Jamming pair of general run-and-tumble particles: Exact results, symmetries and steady-state universality classes

TL;DR

This work presents a microscopic, continuous-space and continuous-time analysis of a pair of interacting Run-and-Tumble Particles on a 1D ring, using a piecewise deterministic Markov process (PDMP) to derive the invariant measure and identify two nonequilibrium universality classes: detailed-jamming and global-jamming. By formulating a complete PDMP with bulk dynamics, periodic and jamming boundaries, and a jam kernel $K$, the authors establish conservation constraints that reveal a Dirac mass at jammed configurations and a boundary-driven relaxation that can be either detailed-symmetric or fully broken. The detailed-jamming class features a Dirac jam mass plus a bulk uniform density, consistent with near-equilibrium bulk behavior, while the global-jamming class exhibits a structured bulk relaxation described by a matrix $B$ and a catenary-like solution with exponential and polynomial contributions, yielding diffusive or ballistic regimes depending on spectral properties. The paper further develops explicit subclasses for two-state and three-state RTP systems, showing how symmetry dictates the relaxation lengthscales and how ballisticity, jamming, and boundary conditions interplay to shape the steady state. Overall, the PDMP framework provides a rigorous, scalable route to understanding microscopic origins of nonequilibrium steady states and potential universality in motility-induced phenomena across larger RTP systems.

Abstract

While run-and-tumble particles are a foundational model for self-propelled particles as bacteria or Janus particles, the analytical derivation of their steady state from the microscopic details is still an open problem. By directly modeling the system at the continuous-space and -time level thanks to piecewise deterministic Markov processes (PDMP), we derive the conservation conditions which sets the invariant distribution and, more importantly, explicitly construct the two universality classes for the steady state, the detailed-jamming and the global-jamming classes. They respectively identify with the preservation or not in a detailed manner of a symmetry at the level of the dynamical internal states between probability flows entering and exiting jamming configurations. We call such symmetry active global balance, as it is the true nonequilibrium counterpart of the equilibrium global balance. Thanks to a spectral analysis of the tumble kernel, we give explicit expressions for the invariant measure in the general case. We show that the non-equilibrium features exhibited by the steady state include positive mass for the jammed configurations and, for the global-jamming class, exponential decay and growth terms, potentially modulated by polynomial terms. Interestingly, we find that the invariant measure follows, away from jamming configurations, a catenary-like constraint, which results from the interplay between probability conservation and the dynamical skewness introduced by the jamming interactions, seen now as a boundary constraint. This work shows the powerful analytical approach PDMP provide for the study of the stationary behaviors of RTP systems and motivates their future applications to larger systems, with the goal to derive microscopic conditions for motility-induced phase transitions.

Figures (8)

• Figure 1: Trajectory of two RTPs ( left) undergoing instantaneous tumble ($\omega = 0.2$, $L=10$), and of their corresponding periodic interdistance ( right), which evolution undergoes several regimes in the bulk (increasing, stalling and decreasing (a)) and boundary conditions (periodic at $L/2$ (b) and jamming at $0$ (c)).
• Figure 2: Transitions between the internal states at the single particle ($\omega(\cdot)q(\cdot,\cdot)$, left) and interdistance level ($\tilde{\omega}(\cdot)Q(\cdot,\cdot)$, right) in the case of instantaneous ( top, and finite tumble ( left), in a reduced representation for $s$ thanks to the single-particle tumble isotropy (instantaneous: $s$ can be omitted, finite: $s$ codes for the number of moving particles).
• Figure 3: Density of the invariant measure in the bulk for $\alpha=1$, $\beta=4$ and $L=1$
• Figure 4: Respective contributions of the Dirac, exponential and uniform terms towards the invariant measure
• Figure 5: Transition rates (i.e. $\omega(\cdot)q(\cdot,\cdot)$) between the internal states of a single symmetrical 3-state RTP ( left) and corresponding transition rates (i.e. $\tilde{\omega}(\cdot)Q(\cdot,\cdot)$) at the interdistance level ( bottom) in the representation $s_t = \mathbb{1}_{\{v_1,v_3\}}(v_{t,1}) + \mathbb{1}_{\{v_1,v_3\}}(v_{t,2})$ ( right). The notations $\omega(v^1_3)=\omega(v_1)=\omega(v_3)$ and $q^1_3= q(v_1,v_3)=1(v_3,v_1)$ are used.