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Random walks and search processes

Vejleder:  Michael Andersen Lomholt


Random walks occur in many natural processes, and in many of these processes it is important that the random walk will result in some target being quickly found. Examples are molecules diffusing around in biological cells until they reach the specific place where they perform their function, or bacteria or animals searching for scarce food. In many of these cases the random walk switches between different modes. For instance a protein searching for a specific site on a long DNA molecule. In this case there is the option of diffusion in the three dimensional volume of the cell, or loosely binding to the DNA molecule and diffuse along it. It turns out that combining the two options leads to a faster search process than one in which the molecule just uses one of the modes. Similar results arise in many other complex random search processes.

As the complexity of the random walk increases it becomes increasingly difficult to obtain analytic mathematical solutions for the search time to find some target. However, results can often be obtained through computer simulations of the random walk process. A bachelor project along the lines described here could involve, on top the simulations of the random walk itself, the generation of the landscape in which the walk takes place. For instance, the generation of thermally equilibrated self-avoiding random walks to model a DNA polymer. Also, there will often be limits of the random walk model in which analytic solutions can be worked out. The simulations can then be checked against these solutions. The specific problem to be examined can be adjusted according to the interests of the student.

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Sidst opdateret: 23.08.2021