Laboratoire de Physique Théorique

de la Matière Condensée

Nicolas Levernier (IUSTI Marseille)

First-passage time of non-markovian random walks

In this talk, I will recall some results I got during my PhD and show new results obtained since then. In a second part, I will present part of the results I got during my postdoc in Geneva. The question of first-passage time, typically the time needed for two particles to encounter, is of importance to quantify the rate of reaction of any diffusion-limited process. If the problem is essentially solved for markovian (that is memoryless) random walks such as brownian motion, only sparse results do exist for non-markovian random walks. I will present a formalism we have developed to quantify the mean first passage time to a target for a confined non-markovian random walker, whose position statistics is gaussian and with stationary increments. Then, I will present some features of confined ageing processes (that is : if the stationary increments hypotheses is relaxed), and decipher the links with the associated unconfined processes. In particular, the importance of the so-called persistance exponent will be emphasised. I will finally show how our gaussian formalism can be extended to ageing unconfined processes in order to get the value of the persistance exponent. In a second part, I will present a physical description of the actin cytoskeleton, which is a layer of fibers in interaction with molecular motors. This layer is polymerizing along the plasma membrane. Usual model typically see it as a thin layer, that is neglecting the dynamics in the direction orthogonal to the membrane. I will show that a more precise description that account for this direction leads to a new instability mechanism that cannot be observed within the thin-layer approach. Finally, I will briefly mention experimental works in agreement with this predicted instability.