[Paper Review] On the prevalence of non-Gibbsian states in mathematical physics

Aernout C. D. van Enter|arXiv (Cornell University)|Apr 26, 2012

Theoretical and Computational Physics13 references4 citations

TL;DR

This paper investigates the prevalence of non-Gibbsian measures in mathematical physics, demonstrating that such measures arise naturally in renormalization group transformations, stochastic dynamics, and coarse-graining procedures. Despite Gibbs measures being central to equilibrium statistical mechanics, the paper shows that quasilocality—key to Gibbsianness—is frequently violated, leading to non-Gibbsian behavior even in physically relevant systems.

ABSTRACT

Gibbs measures are the main object of study in equilibrium statistical mechanics, and are used in many other contexts, including dynamical systems and ergodic theory, and spatial statistics. However, in a large number of natural instances one encounters measures that are not of Gibbsian form. We present here a number of examples of such non-Gibbsian measures, and discuss some of the underlying mathematical and physical issues to which they gave rise.

Motivation & Objective

• To investigate the widespread occurrence of non-Gibbsian measures in mathematical physics, particularly in systems involving coarse-graining or effective descriptions.
• To clarify the conditions under which Gibbsianness fails, especially in renormalization group transformations and stochastic dynamics.
• To examine the physical and mathematical implications of non-Gibbsianness, including violations of quasilocality and the breakdown of effective Hamiltonians.
• To assess the reliability of renormalization group methods when the resulting measures are non-Gibbsian, especially in critical phenomena and phase transitions.
• To explore the limitations of extending Gibbsian concepts to quantum systems and continuous-spin models, where conditional probabilities lack direct analogues.

Proposed method

• Analyzes the DLR equations and quasilocality condition as criteria for Gibbsianness, using conditional probabilities in the product topology.
• Applies renormalization group maps—especially decimation and block-spin transformations—to Gibbs measures, showing that the resulting measures often violate quasilocality.
• Examines stochastic dynamics (e.g., Glauber dynamics) by studying the time-evolved marginal of an initial Gibbs measure, proving non-Gibbsianness after finite time.
• Uses path-space analysis to include the full dynamics in the measure, enabling detection of non-Gibbsianness in time-evolved systems.
• Considers discretization of continuous spins (e.g., XY model) and shows that low-temperature regimes can lead to non-Gibbsian clock-spin measures.
• Applies alternative criteria such as anomalous large deviations and violation of the non-nullness (finite-energy) condition to prove non-Gibbsianness in specific cases.

Experimental results

Research questions

• RQ1 Under what conditions does a renormalization group transformation produce a non-Gibbsian measure, even when starting from a Gibbs measure?
• RQ2 How does stochastic dynamics, such as Glauber spin-flip processes, lead to non-Gibbsian states in finite time?
• RQ3 To what extent can non-Gibbsian measures still be considered physically meaningful or mathematically tractable?
• RQ4 What are the implications of non-Gibbsianness for the universality class and critical behavior in statistical systems?
• RQ5 Can effective theories based on coarse-graining or discretization still be described by a regular Hamiltonian when quasilocality fails?

Key findings

• Non-Gibbsianness arises generically in renormalization group transformations, particularly in first-order transitions and decimation maps.
• Even after finite-time stochastic dynamics, initial Gibbs measures can become non-Gibbsian, violating the quasilocality condition.
• Discretization of continuous spins, such as in the XY model, leads to non-Gibbsian behavior at low temperatures due to non-summable interactions.
• Measures such as random-cluster models, invariant measures of stochastic dynamics, and sign-fields of massless Gaussians are frequently non-Gibbsian.
• The failure of quasilocality implies long-range, non-summable interactions, which may signal a change in universality class.
• Weakly Gibbsian or almost-Gibbsian measures do not always satisfy the variational principle, indicating they can be significantly less well-behaved than standard Gibbs measures.