Mean area of the convex hull of a run and tumble particle in two dimensions
Prashant Singh, Anupam Kundu, Satya N. Majumdar, Hendrik Schawe
Abstract
Abstract We investigate the statistics of the convex hull for a single run-and-tumble particle (RTP) in two dimensions. 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 γt (with γ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits a crossover from ∼ t 3 scaling at small times <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≪</mml:mo> <mml:msup> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> to ∼ t scaling at large times <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≫</mml:mo> <mml:msup> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> . On the other hand, for fixed- n ensemble, the mean expectedly grows linearly with n for n ≫ 1. All our analytical findings are supported with the numerical simulations.