle Systems , 2013 .