Lattice gas simulations of dynamical geometry in two dimensions

We present a hydrodynamic lattice gas model for two-dimensional flows on curved surfaces with dynamical geometry. This model is an extension to two dimensions of the dynamical geometry lattice gas model previously studied in one dimension. We expand upon a variation of the two-dimensional flat space Frisch-Hasslacher-Pomeau (FHP) model created by Frisch et al. [Phys. Rev. Lett. 56, 1505 (1986)] and independently by Wolfram, and modified by Boghosian et al. [Philos. Trans. R. Soc. London, Ser. A 360, 333 (2002)]. We define a hydrodynamic lattice gas model on an arbitrary triangulation whose flat space limit is the FHP model. Rules that change the geometry are constructed using the Pachner moves, which alter the triangulation but not the topology. We present results on the growth of the number of triangles as a function of time. Simulations show that the number of triangles grows with time as $t^{1/3}$, in agreement with a mean-field prediction. We also present preliminary results on the distribution of curvature for a typical triangulation in these simulations.

Figure 1

(Color online) A triangle whose neighbors are neighbors belongs to a tetrahedron. This is an intrinsic definition of a tetrahedron. On the left there are three triangles sharing one edge each with the black triangle. They are the black triangle’s neighbors. If the blue (top), red (bottom left), and yellow (bottom right) triangles in the left figure are also neighbors as designated by the arrows above, the three triangles fold into a tetrahedron (right).

Figure 2

An FHP lattice site has six possible velocities labeled 0–5, each of which represents the velocity of a particle. Each vector can hold at most one particle, so that each site has $2^6=64$ states. The traditional representation of a site in the FHP model is the star shown on the left. By moving the vectors to the edges of a triangle, as shown in the center picture, we convert the site from a single point to the face of a triangle. These two sites are equivalent. If we remove the arrow heads from the vectors, we produce the notationally convenient right-hand figure. We refer to the conversion between the star and the triangle as inflation.

Figure 3

(Color online) A kagome lattice on an icosahedron. The nodes of the triangulation are depicted as unconnected spheres. This image was generated with visual python [36].

Figure 4

(Color online) A kagome lattice on a cylinder. The nodes of the triangulation are depicted as unconnected spheres. Particles are allowed to move along the lines of the kagome lattice and are shown in black. This image was generated with visual python [36].

Figure 5

Pachner moves. A sequence of Pachner moves can connect any pair of triangulations of a manifold, but cannot change the topology. The two-to-two move (left) changes the orientation of two triangles as shown above, and the one-to-three move (right) replaces one triangle with three, creating a tetrahedron, or vice versa.

Figure 6

If new geometry is not created empty one state may have two preimages, a problem illustrated in this figure. Both figures on the left would produce the figure on the right if they were allowed to time evolve. The particles on the triangle in the upper left-hand corner are therefore required to propagate off before the new geometry is created. This allows rules with dynamical geometry in which every state has a unique preimage, and which are therefore invertible.

Figure 7

(a) A two-particle collision at a site. The particles switch with equal probability to one of the other two directions of the lattice. This rule applies to any two particles entering a site with opposite velocities. (b) A three-particle collision at a site. The particles switch to the unoccupied vectors of the site. This move breaks the separate conservation of momentum along each direction of the lattice.

Figure 8

(a) Creation: a triangle is replaced with a tetrahedron. This collision is triggered by a three-particle collision, where three particles enter a site on every other vector (all odd numbered or all even numbered vectors) such that the combined momentum of the three particles is zero. The particles then propagate away, and the tetrahedron is formed. (b) Deletion: three triangles forming a tetrahedron are replaced with a single triangle. This collision is triggered when three particles propagate off the tetrahedron, as shown on the left above. This also happens when the particles leave the tetrahedron on the empty vectors that point to the surrounding triangles.

Figure 9

The two-to-two move is triggered by two different states: (a) the four-particle state and (b) the two-particle state. The particles remain where they are during these moves.

Figure 10

An example evolution under the rules defined above. The evolution begins with a preimage of propagation, which leads to deletion of a tetrahedron. Two-body and three-body collisions occur on the neighboring triangles. No collisions occur at the subsequent step, but a two-to-two move is triggered at the following step, followed by two- and three-body collisions. Particles propagate on and off the triangles shown here during the propagation steps.

Figure 11

The two-to-two Pachner move. The pair of triangles is rotated in the lattice, so that vertex 1 goes to vertex 2, vertex 2 goes to vertex 3, vertex 3 goes to vertex 4, and vertex 4 goes to vertex 1. The relationship between triangles $\mathbf{a}$ and $\mathbf{b}$ stays the same; only the connectivity between each triangle and the surrounding triangles is redefined, along with the vertices of each triangle.

Figure 12

If the new edge $L$ is more than a fraction $x$ different from $L_0$, the two-to-two move is prevented.

Figure 13

(Color online) The one-to-three Pachner move (creation). The triangles are labeled $\mathbf{x}$, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, and the vertices are labeled by 0, 1, and 2. (${\mathbf{m}}^n$ refers to triangle $m$’s $n\text{th}$ vertex). When triangle $\mathbf{x}$ is triggered for addition, copies of triangle $\mathbf{x}$ are rotated out of the plane of the paper along each edge. For example, to produce triangle $\mathbf{b}$ triangle $\mathbf{x}$ is copied and rotated along the edge joined by the vertices ${\mathbf{x}}^1$ and ${\mathbf{x}}^0$, matching vertex ${\mathbf{x}}^2$ with the apex. Acting similarly for the other two sides, triangle $\mathbf{x}$ on the left transitions to the tetrahedron on the right. Deletion is the inverse of this process, and intersections of common vertices and triangle neighbors are used to identify the relevant vertices. The apex is identified as the intersection of the vertices of triangles $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, since the apex is the only vertex shared by all three triangles.

Figure 14

When a tetrahedron undergoes deletion, the connectivity must be redefined. Here, the tetrahedron is being replaced by triangle $\mathbf{a}$.

Figure 15

A tetrahedron attached by an edge to the main body of the manifold. If deletion occurs to any triplet of the triangles $w\text{−}x\text{−}y\text{−}z$, a flap will be created.

Figure 16

(Color online) All Pachner moves allowed. Number of triangles as a function of time averaged over 30 realizations for $100000$ time steps each with no restrictions on the curvature or embeddability of the triangulation, and all Pachner moves utilized. The symbols in (a) show the simulation data every 1000 time steps and are larger than one standard deviation of the mean. The solid line is a fit created in ORIGIN 7.0 with a Levenberg-Marquardt method for $L\left(t\right)=a{t}^b$ to the complete data set of $100000$ points. (b) shows a contour plot of ${\chi }^2$ using a sampling of 200 points evenly spaced along the range of $a$ and $b$. The minimum ${\chi }^2$ value lies at ${\chi }^2=0.0319$ at $a=196$ and $b=0.33618$ in agreement with the fit found by ORIGIN 7.0, and the outermost contour represents a deviation of 1.0 from this minimum. The fitted value of the exponent is $b=0.33618\pm 0.0065$, consistent with a power-law exponent of 1/3.

Figure 17

(Color online) Only addition of tetrahedra. Number of triangles as a function of time averaged over 30 realizations for $100000$ time steps each. The simulations included only the one-to-three Pachner move with no growth or embedding control. The symbols in (a) show the simulation data every 1000 time steps and are larger than one standard deviation of the mean. The solid line is a fit created in ORIGIN 7.0 with a Levenberg-Marquardt method for $L\left(t\right)=a{t}^b$ to the complete data set of $100000$ points. (b) shows a contour plot of ${\chi }^2$ using a sampling of 200 points evenly spaced along the range of $a$ and $b$. The minimum ${\chi }^2$ value lies at ${\chi }^2=1.106$ at $a=190.864$ and $b=0.343$ in agreement with the fit found by ORIGIN 7.0, and the outermost contour represents a deviation of 1.0 from this minimum. The fitted value of the exponent is $0.337\le b=0.344\le 0.350$, consistent with a power-law exponent of 1/3.

Figure 18

(Color online) No deletion. Number of triangles as a function of time averaged over 30 realizations for $100000$ time steps each. The simulations included both two-to-two and one-to-three Pachner moves with no growth or embedding control. The symbols in (a) show the simulation data every 1000 time steps and are larger than one standard deviation of the mean. The solid line is a fit of $L\left(t\right)=a{t}^b$ to the complete data set of $100000$ points. The fit was created in ORIGIN 7.0 using a Levenberg-Marquardt method. (b) shows a contour plot of ${\chi }^2$ using a sampling of 200 points evenly spaced along the range of $a$ and $b$. The minimum ${\chi }^2$ value lies at ${\chi }^2=0.878$ at $a=194.402$ and $b=0.343$ in agreement with the fit found by ORIGIN 7.0. The outermost contour represents a deviation of 1.0 from this minimum. The fitted value of the exponent is $0.336\le b=0.343\le 0.349$, consistent with a power-law exponent of 1/3.

Figure 19

(Color online) No two-to-two moves allowed. Number of triangles as a function of time averaged over 30 realizations for $100000$ time steps each. The simulations included both the one-to-three and three-to-one Pachner moves with no two-to-two moves and without restrictions on the curvature or embeddability of the triangulation. The symbols in (a) show the simulation data every 1000 time steps and are larger than one standard deviation of the mean. The solid line is a fit of $L\left(t\right)=a{t}^b$ to the complete data set of $100000$ points. The fit was created in ORIGIN 7.0 using a Levenberg-Marquardt method. (b) shows a contour plot of ${\chi }^2$ using a sampling of 200 points evenly spaced along the range of $a$ and $b$. The minimum ${\chi }^2$ value lies at ${\chi }^2=0.666$ at $a=197.060$ and $b=0.335$ in agreement with the fit found by ORIGIN 7.0. The outermost contour represents a deviation of 1.0 from this minimum. The fitted value of the exponent is $0.330\le b=0.335\le 0.342$, consistent with a power-law exponent of 1/3.

Figure 20

(Color online) Histogram of number of triangles meeting at each vertex. Results of four separate simulations for $100000$ time steps. The initial geometry was an icosahedron with each of its faces subdivided into 16 equilateral triangles. Data are shown for one simulation with all Pachner moves implemented (black circles), one simulation with only the three-to-one addition move implemented (red upward-pointing triangles), one simulation with the one-to-three addition move and the two-to-two move but no deletion (blue stars), and one simulation with addition and deletion moves but no two-to-two move (green sideway-pointing triangles).

Figure 21

(Color online) Visualizations of the triangulation for a typical realization. (a) Initial condition for all simulations. (b) Typical geometry after $100000$ time steps for a simulation with only the three-to-one Pachner move (addition) implemented, with no restriction on the curvature. The faces shown here are individual triangles—with only the three-to-one Pachner move implemented the immersion being always isometric. The appearance of many smaller triangles is due to the fact that the surface is extensively self-intersected.