# Archive ouverte HAL – Velocity and diffusion constant of an active particle in a one-dimensional force field

### Pierre Le Doussal 1 Satya N. Majumdar 2 Satya Majumdar 2 Gregory Schehr 2

#### Pierre Le Doussal, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Velocity and diffusion constant of an active particle in a one-dimensional force field. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2020, 130 (4), pp.40002. ⟨10.1209/0295-5075/130/40002⟩. ⟨hal-02881224⟩

We consider a run an tumble particle with two velocity states $\pm v_0$, in an inhomogeneous force field $f(x)$ in one dimension. We obtain exact formulae for its velocity $V_L$ and diffusion constant $D_L$ for arbitrary periodic $f(x)$ of period $L$. They involve the "active potential" which allows to define a global bias. Upon varying parameters, such as an external force $F$, the dynamics undergoes transitions from non-ergodic trapped states, to various moving states, some with non analyticities in the $V_L$ versus $F$ curve. A random landscape in the presence of a bias leads, for large $L$, to anomalous diffusion $x \sim t^\mu$, $\mu<1$, or to a phase with a finite velocity that we calculate.

• 1. Champs Aléatoires et Systèmes hors d'Équilibre
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques