反应扩散系统斑图形成的格子Boltzmann研究
Lattice Boltzmann study of pattern formation in reaction-diffusion systems
摘要：
反应扩散系统中斑图形成在表面微图案化中是非常重要的[Grzybowski等人，软物质. 1，114（2005）]、细胞微生物的自组织[Schulz等人，Annu. Rev. Microbiol. 55，105（2001）]和发育生物学[Barkai等人FEBS Journal 276，1196（2009）]。在这项工作中，我们应用格子玻尔兹曼方法（LBM）来研究反应扩散系统中斑图形成。作为方法的第一步，我们考虑的情况下，一个单一的物种进行转化反应和扩散。在这种情况下，我们进行了三阶Chapman-Enskog多尺度展开和研究的晶格玻尔兹曼截断误差的扩散系数和反应速率的依赖性。这些发现与数值模拟结果吻合良好。此外，以Gray-Scott模型为例，证明了LBM在研究非线性反应扩散系统斑图形成方面的成熟性.为此，我们进行了格雷-斯科特模型的线性稳定性分析，并确定了斑图形成的相关参数范围。格子玻尔兹曼模拟不仅允许测试包括图灵和霍普夫不稳定性的线性稳定性相图的有效性，而且允许超越线性稳定性制度，其中大的扰动引起有趣的动力学行为，如所谓的自我复制点。我们还表明，图案的长度尺度可以通过调整所有相关的扩散系数在系统中具有相同的因素，而让所有的反应常数不变。
英文摘要：
Pattern formation in reaction-diffusion systems is of great importance in surface micro-patterning [Grzybowski et al. Soft Matter. 1, 114 (2005)], self-organization of cellular micro-organisms [Schulz et al. Annu. Rev. Microbiol. 55, 105 (2001)] and in developmental biology [Barkai et al. FEBS Journal 276, 1196 (2009)]. In this work, we apply the Lattice Boltzmann method (LBM) to study pattern formation in reaction-diffusion systems. As a first methodological step, we consider the case of a single species undergoing transformation reaction and diffusion. In this case, we perform a third-order Chapman-Enskog multiscale expansion and study the dependence of the Lattice Boltzmann truncation error on the diffusion coefficient and the reaction rate. These findings are in good agreement with numerical simulations. Furthermore, taking the Gray-Scott model as a prominent example, we provide evidence for the maturity of the LBM in studying pattern formation in non-linear reaction-diffusion systems. For this purpose, we perform linear stability analysis of the Gray-Scott model and determine the relevant parameter range for pattern formation. Lattice Boltzmann simulations allow not only to test the validity of the linear stability phase diagram including Turing and Hopf instabilities, but also permit going beyond the linear stability regime, where large perturbations give rise to interesting dynamical behavior such as the so called self replicating spots. We also show that the length scale of the patterns may be tuned by rescaling all relevant diffusion coefficients in the system with the same factor while letting all the reaction constants unchanged.
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