Activity-driven clustering and many-body steady state of jamming run-and-tumble particles

Abstract

We exactly resolve the three-particle steady state of run-and-tumble particles with jamming interactions, providing the first microscopic description beyond two bodies. The invariant measure, derived via a piecewise-deterministic Markov process description and symmetry principles, reveals persistent, separated, and diffusive regimes ruled by the activity parameter. A geometric cascade of scales in the activity parameter organizes the structural weights, showing the separated phase dominates at finite activity, while non-uniformity plays only a minor role. Extending these results to larger systems, we show that the $N$-body steady state inherits the same organization: the number of clusters becomes sharply defined by the activity value, with crossover boundaries whose slopes diverge with $N$. We also show how the activity plays a role similar to a fugacity conjugate to cluster number, yielding a grand-canonical-like structure emerging directly from the microscopic dynamics. This framework lays the groundwork for a systematic microscopic theory of active many-body steady states.

Figures (5)

• Figure 1: (a,c) State space for the 2-RTP (Bulk and Jammed states) and 3-RTP (Bulk, Separated and Jammed states) systems respectively. (b,d) Weights in the steady state (jammed $w_{\mathcal{J}}$ (green, long dashes), bulk $w_{\mathcal{B}}$ (blue, solid) and separated $w_{\mathcal{S}}$ (orange, dash-dotted)) as a function of $\omega L$ for the 2-RTP and 3-RTP systems respectively. Grey cross markers denote simulation results and exhibit excellent agreement (see Appendix \ref{['sec:simu']} for details). For the 3-RTP system, $w_{\mathcal{S}}$ decomposes into its uniform $w_\mathrm{eq}w_{\mathcal{S}}$ (yellow, dashed) and exponential part $w_\mathrm{rel}w_{\mathcal{S}}$ (maroon, dotted); $w_{\mathcal{S}}$ achieves a maximum of $\approx 0.582$ at $\omega L \approx 2.16$. For $\omega L \le 3w_{\mathrm{eq}}/(5-2w_{\mathrm{eq}})\approx 0.759$, jammed states dominate, then for $0.759\approx \le \omega L \le 6/w_{\mathrm{eq}}\approx 6.16$ separated ones do, after which the free states dominate. In the 2-RTP case, $w_{\mathcal{J}}=w_{\mathcal{B}}$ at $\omega L = 2$.
• Figure 2: (a) Detailed-jamming is realized for instantaneous tumbles ($\mathrel{ } \leftrightarrow \mathrel{ }$). (b) Only global-jamming is achieved for finite tumbles ($\mathrel{ }\leftrightarrow \mathrel{ } \leftrightarrow \mathrel{ }$): exiting with $\mathrel{ } \mathrel{ }$ is impossible as $\mu_{\mathcal{J}}( \mathrel{ } \mathrel{ }) =\mu_{\mathcal{J}}( \mathrel{ } \mathrel{ })=0$.
• Figure 3: Probability weights to observe $n_C$ clusters as a function of $\omega L$ for different system sizes $N$ (simulation details in the Appendix \ref{['sec:simu']}). Each curve gathers configurations with $n_c/N$ within intervals $[0.2\, m,0.2\, (m+1)]$. A progressive sharpening of the crossovers between $n_c$ intervals is observed as $N$ increases. Inset: Same data for $N=20$ (green), $40$ (red), $80$ (blue), and $160$ (black), with $\omega L$ rescaled by $N$; the curves point at an asymptotic collapse onto a single master curve.
• Figure 4: Diagram of configuration relationships at fixed $r$ within the separated phase and at the separated–free boundary. Black contours denote separated states, red indicate configurations becoming jammed, and green mark unjamming states. Grey shading encodes the relative speed of the free particle in the separated phase (white for speed 0, light grey for speed 1, dark grey for speed 2).
• Figure 5: Diagram of configuration relationships within the jammed phase and at the jammed–separated boundary. Configurations with black contours correspond to jammed states, red contours indicate states becoming jammed, and green contours represent unjamming states. The grey shading denotes the relative speed of the jamming or unjamming free particle (light grey for speed 1, dark grey for speed 2).