Jamming Transition in Kinetically Constrained Models with Reflection Symmetry

Abstract

A class of kinetically constrained models with reflection symmetry is proposed as an extension of the Fredrickson–Andersen model. It is proved that the proposed model on the square lattice exhibits a freezing transition at a non-trivial density. It is conjectured by numerical experiments that the known mechanism of the singular behaviors near the freezing transition in a previously studied model (spiral model) is not responsible for that in the proposed model.

Appendix: Sketch of the Proof for the Absence of Locally Frozen Particles

In Sect. 3.1, we assumed the fact that the clusters of frozen particles in the bulk are always constrained partly by the boundary sites. In order to explicitly state such an argument, as preliminary, we start with the following definitions and will give a sketch of the proof for the argument.

Definition 0 (frozen particles) ${\fancyscript{F}_\mathrm{all}}(\sigma )$ denotes a set of all the frozen sites (particles) for a given configuration $\sigma$, where for $^\forall i\in {\fancyscript{F}_\mathrm{all}}$, site i is constrained only by sites in ${\fancyscript{F}_\mathrm{all}}$ and boundary sites. ${\fancyscript{B}_\mathrm{d}}$ denotes a set of boundary sites.

Definition 1 (frozen cluster) $\fancyscript{F}\subset {\fancyscript{F}_\mathrm{all}}$ denotes a set “frozen cluster” of frozen particles, where for $^\forall i\in \fancyscript{F}$, $^\exists j\in {\fancyscript{F}}\cup {\fancyscript{B}}_\mathrm{d}$ such that $|i-j|\le 3$ and (maximum property) for $^\forall k\notin \fancyscript{F},\,k\cup \fancyscript{F}$ is not a frozen cluster. See Fig. 9 for helping to imagine the frozen particles and frozen clusters.

Fig. 9

Frozen particles and clusters. The configurations outside of the figures are assumed to be consistent.

Definition 2 (frozen links) ${\fancyscript{L}}({\fancyscript{F}})$ denotes a set of “frozen links” for ${\fancyscript{F}}$, consisting of straight line segments (i,j) between i and j for $^\forall i,j\in \fancyscript{F}$.

Definition 3 (outer sites) $\overline{{\fancyscript{F}}}$ denotes a set of outer sites for ${\fancyscript{F}}$, where for $^\forall i\in \overline{{\fancyscript{F}}}$, i has a path to a boundary site without crossing $^\forall k\in \fancyscript{L}({\fancyscript{F}})$.

Definition 4 (outer links) $\partial \fancyscript{L}({\fancyscript{F}})$ denotes a set of outer links, where arbitrary points on $k\in \partial \fancyscript{L}({\fancyscript{F}})$ has a path to a boundary site without crossing $^\forall k\in \fancyscript{L}({\fancyscript{F}})$.

Definition 5 (edge sites) $\partial {\fancyscript{F}}\equiv {\fancyscript{F}}\cap \overline{{\fancyscript{F}}}$ is a set of edge sites. By the definition of edge sites and outer links, for $^\forall (i,j)\in \partial \fancyscript{L}(\fancyscript{F})$, $i,j\in \partial {\fancyscript{F}}$. See Fig. 10 for helping to image outer sites, outer links, and edges sites.

Fig. 10

Outer sites, outer links, and edges sites. The configurations outside of the figures are assumed to be consistent

On the basis of those definitions, one can obtain the following property:

Geometric property 1: The figure generated by all the outer links in $\partial \fancyscript{L}({\fancyscript{F}})$ are locally convex toward $\fancyscript{F}$.

Assume that the figure generated by $(i,j),(j,k)\in \partial \fancyscript{L}(\fancyscript{F})$ is not convex toward $\fancyscript{F}$. Then, one can construct another outer link connecting $i$ and $k$, and $(i,j),(j,k)\notin \partial \fancyscript{L}(\fancyscript{F})$ by the definition of the outer links. This contradiction immediately leads to Geometric property 1.

By the local convexity in Geometric property 1, one may immediately conclude the following property:

Geometric property 2: The figure generated by all the outer links in $\partial \fancyscript{L}({\fancyscript{F}})$ is a convex polygon or a convex polyline toward $\fancyscript{F}$.

Finally, we explicitly state about the absence of locally frozen particles in the following.

Statement: There exist frozen particles constrained by the boundary sites for arbitrary frozen clusters: For $^\forall {\fancyscript{F}}\ne \emptyset ,\,^\exists i\in {\fancyscript{F}}$ such that $\min _{j\in {\fancyscript{B}}_\mathrm{d}}|i-j|\le 3$.

Sketch of the proof: Assume that there exists a frozen cluster ${\fancyscript{F}}$ such that $^\forall i\in {\fancyscript{F}}$, $\min _{j\in {\fancyscript{B}}_\mathrm{d}}|i-j|> 3$. Then let us consider what kinds of configuration could appear near edge sites in $\partial {\fancyscript{F}}$. Remembering the maximum property of frozen clusters, one can easily find that each site in $\partial {\fancyscript{F}}$ has at least two particles, but less than four particles at the nearest neighbors. The case with three particles is illustrated in Fig. 11, and it can been seen that one immediately has to consider the case with two particles.

Fig. 11

The case with $3$ particles at sites in $\mathrm{B}_0$. In this case, the slope of outer links $(0,1)\in \partial \fancyscript{L}({\fancyscript{F}})$ clearly does not change, comparing to the previous outer link. As far as the edge sites continue to have three particles at the nearest neighbors, this situation does not change

Thus, we focus on the case with two particles at the nearest neighbor of an edge site. According to Geometric property 1, one can illustrate two possible and nontrivial configurations as illustrated in Fig. 12. Concretely, we first pick up an edge site $0$ in $\partial {\fancyscript{F}}$, and consider the slope of outer links $(0,k)\in \partial \fancyscript{L}({\fancyscript{F}})$ to curve downward in order to make a polygon, without loss of generality because one has to consider these cases in the end at the latest. However, it turns out that the slope of outer links $(0,k)\in \partial \fancyscript{L}({\fancyscript{F}})$ in both configurations cannot be changed to make any polygons (see the caption in Fig. 12). Note that the cases where there is a particle at site $a$ or $b$ to make site $0$ constrained are also possible, but the slope of the outer link $(0,k)\in \partial \fancyscript{L}({\fancyscript{F}})$ does not change downward at all. As explained in the caption of Fig. 12, those cases are enough to consider impossibility for the outer links to make them being a polygon on the basis of the assumption we have made. Therefore, one may conclude that the figure generated by all the outer links in $\partial \fancyscript{L}({\fancyscript{F}})$ is not a polygon on the assumption we have made.

Fig. 12

The situations where an edge site $0$ has two particles in $\mathrm{B}_0$ and a particle at site $1$ makes site $0$ constrained ($a$ and $b$ are outer sites in $\overline{{\fancyscript{F}}}$). It is necessary to consider these rather particular situations because even if we consider the cases where site $0$ is constrained by the other sites such as sites $a$ and $b$, one clearly has to consider these situations again in order to make a polygon. (left) Assume that site $1$ is an edge site in $\partial {\fancyscript{F}}$. In this case, the situation that there are particles at sites $2$ and $3$ is only the possibility to be consistent with the definitions made in the text. Finally, the edge site $1$ has the same situation as that of site $0$. Therefore, The slope of outer link $(0,1)$ cannot become sharper than that of the next outer link $(1,k)\in \partial \fancyscript{L}({\fancyscript{F}})$. (right) Assume that site $1$ is not an edge site. In this case, there must exist a particle at site $2$. Assume that site $2$ is an edge site in $\partial {\fancyscript{F}}$. In this case, the situation that there are particles at sites $1$ and $4$, is only the possibility to be consistent with the definitions made in the text. Finally, the edge site $2$ has the same situation as that of site $0$, and the slope of outer link $(0,2)$ is not sharper than that of outer link $(0,1)$. Finally, if one consider the last possibility that site $2$ is not an edge site, the slope of the outer link $(1,k)\in \partial \fancyscript{L}({\fancyscript{F}})$ cannot be sharper than that of outer link $(0,2)$ by the definition of the outer links. Concretely, this is because if the slope of outer link $(1,k)$ would be sharper than that of outer link $(0,2)$, site $2$ would be an edge site, which leads to the contradiction

Therefore, another possibility is that the figure generated by all the outer links in $\partial \fancyscript{L}({\fancyscript{F}})$ is a convex polyline according to Geometric property 2. However, in this case, the tip of the line has to be constrained by a boundary site because the tip of the line should have at least one more particle at the nearest neighbors by the definition. Thus, one may conclude there are contradictions, leading to Statement.