Diffusion and random search in heterogeneous media: theory and applications

点击翻译按钮获取中文摘要

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.