# Archive ouverte HAL – Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties

### Abhishek Dhar 1 Anupam Kundu 1 Satya N. Majumdar 2 Sanjib Sabhapandit 3 Gregory Schehr 2

#### Abhishek Dhar, Anupam Kundu, Satya N. Majumdar, Sanjib Sabhapandit, Gregory Schehr. Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties. Physical Review E , American Physical Society (APS), 2019, 99 (3), ⟨10.1103/PhysRevE.99.032132⟩. ⟨hal-02102138⟩

We study the dynamics of a one-dimensional run and tumble particle subjected to confining potentials of the type $V(x) = \alpha \, |x|^p$, with $p>0$. The noise that drives the particle dynamics is telegraphic and alternates between $\pm 1$ values. We show that the stationary probability density $P(x)$ has a rich behavior in the $(p, \alpha)$-plane. For $p>1$, the distribution has a finite support in $[x_-,x_+]$ and there is a critical line $\alpha_c(p)$ that separates an active-like phase for $\alpha > \alpha_c(p)$ where $P(x)$ diverges at $x_\pm$, from a passive-like phase for $\alpha < \alpha_c(p)$ where $P(x)$ vanishes at $x_\pm$. For $p<1$, the stationary density $P(x)$ collapses to a delta function at the origin, $P(x) = \delta(x)$. In the marginal case $p=1$, we show that, for $\alpha < \alpha_c$, the stationary density $P(x)$ is a symmetric exponential, while for $\alpha > \alpha_c$, it again is a delta function $P(x) = \delta(x)$. For the special cases $p=2$ and $p=1$, we obtain exactly the full time-dependent distribution $P(x,t)$, that allows us to study how the system relaxes to its stationary state. In addition, in these two cases, we also study analytically the full distribution of the first-passage time to the origin. Numerical simulations are in complete agreement with our analytical predictions.

• 1. International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 3. Raman Research Insitute