[Paper Review] Gibbsianness versus Non-Gibbsianness of time-evolved planar rotor models

Aernout C. D. van Enter, Wioletta M. Ruszel|ArXiv.org|Nov 22, 2007

Stochastic processes and statistical mechanics9 references4 citations

TL;DR

This paper investigates the Gibbsianness of time-evolved planar rotor systems on ℤᵈ (d ≥ 2) under stochastic dynamics, modeling spins as Brownian diffusions on circles. It proves that the evolved measure remains Gibbsian for small times or high/zero-temperature initial and dynamics, but fails to be Gibbsian for low-temperature initial measures under infinite-temperature dynamics in d=2, due to a hidden phase transition induced by dynamic fields.

ABSTRACT

We study the Gibbsian character of time-evolved planar rotor systems on Z^d, d at least 2, in the transient regime, evolving with stochastic dynamics and starting with an initial Gibbs measure. We model the system by interacting Brownian diffusions, moving on circles. We prove that for small times and arbitrary initial Gibbs measures ν, or for long times and both high- or infinite-temperature measure and dynamics, the evolved measure ν^t stays Gibbsian. Furthermore we show that for a low-temperature initial measure ν, evolving under infinite-temperature dynamics thee is a time interval (t_0, t_1) such that ν^t fails to be Gibbsian in d=2.

Motivation & Objective

• To determine whether time-evolved measures of planar rotor systems remain Gibbsian under stochastic dynamics.
• To analyze the interplay between initial temperature, dynamics temperature, and Gibbsianness in the transient regime.
• To identify conditions under which the Gibbs property is lost or preserved during stochastic time evolution.
• To extend known results on discrete spins to continuous compact spins (on the circle) using reflection positivity and conditional probability techniques.

Proposed method

• Model the system as interacting Brownian diffusions on circles, with spins evolving stochastically on ℤᵈ.
• Use reflection positivity to analyze symmetry breaking and construct orthogonal Gibbs measures in the infinite-volume limit.
• Define conditional probabilities given the existence of a unique infinite cluster in either the right or left half-plane.
• Apply the DLR formalism to characterize Gibbsianness via the existence of a summable interaction potential.
• Leverage the symmetry of the system under reflection r₁ to show that broken symmetry implies non-Gibbsianness.
• Use the fact that tail events are invariant under even translations to establish that conditional measures are Gibbsian.

Experimental results

Research questions

• RQ1 Under what conditions does the time-evolved measure of a planar rotor system remain Gibbsian?
• RQ2 Can non-Gibbsianness emerge in continuous compact-spin systems under infinite-temperature dynamics?
• RQ3 Does the Gibbs property persist for low-temperature initial measures evolving under infinite-temperature dynamics in d=2?
• RQ4 Is the loss of Gibbsianness reversible in continuous compact-spin models, as it is in discrete-spin systems?
• RQ5 Can reflection positivity be used to detect phase transitions in time-evolved continuous-spin systems?

Key findings

• For small times t and arbitrary initial Gibbs measures ν, the time-evolved measure νᵗ remains Gibbsian.
• For long times and high- or infinite-temperature dynamics, νᵗ stays Gibbsian regardless of the initial temperature.
• In d=2, a low-temperature initial measure ν evolving under infinite-temperature dynamics leads to non-Gibbsianness in a time interval (t₀, t₁).
• The non-Gibbsianness arises from a hidden phase transition due to dynamic fields, breaking the reflection symmetry r₁.
• Conditional measures νᵗ,ri and νᵗ,le given infinite clusters in the right or left half-plane are orthogonal and Gibbsian, indicating symmetry breaking.
• The result holds for any translation-invariant initial Gibbs measure, including those with infinitely many pure states.