Run-and-Tumble particle in inhomogeneous media in one dimension

Abstract

We investigate the run and tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise $\sigma (t)$ drives the particle which changes between $\pm 1$ values with some rates. Denoting the rate of flip from $1$ to $-1$ as $R_1$ and the converse rate as $R_2$, we consider the position and direction dependent rates of the form $R_1(x)={\left(\frac{\mid x\mid }{l}\right)}^{\alpha }\left[{\gamma }_1\theta (x)+{\gamma }_2\theta (-x)\right]$ and $R_2(x)={\left(\frac{\mid x\mid }{l}\right)}^{\alpha }\left[{\gamma }_2\theta (x)+{\gamma }_1\theta (-x)\right]$ with $\alpha \ge 0$. For ${\gamma }_1>{\gamma }_2$, we find that the particle exhibits a steady-state probability distriution even in an infinite line whose exact form depends on $\alpha$. For $\alpha =0$ and $1$, we solve the master equations exactly for arbitrary ${\gamma }_1$ and ${\gamma }_2$ at large $t$. From our explicit expression for time-dependent probability distribution $P(x,t)$ we find that it exponentially relaxes to the steady-state distribution for ${\gamma }_1>{\gamma }_2$. On the other hand, for ${\gamma }_1<{\gamma }_2$, the large $t$ behaviour of $P(x,t)$ is drastically different than ${\gamma }_1={\gamma }_2$ case where the distribution decays as $t^{-\frac{1}{2}}$. Contrary to the latter, detailed balance is not obeyed by the particle even at large $t$ in the former case. For general $\alpha$, we argue that the approach to the steady state in ${\gamma }_1>{\gamma }_2$ case is exponential which we numerically demonstrate....

44 pages, 32 figures

References in corpus (17)

• Interaction Ruling Animal Collective Behaviour Depends on Topological rather than Metric Distance: Evidence from a Field Study
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Active matter
• Sedimentation, trapping, and rectification of dilute bacteria
• Pattern formation in self-propelled particles with density-dependent motility
• Spontaneous velocity alignment in Motility-induced Phase Separation
• Multidimensional Stationary Probability Distribution for Interacting Active Particles
• Generalized energy equipartition in harmonic oscillators driven by active baths
• Motility-induced phase separation and coarsening in active matter
• Run-and-Tumble Dynamics of Self-Propelled Particles in Confinement
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• First-passage time of run-and-tumble particles
• 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
• Exact Solution of Two Interacting Run-and-Tumble Random Walkers with Finite Tumble Duration
• Run-and-tumble motion with step-like responses to a stochastic input

Cited by in corpus (19)

• Run-and-tumble motion: field theory and entropy production
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Local time for run and tumble particle
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Attractor-driven matter
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• First hitting times between a run-and-tumble particle and a stochastically-gated target
• Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle
• Run-and-tumble particle in one-dimensional potentials: mean first-passage time and applications
• Generating constrained run-and-tumble trajectories
• Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes
• Long time behavior of run-and-tumble particles in two dimensions
• Target search by active particles
• Tracer dynamics in the active random average process
• Coarse-grained Stochastic Model of Myosin-Driven Vesicles into Dendritic Spines
• Proxitaxis: an adaptive search strategy based on proximity and stochastic resetting
• Universal framework for the long-time position distribution of free active particles
• Run-and-tumble particle with saturating rates