Diffusion and random search in heterogeneous media: theory and applications

异构介质中的扩散和随机搜索：理论与应用

• 批准号: 452796171
• 负责人: Professor Dr. Ralf Metzler
• 金额: --
• 依托单位: Institut für Physik und Astronomie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/452796171?language=en
• 关键词: Diffusion; random; search; heterogeneous; media

Various studies of active and passive tracer diffusion, e.g., in biologicalcells or in disordered and porous media showed that the underlying structureof the environment has a strong effect on the particle movement, leading toconstrained particle motion or variations of the local diffusion coefficientand potential energy function. Therefore, the development of models accountingfor the structure of the environment is needed for the analysis of diffusiveparticle motion in such media. This gives rise to heterogeneous diffusionmodels which describe the heterogeneity of the environment by space and/ortime-dependent diffusion coefficients, reflecting different forms of disorder(e.g., quenched, energetic). The heterogeneous media we have in mind includematerials with impurities, amorphous media, fractal and random heterogeneousstructures, crowded environments, weighted graphs, etc. Thus the modelsproposed in this project will be of interest for the modelling of diffusionin quite different complex systems.Moreover, to determine optimal search strategies is central for diverse fields,from biology to robotics, from physics to computer science. In particular,random search strategies have been widely observed, for animals foragingin an environment with scarce resources, reaction pathways of DNA-bindingproteins, intracellular transport, search operations in disaster zones,etc. The famed debate on the Levy flight of albatrosses and optimality of Levysearch have triggered a vast amount of theoretical and experimental researchactivities in the field of animal motility patterns and optimal strategies. Amore recent addition to optimal search is the resetting mechanism, theoccasional reset of the particle back to its starting point (the beehive,for instance), increasing the performance of computer algorithms or searchprocesses of foraging animals, and reducing the mean first-passage time ofthe diffusing particle to a target. Experimental realisations of colloidalparticle diffusion and resetting via holographic optical tweezers, requiresfurther theoretical investigation of diffusion and random search processeswith stochastic resetting in heterogeneous media, particularly in higherdimensional settings. These are key objectives in our project.These problems of heterogeneous diffusion are closely related toinhomogeneous advection-diffusion and turbulent diffusion problems. Theyare also heavily used in financial mathematics, especially the theory ofgeometric Brownian motion underlying the Black-Scholes model, or the CIRand Heston models. Therefore, the results obtained for heterogeneous andturbulent diffusion with resetting will be of interest in economic sciencesas well. The presence of stochastic resetting in such economic systems mayoccur due to natural disasters, epidemics or due to stock market crashes.

对主动和被动示踪剂扩散的各种研究，例如，在生物细胞或无序和多孔介质中，表明环境的潜在结构对颗粒运动有很强的影响，导致颗粒运动受限或局部扩散系数和势能函数的变化。因此，需要开发考虑环境结构的模型来分析这种介质中的扩散粒子运动。这就产生了异质性扩散模型，该模型通过空间和/或时间相关的扩散系数来描述环境的异质性，反映不同形式的无序(例如：熄灭的，精力充沛的)。我们所想到的非均质介质包括含杂质的材料、非晶介质、分形和随机非均质结构、拥挤环境、加权图等。因此，在这个项目中提出的模型将对完全不同的复杂系统中的扩散建模感兴趣。此外，确定最佳搜索策略是各个领域的核心，从生物学到机器人，从物理学到计算机科学。其中，随机搜索策略在资源稀缺环境下的动物觅食、dna结合蛋白的反应途径、细胞内运输、灾区的搜索行动等方面得到了广泛的观察。关于信天翁的利维飞行和利维搜索的最优性的著名争论引发了动物运动模式和最优策略领域的大量理论和实验研究活动。最优搜索最近增加的是重置机制，偶尔将粒子重置回其起始点（例如蜂巢），提高计算机算法或觅食动物搜索过程的性能，并减少扩散粒子到达目标的平均首次通过时间。通过全息光学镊子实现胶体颗粒扩散和重置的实验实现，需要对异质介质中具有随机重置的扩散和随机搜索过程进行进一步的理论研究，特别是在高维环境中。这些是我们项目的关键目标。这些非均匀扩散问题与非均匀平流扩散和湍流扩散问题密切相关。它们在金融数学中也被大量使用，尤其是作为布莱克-斯科尔斯模型基础的几何布朗运动理论，或者CIRand Heston模型。因此，对于具有重置的非均质和湍流扩散所获得的结果也将在经济科学中引起兴趣。在这种经济系统中，由于自然灾害、流行病或股票市场崩溃，可能会出现随机重置。