论文标题
在重置的情况下，平面布朗尼运动的平均周长和面积
Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
论文作者
论文摘要
我们准确地计算出$ 2 $ -D的持续时间$ t $和扩散常数$ d $的凸壳的平均周长和平均面积，并以恒定的速率$ r $ $ $ $ $。我们表明，对于任何$ t $，平均周长均由$ \ langle l（t）\ rangle =2π\ sqrt {\ frac {\ frac {d} {r}}} \，f_1（rt）$，平均面积和均值由$ \ langle a（t）\ langle a（t） f_2（rt）$，其中缩放功能$ f_1（z）$和$ f_2（z）$是明确计算的。对于大型$ t \ gg 1/r $，随着时间的推移，平均周长的生长非常缓慢，因为$ \ langle l（t）\ rangle \ propto \ ln（rt）$随着时间的推移。同样，平均区域也随着$ \ langle a（t）\ rangle \ propto \ ln^2（rt）$的$ \ langle a（t）$ $ t \ gg 1/r $ $ $而缓慢生长。我们的确切结果表明，在存在重置的情况下，凸壳在后期接近圆形。数值模拟与我们的分析预测完全一致。
We compute exactly the mean perimeter and the mean area of the convex hull of a $2$-d 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 $\langle L(t)\rangle= 2 π\sqrt{\frac{D}{r}}\, f_1(rt)$ and the mean area is given by $\langle A(t) \rangle= 2π\frac{D}{r}\, f_2(rt)$ where the scaling functions $f_1(z)$ and $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, the mean perimeter grows extremely slowly as $\langle L(t)\rangle \propto \ln (rt)$ with time. Likewise, the mean area also grows slowly as $\langle A(t)\rangle \propto \ln^2(rt)$ for $t\gg 1/r$. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times. Numerical simulations are in perfect agreement with our analytical predictions.
