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Metastable Hierarchy in Abstract Low-Temperature Lattice Models

Seonwoo Kim

TL;DR

The paper analyzes metastability in abstract low-temperature lattice models under Metropolis-type dynamics by introducing a hierarchical decomposition of stable plateaux into levels $\mathscr{P}^{1},\dots,\mathscr{P}^{\mathfrak{m}}$, with metastable transitions at each level converging to a Markov chain as $\beta\to\infty$. It develops a general framework based on stable plateaux, cycles, and energy barriers using the communication height $\Phi(\cdot,\cdot)$ and the maximal barrier $\overline{\Phi}$, and constructs a multi-level Markovian description via trace processes accelerated by $e^{\Gamma^{\star,\mathfrak{h}}\beta}$, culminating in a terminal level where ground states form a unique irreducible class. The framework is instantiated for the Ising model under four dynamics (Glauber with positive/zero field and Kawasaki with few/many particles), revealing explicit hierarchical structures, level-specific depths $\Gamma^{\star,\mathfrak{h}}$, and limiting Markov chains that describe tunneling between metastable plateaux on progressively longer timescales. These results provide a rigorous, scalable description of metastable transitions in complex energy landscapes and offer a blueprint for analyzing similar systems beyond reversible Markov chains. The work advances the quantitative understanding of metastability in lattice systems and informs predictions of transition timescales relevant to statistical physics and related fields.

Abstract

In this article, we review the metastable hierarchy in low-temperature lattice models. In the first part, we state that for any abstract lattice system governed by a Hamiltonian potential and evolving according to a Metropolis-type dynamics, there exists a hierarchical decomposition of the collection of stable plateaux in the system into multiple $\mathfrak{m}$ levels, such that at each level there exist tunneling metastable transitions between the stable plateaux, which can be characterized by convergence to a simple Markov chain as the inverse temperature $β$ tends to infinity. In the second part, we collect several examples that realize this hierarchical structure of metastability. In order to fix the ideas, we select the Ising model as our lattice system and discuss its metastable behavior under four different types of dynamics, namely the Glauber dynamics with positive/zero external fields and the Kawasaki dynamics with few/many particles. This review article is submitted to the proceedings of the event PSPDE XII, held at the University of Trieste from September 9-13, 2024.

Metastable Hierarchy in Abstract Low-Temperature Lattice Models

TL;DR

The paper analyzes metastability in abstract low-temperature lattice models under Metropolis-type dynamics by introducing a hierarchical decomposition of stable plateaux into levels , with metastable transitions at each level converging to a Markov chain as . It develops a general framework based on stable plateaux, cycles, and energy barriers using the communication height and the maximal barrier , and constructs a multi-level Markovian description via trace processes accelerated by , culminating in a terminal level where ground states form a unique irreducible class. The framework is instantiated for the Ising model under four dynamics (Glauber with positive/zero field and Kawasaki with few/many particles), revealing explicit hierarchical structures, level-specific depths , and limiting Markov chains that describe tunneling between metastable plateaux on progressively longer timescales. These results provide a rigorous, scalable description of metastable transitions in complex energy landscapes and offer a blueprint for analyzing similar systems beyond reversible Markov chains. The work advances the quantitative understanding of metastability in lattice systems and informs predictions of transition timescales relevant to statistical physics and related fields.

Abstract

In this article, we review the metastable hierarchy in low-temperature lattice models. In the first part, we state that for any abstract lattice system governed by a Hamiltonian potential and evolving according to a Metropolis-type dynamics, there exists a hierarchical decomposition of the collection of stable plateaux in the system into multiple levels, such that at each level there exist tunneling metastable transitions between the stable plateaux, which can be characterized by convergence to a simple Markov chain as the inverse temperature tends to infinity. In the second part, we collect several examples that realize this hierarchical structure of metastability. In order to fix the ideas, we select the Ising model as our lattice system and discuss its metastable behavior under four different types of dynamics, namely the Glauber dynamics with positive/zero external fields and the Kawasaki dynamics with few/many particles. This review article is submitted to the proceedings of the event PSPDE XII, held at the University of Trieste from September 9-13, 2024.

Paper Structure

This paper contains 9 sections, 17 theorems, 59 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Metastable Hierarchy in Low-Temperature Lattice Models
  3. Hierarchical structure of stable plateaux
  4. Metastable hierarchy of tunneling transitions between stable plateaux
  5. Applications: Glauber/Kawasaki Dynamics on the Ising Model
  6. Case 1: Glauber dynamics, positive external field
  7. Case 2: Glauber dynamics, zero external field
  8. Case 3: Kawasaki dynamics, few particles
  9. Case 4: Kawasaki dynamics, $\mathscr{N}>L^2/4$

Key Result

lemma 1

For every cycle $\mathcal{C}$, its bottom $\mathcal{F}(\mathcal{C})$ is a union of stable plateaux.

Figures (5)

  • Figure 1: Example of a hierarchical decomposition of stable plateaux in $\mathscr{P}^{1}$ with $\mathfrak{m}=3$. At each level $h\in[1,3]$, bold-faced elements are recurrent and the rest are transient with respect to $\{\mathfrak{X}^{\star,h}(t)\}_{t\ge0}$. At level $1$, we have $\mathscr{P}_{1}^{\star,1}=\{\mathcal{P}_{1}^{1},\mathcal{P}_{2}^{1}\}$, $\mathscr{P}_{2}^{\star,1}=\{\mathcal{P}_{3}^{1}\}$, $\mathscr{P}_{3}^{\star,1}=\{\mathcal{P}_{5}^{1},\mathcal{P}_{6}^{1}\}$, $\mathscr{P}_{4}^{\star,1}=\{\mathcal{P}_{7}^{1}\}$ and $\mathscr{P}_{{\rm tr}}^{\star,1}=\{\mathcal{P}_{4}^{1}\}$. At level $2$, we have $\mathscr{P}_{1}^{\star,2}=\{\mathcal{P}_{1}^{2},\mathcal{P}_{2}^{2}\}$, $\mathscr{P}_{2}^{\star,2}=\{\mathcal{P}_{4}^{2}\}$ and $\mathscr{P}_{{\rm tr}}^{\star,2}=\{\mathcal{P}_{3}^{2}\}$. Finally, at level $\mathfrak{m}=3$, we have $\mathscr{P}^{\star,3}=\mathscr{P}_{1}^{\star,3}=\{\mathcal{P}_{1}^{3},\mathcal{P}_{2}^{3}\}$ which is exactly composed of the ground states.
  • Figure 2: Examples of configurations in $\mathscr{P}^1$ if $h>0$. We illustrate each configuration in the dual lattice in the sense that the gray (resp. white) faces indicate the $+1$ (resp. $-1$) spins.
  • Figure 3: Configurations $\xi_{2,11}$ and $\xi_{7,5}$ which belong to $\mathscr{P}^1$ (left), and examples of atypical saddle configurations (right) if $h=0$.
  • Figure 4: Configurations belonging to $\Omega^{0}$, $\Omega^{1}$, $\Omega^{2}$, $\Omega^{3}$, and $\Omega^{4}$, respectively.
  • Figure 5: Configuration $\sigma^{k}$ and examples of other stable plateaux.

Theorems & Definitions (29)

  • definition 1
  • remark 1
  • definition 2: Stable plateau
  • definition 3: Cycle
  • lemma 1
  • proof
  • definition 4: Construction of Markovian jumps between cycles
  • lemma 2
  • proof
  • definition 5
  • ...and 19 more