On the Zero-Temperature Limit of Gibbs States

2009/07/31 by Jean-René Chazottes, J. -R. Chazottes, Michael Hochman +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Statistical Mechanics and Entropy #Theoretical and Computational Physics #math-ph #math.DS #math.MP

paper · pdf · doi:10.1007/s00220-010-0997-8

published as Commun. Math. Phys vol. 297 (2010) 265-281 · The statement of Theorem 1.2 is more accurate and some new comment follow it

openalex publication_date 2010/03/12 · arxiv created 2010/07/06 · arxiv updated 2010/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We exhibit Lipschitz (and hence Hölder) potentials on the full shift \0,1\^ℕ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space \0,1\^{\mathbb Z} for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space \0,1\^ℤ^d, d≥3, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.

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