Extremal statistics of a one-dimensional run and tumble particle with an absorbing wall
Prashant Singh, Saikat Santra, Anupam Kundu
Abstract
Abstract We study the extreme value statistics of a run and tumble particle (RTP) in one dimension till its first passage to the origin starting from the position <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . This model has recently drawn a lot of interest due to its biological application in modelling the motion of certain species of bacteria. Herein, we analytically study the exact time-dependent propagators for a single RTP in a finite interval with absorbing conditions at its two ends. By exploiting a path decomposition technique, we use these propagators appropriately to compute the joint distribution of the maximum displacement M till first-passage and the time t m at which this maximum is achieved exactly. The corresponding marginal distributions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi>M</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> are studied separately and verified numerically. In particular, we find that the marginal distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> has interesting asymptotic forms for large and small t m . While for small t m , the distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> depends sensitively on the initial velocity direction σ i and is completely different from the Brownian motion, the large t m decay of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> is same as that of the Brownian motion although the amplitude crucially depends on the initial conditions x 0 and σ i . We verify all our analytical results to high precision by numerical simulations.