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 FrischHasslacher-Pomeau ͑FHP͒ model created by Frisch et al. ͓Phys. Rev. Lett. 56, 1505 ͑1986͔͒ and independently by Wolfram, and modified by Boghosian et al. ͓Philos. Trans. R. Soc. London, Ser. A 360, 333 ͑2002͔͒. 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.

We present numerical results obtained using a lattice gas model with dynamical geometry. The (irreversible) macroscopic behaviour of the geometry (size) of the lattice is discussed in terms of a simple scaling theory and obtained numerically. The emergence of irreversible behaviour from the reversible microscopic lattice gas rules is discussed in terms of the constraint that the macroscopic evolution be reproducible. The average size of the lattice exhibits power-law growth with exponent at late times. The deviation of the macroscopic behaviour from reproducibility for particular initial conditions ('rogue states') is investigated as a function of system size. The number of such 'rogue states' is observed to decrease with increasing system size. Two mean-field analyses of the macroscopic behaviour are also presented.