Paper detail

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

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 $μ.$ The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate $γ$. We compute the stationary position distribution exactly for arbitrary values of $μ$ and $γ$ which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as $β=γ/μ$ is changed. For $β<1,$ the distribution has a double-concave shape and shows algebraic divergences with an exponent $(β-1)$ both at the origin and at the boundaries. For $β>1,$ 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 $β=1,$ the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries.