[Paper Review] Sensitive dependence of Gibbs measures

Daniel Coronel, Juan Rivera‐Letelier|arXiv (Cornell University)|Nov 12, 2014

Mathematical Dynamics and Fractals21 references3 citations

TL;DR

This paper demonstrates sensitive dependence of Gibbs measures on interactions in classical lattice systems at low temperatures, showing that arbitrarily small perturbations in smooth or Lipschitz-continuous potentials can induce large fluctuations in the resulting Gibbs measures. It establishes chaotic temperature dependence in 2D for finite-state systems and resolves open problems by Chazottes and Hochman.

ABSTRACT

Gibbs measures can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there is a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant fluctuations of low-temperature Gibbs measures. For the XY model we exhibit the sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit the sensitive dependence of Gibbs measures for classical lattice systems with finite-state space. The interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we obtain for the first time the chaotic temperature dependence in dimension 2 for a finite-state space model, and solve some problems stated by Chazottes and Hochman.

Motivation & Objective

• To investigate how small changes in interaction potentials affect Gibbs measures at low temperatures in classical lattice systems.
• To demonstrate that Gibbs measures can exhibit chaotic dependence on interactions even under arbitrarily small perturbations in the smooth or Lipschitz topology.
• To resolve open problems posed by Chazottes and Hochman concerning chaotic temperature dependence in finite-state lattice models.
• To establish the first example of chaotic temperature dependence in two dimensions for finite-state systems with exponentially decaying interactions.

Proposed method

• Analyzing Gibbs measures for the XY model with smooth, nearest-neighbor interactions under small perturbations in the smooth category.
• Constructing finite-state lattice systems with exponentially decaying interactions and Lipschitz-continuous potentials in the configuration space.
• Using perturbation theory and thermodynamic formalism to study the sensitivity of Gibbs measures to changes in interaction parameters.
• Employing the Lipschitz topology to quantify small perturbations and assess their impact on the resulting equilibrium measures.
• Applying results from statistical mechanics and dynamical systems to demonstrate the emergence of chaotic fluctuations in low-temperature regimes.

Experimental results

Research questions

• RQ1 Can arbitrarily small perturbations in the interaction potential lead to significant changes in the low-temperature Gibbs measures of classical lattice systems?
• RQ2 Does the XY model exhibit sensitive dependence of Gibbs measures under small smooth perturbations of its interaction?
• RQ3 Can chaotic temperature dependence be observed in two-dimensional finite-state lattice models with exponentially decaying interactions?
• RQ4 What is the role of the Lipschitz topology in characterizing the sensitivity of Gibbs measures to interaction changes?
• RQ5 How do the results resolve the open problems on chaotic temperature dependence posed by Chazottes and Hochman?

Key findings

• The paper establishes that Gibbs measures in the XY model exhibit sensitive dependence on interactions under arbitrarily small smooth perturbations.
• For finite-state lattice systems with exponentially decaying interactions, small perturbations in the Lipschitz topology lead to significant fluctuations in low-temperature Gibbs measures.
• The study provides the first example of chaotic temperature dependence in two dimensions for a finite-state system.
• The results resolve long-standing open problems by Chazottes and Hochman concerning the existence of chaotic temperature dependence in finite-state models.
• The analysis confirms that the sensitivity of Gibbs measures is robust even under minimal changes in interaction structure, highlighting intrinsic instability in low-temperature equilibrium states.