LSI for Kawasaki dynamics with weak interaction
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Submission date: 07. Jun. 2010
published in: Communications in mathematical physics, 307 (2011) 3, p. 817-860
DOI number (of the published article): 10.1007/s00220-011-1326-6
MSC-Numbers: 60K35, 60J25, 82B21
Keywords and phrases: Logarithmic Sobolev inequality, Spin system, Kawasaki dynamics, Canonical ensemble, coarse-graining
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We consider a large lattice system of unbounded continuous spins that are governed by a Ginzburg-Landau type potential and a weak quadratic interaction. We derive the logarithmic Sobolev inequality (LSI) for Kawasaki dynamics uniform in the boundary data. The scaling of the LSI constant is optimal in the system size and our argument is independent of the geometric structure of the system. The proof consists of an application of the two-scale approach of Grunewald, Otto, Westdickenberg & Villani. Several ideas are needed to solve new technical difficulties due to the interaction. Let us mention the application of a new covariance estimate, a conditioning technique, and a generalization of the local Cramér theorem.