Invariant measures for Glauber dynamics of continuous systems

2003/07/24 by Yuri G. Kondratiev, Kondratiev, Yuri G., Maria João Oliveira +1
Mathematics · Physics and Astronomy · #37L40 #60J75 #60J80 #60K35 #82C21 #82C22 #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:37L40 #msc:60J75 #msc:60J80 #msc:60K35 #msc:82C21 #msc:82C22

paper · pdf · doi:10.48550/arxiv.math-ph/0307050

19 pages

openalex publication_date 2003/07/24 · arxiv created 2003/08/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Glauber-type stochastic dynamics of continuous systems \citeBCC02, \citeKL03, a particular case of spatial birth-and-death processes. The dynamics is defined by a Markov generator in such a way that Gibbs measures of Ruelle type are symmetrizing, and hence invariant for the stochastic dynamics. In this work we show that the converse statement is also true. Namely, all invariant measures satisfying Ruelle bound condition are grand canonical Gibbsian for the potential defining the dynamics. The proof is based on the observation that the well-known Kirkwood-Salsburg equation for correlation functions is indeed an equilibrium equation for the stochastic dynamics.

Related

• Glauber dynamics of continuous particle systems
• In Markov process, an extremal reversible measure is an extremal invariant measure
• Spectral gap for Glauber type dynamics for a special class of potentials
• A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes
• Glauber Dynamics: An Approach in a Simple Physical System