Stationary nonequilibrium bound state of a pair of run and tumble particles – Archive ouverte HAL

Pierre Le Doussal ¹ Satya N. Majumdar ² Grégory Schehr ³ Pierre Le Doussal ¹ Satya Majumdar ²

Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr, Pierre Le Doussal, Satya Majumdar. Stationary nonequilibrium bound state of a pair of run and tumble particles. Physical Review E , American Physical Society (APS), 2021, 104 (4), ⟨10.1103/PhysRevE.104.044103⟩. ⟨hal-03389938⟩

We study two interacting identical run and tumble particles (RTP’s) in one dimension. Each particle is driven by a telegraphic noise, and in some cases, also subjected to a thermal white noise with a corresponding diffusion constant $D$. We are interested in the stationary bound state formed by the two RTP’s in the presence of a mutual attractive interaction. The distribution of the relative coordinate $y$ indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability $P(y)$ of $y$ for two examples of interaction potential $V(y)$. The first one corresponds to $V(y) \sim |y|$. In this case, for $D=0$ we find that $P(y)$ contains a delta function part at $y=0$, signaling a strong clustering effect, together with a smooth exponential component. For $D>0$, the delta function part broadens, leading instead to weak clustering. The second example is the harmonic attraction $V(y) \sim y^2$ in which case, for $D=0$, $P(y)$ is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single RTP model in one dimension, as well as with a four-state single RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.

• 1. LPENS (UMR_8023) – Laboratoire de physique de l’ENS – ENS Paris
• 2. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
• 3. LPTHE – Laboratoire de Physique Théorique et Hautes Energies