Lattice gas simulations of dynamical geometry in two dimensions

TL;DR

This paper introduces a 2D hydrodynamic lattice gas model on curved surfaces with dynamic geometry, extending previous 1D models, and demonstrates how the triangulation evolves over time with growth proportional to the cube root of simulation steps.

Contribution

It develops a novel 2D lattice gas model on arbitrary triangulations with dynamic geometry, expanding on prior 1D models and incorporating Pachner moves to alter triangulation.

Findings

Number of triangles grows as the cube root of time steps.

Simulation results align with mean field predictions.

Preliminary curvature distribution analysis provided.

Abstract

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 FHP model created by Frisch, Hasslacher and Pomeau, and independently by Wolfram, and modified by Boghosian, Love, and Meyer. 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 lattice grows with time as the cube root of the number of time steps, in agreement a mean field prediction. We also present preliminary results on the distribution of curvature over a typical triangulation for these simulations.