Kawasaki Dynamics in Continuum: Micro- and Mesoscopic Descriptions

Abstract

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, +∞).

Evolution of states in a continuum migration model

“…Indeed, for each t > 0 and θ ∈ Θ, cf (2.1), we have that θψ t ∈ Θ, and hence B μ 0 (θ ψ t ) is the Bogoliubov functional of a certain state. 1 The same is true for the left-hand side of (2.23), and the state μ t contained therein can be condidered as a weak solution of the corresponding Fokker-Planck equation (1.5).…”

“…Fix σ > 0 and then, for given n and N ∈ N, obtain q n ,N 0 from k μ 0 ∈ K ϑ 0 by (3.33), (3.40). As in [1,Appendix] one can show that, for each G ∈ G ϑ 0 , the following holds…”

“…Another important observation regarding the competition in this model is based on (4.15). Let be compact and k (1) t satisfy (4.19). Then for an arbitrary ε > 0, one…”

A Widom–Rowlinson Jump Dynamics in the Continuum

“…In this case, the Banach space where the Cauchy problem in (1.3) is defined can be the space of signed measures with finite variation. For infinite systems, the evolution of states is constructed indirectly, by employing correlation functions and or the related Bogoliubov functionals, see [3,[6][7][8][9][10] and the references quoted therein. In this paper, in describing the evolution of states, see Theorem 3.5 below, we mostly follow the scheme elaborated in [10].…”

“…Let denote the set of all γ ⊂ R d that are locally finite, i.e., such that γ ∩ is a finite set whenever ⊂ R d is compact. Thus, is a configuration space as defined in [1,3,8,11]. In order to take into account the particle's type we use the Cartesian product 2 = × , see [5,7,9], the elements of which are denoted by γ = (γ 0 , γ 1 ).…”