Kawasaki dynamics in continuum: micro- and mesoscopic descriptions

The dynamics of an infinite system of point particles in ${\mathbb{R}}^d$, 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 ${\mu }_0\mapsto {\mu }_t$ 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 $k_0\mapsto k_t$, $t\in [0,T)$,inascaleofBanachspaces;(b)provingthateach$k_t$ is a correlation function for a unique measure ${\mu }_t$. 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 ${\varrho }_t$, $t\in [0,+\mathrm{\infty })$.