期刊
PHYSICAL REVIEW E
卷 103, 期 2, 页码 -出版社
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevE.103.022135
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资金
- Labex MME-DII
- ANR [11-LABEX-0023]
This study computed the mean perimeter and mean area of the convex hull of Brownian motion with resetting, showing that the convex hull approaches a circular shape at late times. The analytical predictions were confirmed through numerical simulations.
We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by < L(t)> = 2 pi root D/r f(1) (rt) and the mean area is given by < A(t)> = 2 pi D/r f(2)(rt) where the scaling functions f(1)(z) and f(2)(z) are computed explicitly. For large t >> 1/r, the mean perimeter grows extremely slowly as < L(t)> proportional to ln(rt) with time. Likewise, the mean area also grows slowly as < A(t)> proportional to 1n(2) (rt) for t >> 1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.
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