Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions – Archive ouverte HAL

Naftali R. SmithPierre Le Doussal ¹ Satya N. Majumdar ² Grégory Schehr ³ Naftali SmithPierre Le Doussal ¹ Satya Majumdar ²

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr, Naftali Smith, et al.. Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions. Physical Review E , 2022, 106 (5), pp.054133. ⟨10.1103/PhysRevE.106.054133⟩. ⟨hal-03903940⟩

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness $\mu$, and possibly diffuses. We find the exact time-dependent distribution $P\left(x,y,t\right)$ of the particle’s position, and in particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a « free » particle, $\mu=0$. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically-independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

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