Stochastic closed Frank model in two dimensions: Chiral symmetry breaking driven by diffusive control over bounded surfaces

In this work, the closed Frank model was simulated using a stochastic method based on the Ehrenfest urn on a surface (mathematically expressed as a matrix), considering that the reactions and diffusion processes of the species occur within a predetermined, adjusted neighborhood. It investigated the role of neighborhood size in achieving (or not) the global homochiral state (GHS). All simulations started without enantiomeric excess, with the particles randomly distributed over the surface, and other distributions were also considered. It is shown that, spatially, the surface gradually becomes occupied by domains (of the two enantiomers) that are randomly distributed, and as the size of the neighborhood increases, the number of simulations that reach the GHS also increases, with a corresponding decrease in the simulation time required to reach it. Besides, domains continue to form in this case, but over time, fluctuations (in space and time) in concentrations lead to GHS. When the GHS is not reached, the steady state is characterized by the presence of domains of both species on the surface, as previously mentioned. Still, the reactions continue to occur, and the generation and consumption rates of each species are equal. These occur mainly on the interface of the domains, especially in small neighborhood sizes.