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

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.

arXiv comment13 pages, 2 figures

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