Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

TL;DR

This paper derives the exact stationary position distribution of a three-state run-and-tumble particle in a harmonic trap, revealing a shape transition dependent on the ratio of tumbling rate to trap stiffness.

Contribution

It provides an exact analytical solution for the stationary distribution of a three-state run-and-tumble particle in a harmonic potential, including a detailed analysis of the shape transition.

Findings

Distribution has finite support on the real line.

Shape transition occurs at the critical ratio β=1.

Distribution exhibits algebraic divergences or a single peak depending on β.

Abstract

We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness $\mu.$ The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate $\gamma$. We compute the stationary position distribution exactly for arbitrary values of $\mu$ and $\gamma$ which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as $\beta=\gamma/\mu$ is changed. For $\beta<1,$ the distribution has a double-concave shape and shows algebraic divergences with an exponent $(\beta-1)$ both at the origin and at the boundaries. For $\beta>1,$ the position distribution becomes convex, vanishing at the boundaries and with a single, finite, peak at the origin. We also show that for the special case $\beta=1,$ the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries.