Equilibrium Kawasaki dynamics of continuous particle systems. (English) Zbl 1142.60064

Let \(X\) be a Riemannian manifold and let \(\Gamma\) be a set of all subsets of \(X\) which are locally finite. \(\gamma \in \Gamma\) can be then interpreted as a configuration of particles on \(X\). The Kawasaki dynamics is a dynamics of a particle system, where interacting particles randomly hop from one point of \(X\) to another, that is a dynamics whose generator is informally given by the generator
\[ (HF)(\gamma )=\sum_{x\in \gamma}\int_X c(x,y,\gamma) (D_{xy}F)(\gamma) dy, \]
where \(D_{xy}F(\gamma ) = F(\{y\}\cup (\gamma \setminus\{x\}))-F(\gamma)\). The main result of the paper is a theorem on the existence of an equilibrium Kawasaki dynamics as a conservative Markov process on \(\Gamma\) which has the a priori explicitly given Gibbs measure \(\mu \) as symmetrizing measure. It is assumed that the Gibbs measure \(\mu \) corresponds to a relative energy \(E(x,\gamma)\) of interaction between a particle \(x\) and a configuration \(\gamma\) and that \(\mu \) has correlation functions which satisfy the classical Ruelle bound. In order to show the existence of the Kawasaki dynamics several other mild conditions on \(\mu \) are necessary. The authors further give some examples and outline two types of scaling limit of the equilibrium Kawasaki dynamics, one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J75 Jump processes (MSC2010)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text: DOI arXiv


[1] DOI: 10.1006/jfan.1997.3215 · Zbl 0931.58019 · doi:10.1006/jfan.1997.3215
[2] S. Albeverio and M. Röckner, Stochastic Analysis, Proc. of Symposia in Pure Mathematics 57, eds. M. C. Cranston and M. A. Pinsky (Amer. Math. Soc., 1995) pp. 513–528.
[3] DOI: 10.1016/S0246-0203(01)01085-8 · Zbl 0994.82054 · doi:10.1016/S0246-0203(01)01085-8
[4] DOI: 10.1063/1.532662 · Zbl 0935.60086 · doi:10.1063/1.532662
[5] Daley D. J., An Introduction to the Theory of Point Processes (1988) · Zbl 0657.60069
[6] Davies E. B., One-Parameter Semigroups (1980) · Zbl 0457.47030
[7] DOI: 10.1007/BFb0086457 · Zbl 0754.60122 · doi:10.1007/BFb0086457
[8] DOI: 10.1214/aop/1176992156 · Zbl 0623.60119 · doi:10.1214/aop/1176992156
[9] Fukushima M., Dirichlet Forms and Symmetric Markov Processes (1980) · Zbl 0422.31007
[10] DOI: 10.1002/mana.19811020118 · Zbl 0483.60091 · doi:10.1002/mana.19811020118
[11] DOI: 10.1002/mana.19821060107 · Zbl 0514.60099 · doi:10.1002/mana.19821060107
[12] DOI: 10.1016/j.jfa.2006.05.017 · Zbl 1107.60067 · doi:10.1016/j.jfa.2006.05.017
[13] Grothaus M., Ann. Probab. 31 pp 1494–
[14] DOI: 10.1007/BF02392306 · Zbl 0405.60090 · doi:10.1007/BF02392306
[15] Kondratiev Yu. G., Meth. Funct. Anal. Topol. 4 pp 50–
[16] DOI: 10.1016/j.anihpb.2004.05.002 · Zbl 1085.60074 · doi:10.1016/j.anihpb.2004.05.002
[17] Kondratiev Yu. G., Forum Math. 18 pp 9–
[18] Kallenberg O., Random Measures (1975)
[19] Kuna T., Meth. Funct. Anal. Topol. 7 pp 33–
[20] DOI: 10.1007/978-1-4613-8542-4 · doi:10.1007/978-1-4613-8542-4
[21] DOI: 10.1007/978-3-642-77739-4 · doi:10.1007/978-3-642-77739-4
[22] Ma Z.-M., Osaka J. Math. 37 pp 273–
[23] Nguyen X. X., Math. Nachr. 88 pp 105–
[24] DOI: 10.1016/S0764-4442(97)82995-3 · Zbl 0914.60059 · doi:10.1016/S0764-4442(97)82995-3
[25] Ruelle D., Statistical Mechanics. Rigorous Results (1969) · Zbl 0177.57301
[26] DOI: 10.1007/BF01646091 · Zbl 0198.31101 · doi:10.1007/BF01646091
[27] DOI: 10.1007/978-3-642-84371-6 · doi:10.1007/978-3-642-84371-6
[28] DOI: 10.1016/j.anihpb.2003.11.003 · Zbl 1042.60073 · doi:10.1016/j.anihpb.2003.11.003
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