A Topological Glass

TL;DR

This paper introduces a model of triangulations with colored nodes exhibiting glassy dynamics, slow approach to equilibrium, and a hierarchical energy landscape similar to spin glasses, revealing complex emergent behavior.

Contribution

It presents a novel triangulation-based model with glassy properties, analyzing its slow dynamics, unordered steady state, and hierarchical energy landscape.

Findings

System approaches steady state exponentially slowly

Stationary state is unordered

Energy landscape has hierarchical structure similar to spin glasses

Abstract

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with $n$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.

Figures

Equations

$$\epsilon {\left(\frac{{\sigma }_a+{\sigma }_b}{2r}\right)}^{12},$$

$$E(\mathcal{T})=\sum _{i=\mathrm{odd}}(d_i-7{)}^2+\sum _{i=\mathrm{even}}(d_i-5{)}^2,\text{immediate}(1):energy=e01$$

Full text

1 Introduction

Our model of a glass can is inspired by an abstraction of a model and its representation found in Proglass2007. Before we introduce our version, and to make contact with that paper, we describe briefly the model of Proglass2007. One starts with a binary mixture of $n$ disks in the plane, with the small disks having radius $\sigma_{\rm s}=1$ and the large, $\sigma_{\rm\ell}=1.4$. The three pairwise additive interactions are given by purely repulsive soft-core potentials of the form

[TABLE]

with $a,b\in\{{\rm s},{\rm\ell}\}$, $\varepsilon>0$, and $r$ the distance between the centers of the disks. One assumes the interaction vanishes for $r>2.25(\sigma_{a}+\sigma_{b})$. Taking periodic boundary conditions, and a relatively tight volume, the authors of Proglass2007 found that this system shows the characteristics of a glass when the temperature is sufficiently low.

The part of the analysis which is of interest for the present study has to do with a geometric representation of configurations of this system. One draws in the plane a point for the position of the centers of the disks, and proceeds then to use the Voronoi tessellation. This means that lines are drawn between nearest neighbors in the sample, and their normal bisectors are then used to draw polygons around each particle.

These polygons turn out to be mostly pentagons around the small particles and heptagons around the large ones, with a few exceptions as a function of temperature, and by the constraints of Euler’s theorem on polygonal domains on surfaces. See Fig. 1.

In this paper, we consider a purely topological variant of the above model and show it has the properties of a glass. The model is basically obtained by considering instead of the Voronoi tessellation the dual graph, which is a triangulation (of the torus). This triangulation is obtained from Fig. 1 by connecting the centers of the particles normally across the edges of the Voronoi tessellation. (The dual graph is a triangulation since each edge of the polygons meets exactly 2 other edges at its end.)

To simplify things further, we consider instead a triangulation ${\cal T}$ of the sphere. Given $n$, the number of “particles,” we let ${\cal T}$ denote a triangulation of the sphere with $n$ nodes and we let ${\mathbb{T}}_{n,0}$ denote the set of all such triangulations. By this, one means the set of all combinatorially distinct rooted simplicial 3-polytopes. In particular, a triangulation should not have any “double edges”. We further refine the definition, by distinguishing 2 types of nodes in the triangulation: We first number the nodes from 1 to $n$ and then define 2 types of nodes. Those with even index are the “small particles” and those with odd index the “large” ones. This means that the triangulation has the same number of large and small nodes (up to a difference of one). We will also call the two types of nodes two colors.

Definition 1.1**.**

We shall call the odd nodes blue and the even ones red and will refer to the triangulations as colored triangulations.

Once the colors are assigned, the numbers are again forgotten. The set of all colored triangulations with $n$ nodes will be called ${\mathbb{T}}_{n}$. This is our phase space and the dynamics is mapping points in this phase space to other points.

As in the original model, we introduce an energy $E$ for each triangulation ${\cal T}$:

[TABLE]