# Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap – Archive ouverte HAL

### Urna Basu 1 Satya N. Majumdar 2 Alberto Rosso 2 Sanjib Sabhapandit 3 Gregory Schehr 2

#### Urna Basu, Satya N. Majumdar, Alberto Rosso, Sanjib Sabhapandit, Gregory Schehr. Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap. Journal of Physics A: Mathematical and General (1975 - 2006), IOP Publishing, 2020, ⟨10.10083⟩. ⟨hal-02512239⟩

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.

• 1. Theoretical Condensed Matter Physics Division
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 3. Raman Research Institute