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- W4309912046 abstract "We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the long time limit, the maximum converges to the radius of the ball $M_x(t) to R$ for $tto infty$. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations $Delta(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all dimensions. We find that the distribution of $Delta(t)$ exhibits a rich variety of behaviors depending on the dimension $d$. These results are obtained by establishing a connection between this problem and the narrow escape time problem. We apply our results in $d=2$ to study the convex hull of the trajectory of the particle in a disk of radius $R$ with reflecting boundaries. We find that the mean perimeter $langle L(t)rangle$ of the convex hull exhibits a slow convergence towards the perimeter of the circle $2pi R$ with a stretched exponential decay $2pi R-langle L(t)rangle propto sqrt{R}(Dt)^{1/4} ,e^{-2sqrt{2Dt}/R}$. Finally, we generalise our results to other confining geometries, such as the ellipse with reflecting boundaries. Our results are corroborated by thorough numerical simulations." @default.
- W4309912046 created "2022-11-30" @default.
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- W4309912046 date "2021-12-10" @default.
- W4309912046 modified "2023-10-16" @default.
- W4309912046 title "Statistics of the maximum and the convex hull of a Brownian motion in confined geometries" @default.
- W4309912046 doi "https://doi.org/10.48550/arxiv.2112.05663" @default.
- W4309912046 hasPublicationYear "2021" @default.
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