Lattice gas simulations of dynamical geometry in one dimension

Abstract

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.

Lattice gas simulations of dynamical geometry in two dimensions

“…We have presented the first lattice-gas model with dynamical geometry in two dimensions. Our model is an extension of the FHP hydrodynamic two-dimensional lattice gas model, and the one-dimensional dynamical geometry lattice gas [1][2][3][4]. We have defined and implemented rules for dynamical geometry by both Pachner moves.…”

“…In these simulations the fluid represented by the particles is quiescent -there is no forcing applied and because the initial velocities of the particles are assigned randomly the average hydrodynamic velocity field will be zero. In one dimension, where the only geometrical degree of freedom is the size of the lattice, both numerical simulation and calculations for particular sets of initial conditions result in an average growth of the lattice size of t 1 2 [1][2][3]. We perform the comparable calculations and mean field theory treatment of the two dimensional model.…”

“…We allow the geometry so defined to become dynamical by applying the Pachner moves contingent on the particle content. The restriction of time-reversibility is used to restrict the rule space, as in the one dimensional version of this model [1][2][3]. We present a mean-field prediction and simulation results for the growth of the lattice as a function of time, and give preliminary results on the distribution of curvature on the triangulations generated by these simulations.…”

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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 [1][2][3]. We expand upon a variation of the twodimensional flat space FHP model created by Frisch, Hasslacher and Pomeau, and independently by Wolfram [4,5], and modified by Boghosian, Love, and Meyer in [6].

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On Quantum Extensions of Hydrodynamic Lattice Gas Automata

“…Propagation always occurs along straight lines in the lattice composed of 2N vectors for some N. The operator for the D1Q2 model may therefore be directly applied to this subset of bits. Even for models in which the underlying geometry is dynamical [60,61], or not a lattice [62], propagation occurs on a subset of the variables composed of an even number of the bits and the D1Q2 model operator can be applied.…”