Generalised `Arcsine' laws for run-and-tumble particle in one dimension

Abstract

The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time $t_m$ to reach maximum position, the time $t_r$ spent on the positive side and the time $t_{\ell }$ of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times $t_m,t_r$ and $t_{\ell }$ in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities for arbitrary time and find that all three distributions have delta function part and a non-delta function part. Interestingly, we find that the distributions of $t_m$ and $t_r$ are identical (reminiscent of the Brownian particle case) when the initial velocities of the particle are chosen with equal probability. On the other hand, for $t_{\ell }$, only the non-delta function part is same with the other two. In addition, we find explicit expressions of the joint distributions of the maximum displacement and the time at which this maxima occurs. We verify all our analytical results through numerical simulations.

32 pages, 15 figures, Corrected typos

References in corpus (31)

• Random-Matrix Theory of Quantum Transport
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria
• Sedimentation, trapping, and rectification of dilute bacteria
• Active Brownian Particles and Run-and-Tumble Particles: a Comparative Study
• Run and tumble particle under resetting: a renewal approach
• Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension
• Dynamics of a self-propelled particle in a harmonic trap
• Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Random Convex Hulls and Extreme Value Statistics
• Jamming and Attraction of Interacting Run-and-Tumble Random Walkers
• The Local and the Occupation Time of a Particle Diffusing in a Random Medium
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• On the time to reach maximum for a variety of constrained Brownian motions
• Generalized arcsine laws for fractional Brownian motion
• Active processes in one dimension
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Non-crossing run-and-tumble particles on a line
• Time to reach the maximum for a random acceleration process
• Distribution of the time at which N vicious walkers reach their maximal height
• Residence Time Distribution for a Class of Gaussian Markov Processes
• Statistical Properties of Functionals of the Paths of a Particle Diffusing in a One-Dimensional Random Potential
• Exact Solution of Two Interacting Run-and-Tumble Random Walkers with Finite Tumble Duration
• Helix or Coil? Fate of a Melting Heteropolymer
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
• Arcsine Laws in Stochastic Thermodynamics
• Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
• Symmetry breaking between statistically equivalent, independent channels in a few-channel chaotic scattering
• Residence time statistics for $N$ blinking quantum dots and other stochastic processes

Cited by in corpus (50)

• Run-and-tumble particles in two-dimensions : Marginal position distributions
• Long time position distribution of an active Brownian particle in two dimensions
• Active Brownian Motion in two-dimensions under Stochastic Resetting
• Current fluctuations in non-interacting run-and-tumble particles in one-dimension
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Extremal statistics for stochastic resetting systems
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Occupation time of a run-and-tumble particle with resetting
• Time between the maximum and the minimum of a stochastic process
• Distribution of the Time Between Maximum and Minimum of Random Walks
• Position distribution in a generalised run and tumble process
• First-passage Brownian functionals with stochastic resetting
• Survival probability of a run-and-tumble particle in the presence of a drift
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions
• Time to reach the maximum for a stationary stochastic process
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries
• Heat fluctuations in a harmonic chain of active particles
• Local time for run and tumble particle
• Universal survival probability for a correlated random walk and applications to records
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• First-passage functionals for Ornstein Uhlenbeck process with stochastic resetting
• Attractor-driven matter
• Encounter-based model of a run-and-tumble particle
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Smoluchowski flux and Lamb-Lion Problems for Random Walks and Lévy Flights with a Constant Drift
• Nonequilibrium steady state of trapped active particles
• Active Random Walks in One and Two Dimensions
• Generating constrained run-and-tumble trajectories
• First passage statistics of active random walks on one and two dimensional lattices
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Last-passage time for linear diffusions and application to the emptying time of a box
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Occupation time of a system of Brownian particles on the line with steplike initial condition
• Confined run and tumble particles with non-Markovian tumbling statistics
• Experimental verification of Arcsine laws in mesoscopic non-equilibrium and active systems
• Inertial Dynamics of Run-and-Tumble Particle
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Active particle in a harmonic trap driven by a resetting noise: an approach via Kesten variables
• Nonequilibrium fluctuation-response relations for state observables
• Active Brownian particle under stochastic position and orientation resetting in a harmonic trap
• Exact joint distributions of three global characteristic times for Brownian motion
• Tracer dynamics in the active random average process
• Coarse-grained Stochastic Model of Myosin-Driven Vesicles into Dendritic Spines
• Ergodic properties of functionals of Gaussian processes
• Full Record Statistics of 1d Random Walks
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• Records and occupation time statistics for area-preserving maps
• Current fluctuations in finite-sized one-dimensional non-interacting passive and active systems