Finite dimension unravels the structural features at the glass transition

Abstract

Physics Nobel Prize winner P.W. Anderson famously wrote in 1995: “The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of the glass and the glass transition”. Despite much effort in the intervening years, the problem is still unsolved. We contribute a novel mathematical approach to this problem. The main new ingredient is finite dimension, a recently introduced “fractal” dimension defined only for finite sets. Our methods sharply distinguish the glass transition temperature and give hints as to the structural changes that occur in the transition.

Graphic Abstract

Appendices

Appendix A

The first term in Eq. 1 is the Coulomb interaction with the effective charge numbers $q_{i}$; the second term is a dispersive interaction and it presents those involving only oxygen ions; and the last term is a Born–Meyer type potential that takes into account the repulsive short-range interactions.

The system was first equilibrated at 3500 K in a 2 ns run using the microcanonical ensemble (NVE). Then, to reach the working temperature, it was cooled down from 3500 K until it reached both the target temperature and the normal pressure, in 2 successive cooling steps, one of cooling followed by a stabilization. Each cooling step consisted of a 2 ns run using a thermostat to decrease the temperature linearly in the NPT ensemble

Once the temperature was reached, a period of a 2 ns equilibration in the NPT was applied to verify no temperature drift and minimize the typical pressure oscillations in solids.

Appendix B

This appendix contains excerpts from [17], detailing the definition of the Connected Sparse Graph CSG(M), simplified to CS. See Fig. 2 for a 2-dimensional illustration of the definition. We let ${\mathbb {R}}_{>0}$ denote the set of positive real numbers.

Recall that two vertices belong to the same connected component (or simply, component) if there is a path joining them. The definition is inductive: starting with M, new edges are added at each step, obtaining an increasing sequence of graphs:

$$\begin{aligned} S_0 \subset S_1 \subset \cdots \end{aligned}$$

STEP 0. $S_0$ is a graph with no edges, $V(S_0)=M$, and $E(S_0)=\emptyset$. The set of components of $S_0$ is $K_0 =\{\{x\}| x \in M\}$, which we identify with M by setting $d(\{x\},\{y\}){:}{=}d(x,y)$. Thus, $S_0$ has $|K_0|=|M|$ components.

STEP 1. The function $\nu _0:M\rightarrow {\mathbb {R}}_{>0}$, defined by

$$\begin{aligned} \nu _0(x){:}{=} \min _y \Big \{d(x,y)| y\in M, y\ne x \Big \}, \end{aligned}$$

gives the distance to the nearest neighbours of a point. Note that we can consider $\nu _0$ as a function defined on $K_0$, by setting $\nu _0(\{x\}){:}{=} \nu _0(x)$. Let

$$\begin{aligned} E(S_1){:}{=} \bigcup _{x\in M} \Big \{ \{x,y\}| d(x,y)= \nu _0(x) \Big \}. \end{aligned}$$

That is, for every vertex x, we add an edge connecting x to each one of its nearest neighbours. Clearly, $S_0\subset S_1$.

The graph $S_1$ is not connected in general. Let $K_1$ denote the set of its components. If $|K_1|=1$, we stop here and define $CSG(M){:}{=} S_1$. If $|K_1|>1$, we proceed to Step 2. Note that $|K_1|\le |M|/2$, since every $x \in M$ has a nearest neighbour.

STEP $i+1$. In this inductive step, we assume the graphs $S_0\subset S_1 \subset \cdots \subset S_{i}$ are constructed, that $K_{i}$ is the set of connected components of $S_{i}$ and that $1<|K_{i}| \le |M|/2^{i}$.

Let $k,k'\in K_{i}$ be distinct components. The function $\nu _{i}: K_{i} \rightarrow {\mathbb {R}}_{>0}$ , defined by

$$\begin{aligned} \nu _{i}(k)&{:}{=} \min _{x,y} \Big \{d(x,y)| x\in k, y\in k', k\ne k'\Big \}\\&\,\,= \min _{x,y} \Big \{d(x,y)| x\in k, y\notin k\Big \} \end{aligned}$$

gives the distance from k to its nearest components. For $x\in M$, we let $[x]_i$ denote the component of x in $S_i$. Let

$$\begin{aligned} B(S_{i+1}){:}{=} \bigcup _{k\in K_{i}} \Big \{ \{x,y\}| d(x,y)= \nu _{i}(k), x\in k, y\notin k \Big \}. \end{aligned}$$

and define $E(S_{i+1}){:}{=}E(S_{i}) \cup B(S_{i+1})$. Clearly, $S_{i}\subset S_{i+1}$.

Let $K_{i+1}$ denote the set of components of $S_{i+1}$. If $|K_{i+1}|=1$, we stop here and define $CSG(M){:}{=} S_{i+1}$. If $|K_{i+1}|>1$, we proceed to Step ${i+2}$. Note that $|K_{i+1}|\le |K_{i}|/2$, since every $k \in K_{i}$ has a nearest connected component. Hence, $|K_{i+1}|\le |M|/2^{i+1}$.

Since |M| is finite, after a finite number of steps we obtain a connected graph, and this we define to be CSG(M). Also, note that no choices are made at any step, so that CSG(M) is uniquely defined.