This applet simulates a dimer model on the triangular lattice. In this model, dimers move on a on a triangular lattice, and are allowed to translate along any of the six possible lattice directions, but with a hard-core constraint. The hard-core constraint means that dimers can only move into vacancies, also called monomers. This is a kinetically-constrained model; such models can be used to study how steric constraints can lead to slow, potentially glassy, behavior. A paper on this model can be found here.
We have done similar work studying dimers on a square lattice.
Click on the "Step Forward" button to run the system forward. With a "1" in the field to the left of the button, one dimer will be moved at random (if possible). To make more than one move, just change the number in the field to the left of the button.
Dimers that have never moved are colored green. Dimers moved immediately after clicking the "Step Forward" button are colored red, while all other dimers that have moved at some point in the course of the simulation are colored blue.
As discussed in the paper, pairs of monomers are sometimes localized, and sometimes delocalized, and when delocalized, undergo anomalously slow diffusion.
To create a new starting configuration, specify the size of the system, and the number of vacancies in it, and click one of the two initialize buttons. The first button creates a random configuration where all the vacancies are connected, while the second one creates a cofiguration where the vacancies are randomly scattered throughout the system. (Logically, the number of vacancies should have the same parity as the size, but if it doesn't, the program will adjust it for you.)
By changing the "Show:" tab to "Trees," the nodes of the monomer trees of each of the holes (monomers) are shown in pink. If you create a state with exactly two connected monomers, by changing the "Show:" tab to "Swap clusters," the swap cluster of the connected monomers is shown in pink. The terms "monomer tree" and "swap cluster" are explained in the paper.
As stated above, the dynamical rules allow dimers to translate in any of the six possible lattice directions, so long as the move satisfies the hard-core constraint. These translational moves fall into two types. First, "glide" moves, in which the dimer translates parallel to its axis, into a single monomer:
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And second, "swap" moves, in which the dimer translates at an angle to its axis, into two monomers:
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As we discuss in the paper, understanding the distinction between glide and swap moves is essential to explaining why pairs of dimers diffuse anomalously slow. By changing the "Dynamical rules:" tab, you can change the simulation rules so that only glide moves are allowed, or only swap moves are allowed. While pairs of monomers are sometimes delocalized (free to move throughout the system) under the normal dynamical rules, if only one of the two types of moves are allowed, then pairs of monomers are always localized.
The first field in the bottom right gives the number of sites which have not had a dimer move on, off, or over them, since the simulation began. The second one tells how many dimer moves have occurred.
