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 (RTP), also known as the 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 the perimeter, we study the statistical properties of the area of the convex hull for a 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 area 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 a 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 the numerical simulations.

Time to reach the maximum for a stationary stochastic process

“…For a given potential V (x), one needs first to compute the constrained propagator G M (x, t|x 0 ) and the survival probability Q M (x, t). Then, the distribution of t m can be obtained using the formula (40). As shown in the next sections, this can be done in the cases V (x) = α|x| (corresponding to p = 1) and V (x) = αx 2 (corresponding to p = 2).…”

“…The distribution of t m has also been studied in the case of run-andtumble particles (RTP) [25,33,34] and for N vicious walkers [35]. Moreover, the distribution of the time of the maximum plays a central role for computing the mean area of the convex hull of a two-dimensional process [27,[36][37][38][39][40] and for determining the hitting probability for anomalous diffusion processes [31]. However, to the best of our knowledge, before our recent Letter [41], the time of the maximum was never systematically investigated in the case of stationary stochastic processes.…”

From a microscopic solution to a continuum description of active particles with a recoil interaction

“…The properties of a single persistent particle are by now well understood (see e.g. [19][20][21][22][23][24][25][26][27][28]). Here we focus on run-and-tumble motion, which is inspired by that of the E. coli bacterium.…”

Extremal statistics of a one dimensional run and tumble particle with an absorbing wall

“…Moreover, many interesting features have been observed even for the noninteracting RTPs. Examples include -non-trivial large deviations and condensation transitions for the free space position distributions [59][60][61][62][63][64][65], non-Boltzmann steady state distributions [66][67][68][69], universal first-passage properties and extremal statistics [35,36,[70][71][72][73][74], functional statistics [75], convex hull problems [76,77], tagged particle properties [78][79][80][81] and so on.…”