On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

• Main Authors: E.Lytvynov, P.T.Polara
• Format: Article
• Language: English
• Published: Institute for Condensed Matter Physics 2008-06-01
• Series: Condensed Matter Physics
• Subjects: birth-and-death process; continuous system; Gibbs measure; hopping particles; scaling limit
• Online Access: http://dx.doi.org/10.5488/CMP.11.2.223

We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the L²-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.