Mean area of the convex hull of a run and tumble particle in two dimensions

Abstract

We investigate the statistics of the convex hull for a single run-and-tumble particle in two dimensions. Run-and-tumble particle, also known as persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles $n$ or (ii) the total observation time $t$ is kept fixed. Benchmarking the results on perimeter, we study the statistical properties of the area of the convex hull for RTP. Exploiting the connections to extreme value statistics, we obtain exact analytical expressions for the mean area for both ensembles. For fixed-$t$ ensemble, we show that the mean possesses a scaling form in $\gamma t$ (with $\gamma$ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits crossover from $\sim t^3$ scaling at small times $\left(t\ll {\gamma }^{-1}\right)$ to $\sim t$ scaling at large times $\left(t\gg {\gamma }^{-1}\right)$. On the other hand, for fixed-$n$ ensemble, the mean expectedly grows linearly with $n$ for $n\gg 1$. All our analytical findings are supported with numerical simulations.

26 pages, 10 figures

References in corpus (17)

• Self-motile colloidal particles: from directed propulsion to random walk
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Active matter
• Spontaneous velocity alignment in Motility-induced Phase Separation
• Motility-induced phase separation and coarsening in active matter
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• First-passage time of run-and-tumble particles
• Spatial extent of an outbreak in animal epidemics
• Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap
• Non-crossing run-and-tumble particles on a line
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Convex Hulls of Random Walks: Large-Deviation Properties
• Run-and-Tumble particle in inhomogeneous media in one dimension
• Local time for run and tumble particle
• Stationary nonequilibrium bound state of a pair of run and tumble particles
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study