Abstract
We are interested in applying lattice BGK models to the diffusion-driven reactive systemA+B→C, which was investigated by Gálfi and Rácz with an asymptotic analysis and by Chopard and Droz with a cellular automaton model. The lattice BGK model is free from noise and flexible for various applications. We derive the general reaction-diffusion equations for the lattice BGK models under the assumption of local diffusive equilibrium. Two fourth-order terms are derived and verified by numerical simulations. The motivation of this study is to compare the lattice BGK results with existing results before we apply the models to more complicated systems. The scalings concern two exponents α and β appearing in the production rate ofC componentR(x, t)∼t −β G(xt −α). We find the same values for α=1/6 and β=2/3 as Gálfi and Rácz found at the long time limit. AGaussian-like function forG is numerically obtained, which confirms a similar result of Gálfi and Rácz. On the one hand, when compared with the asymptotic analysis, lattice BGK models are easy to apply to cases where no analytic or asymptotic results exist; on the other hand, when compared with cellular automaton models, lattice BGK models are faster, simpler, and more accurate. The discrepancy of the results between the cellular automaton model and the lattice BGK models for the exponents comes from the role of the intrinsic fluctuation. Once the time and space correlation of stochastic stirring is given, we can incorporate a random fluctuating term in lattice BGK models. The Schlögl model is also tested, showing the ability of lattice BGK models for generating Turing patterns, which may stimulate further interesting investigations.
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Qian, Y.H., Orszag, S.A. Scalings in diffusion-driven reactionA+B→C: Numerical simulations by lattice BGK models. J Stat Phys 81, 237–253 (1995). https://doi.org/10.1007/BF02179978
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DOI: https://doi.org/10.1007/BF02179978