  Various lattice gas automata have been proposed in the past decades to
simulate physics and address a host of problems on collective dynamics arising
in diverse fields. In this work, we employ the lattice gas model defined on the
sphere to investigate the curvature driven dynamic structures and analyze the
statistical behaviors in equilibrium. Under the simple propagation and
collision rules, we show that the uniform collective movement of the particles
on the sphere is geometrically frustrated, leading to several non-equilibrium
dynamic structures not found in the planar lattice, such as the emergent bubble
and vortex structures. With the accumulation of the collision effect, the
system ultimately reaches equilibrium in the sense that the distribution of the
coarse-grained speed approaches the two-dimensional Maxwell-Boltzmann
distribution despite the population fluctuations in the coarse-grained cells.
The emergent regularity in the statistical behavior of the system is
rationalized by mapping our system to a generalized random walk model. This
work demonstrates the capability of the spherical lattice gas automaton in
revealing the lattice-guided dynamic structures and simulating the equilibrium
physics. It suggests the promising possibility of using lattice gas automata
defined on various curved surfaces to explore geometrically driven
non-equilibrium physics.
