# Activity-driven clustering of jamming run-and-tumble particles: Exact three-body steady state by dynamical symmetry

Leo Hahn, Arnaud Guillin, Manon Michel

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Leo Hahn, Arnaud Guillin, Manon Michel. Activity-driven clustering of jamming run-and-tumble particles: Exact three-body steady state by dynamical symmetry. 2025. hal-05251588

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# Activity-driven clustering of jamming run-and-tumble particles: Exact three-body steady state by dynamical symmetry

Leo Hahn$^{1,2,*}$ Arnaud Guillin$^1,\dagger$ and Manon Michel$^{1,‡}$

$^1$Laboratoire de Mathématiques Blaise Pascal UMR 6620, CNRS, Université Clermont-Auvergne, Aubière, France. and

$^2$Institut de Mathématiques, Université de Neuchâtel, Switzerland

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. A 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. This approach lays the groundwork for tackling the N-body problem.

**Introduction.** Active matter—ensembles of units that convert stored or ambient energy into directed motion— has emerged as a cornerstone in nonequilibrium statistical physics, now spanning scales from cytoskeletal filaments, pedestrian crowds to robotic swarms [1–8]. Within this field, run-and-tumble particles (RTPs) furnish a fundamental microscopic model [1, 9–22]: persistent ballistic runs punctuated by Poissonian velocity tumbles break detailed balance and give rise to hallmarks of active behavior such as wall accumulation [23] and motility-induced phase separation (MIPS) [11].

When hard-core repulsion is added, RTPs display phenomena qualitatively different from their passive counterparts. Activity-driven jamming interactions, clustering [12, 13, 24, 25] and different critical behaviours [26] emerge. Clustering is often rationalized in terms of an effective attraction: in dense regions particles slow down, further increasing density and triggering aggregation [27, 28]. Yet this picture is essentially coarse-grained [5, 11, 27, 29, 30], and exact microscopic results have remained elusive. Beyond two particles, where the problem reduces to a single effective RTP with more complex velocity dynamics [12, 16, 31], the steady state remains unknown. Despite an important theoretical interest, the absence of exact three-body or many-body solutions has been a major barrier to a first-principles understanding of clustering and to clarifying the microscopic origin of MIPS.

Indeed, a major challenge lies in handling the singular boundary conditions introduced by jamming [32, 33], even in the two-RTP case. Lattice discretizations [12, 19], soft repulsion limits [20], or added noise [14, 16] regularize the problem but at the cost of complicating the original process and adding limits nontrivial to compute [22]. A breakthrough came with the use of piecewise deterministic Markov processes (PDMPs) [21], whose generators encode both deterministic runs and stochastic tumbles, in-

cluding boundary behavior, directly in continuous space. For two RTPs, this framework yielded the explicit steady state for arbitrary tumble mechanisms and revealed the existence of *detailed-* and *global-jamming* system classes. These classes are linked to a dynamical symmetry, whose breaking creates in the steady state relaxation terms, that are often interpreted as effective attractions.

In this work, we resolve the three-body invariant measure, provide the first controlled microscopic description of clustering beyond two bodies and establish a foundation for the N-body problem. Extending the PDMP and symmetry formalism developed for two RTPs [21], we derive the invariant measure of three instantaneous RTPs and identify three structural weights—corresponding to free, separated, and jammed states—that organize persistent, separated, and diffusive regimes through the activity parameter $\omega L$, as displayed in Fig. 1. This reveals a cascade of scales $(\omega L)^k$, with $k$ the number of jammed particles, and crucially shows that the separated phase dominates at finite activity. Relaxation contributions, previously interpreted as effective attractions, are subdominant and do not drive clustering. Thus, our results close the gap of exact solutions beyond two particle and guide numerical studies towards the parameter range most relevant to observe clustering and phase separation in larger systems. These results pave the way toward the N-body problem, where the formation and stability of clusters are also expected to be governed by the dominant structural weights, organized along scales $(\omega L)^k$. They also highlight the delicate nature of clustering: whether a separated phase survives in the $N \to \infty$ limit, and, if so, whether it is confined to a single activity value or extends across a finite regime.

**Jamming and symmetry.** We consider a system of RTP on a 1D torus of length $L > 0$. Each particle $i$ moves with velocity $\sigma_i$ (Run) updated by an independent Markov jump process (Tumble). Collisions lead to jamming, where both particles remain immobile until a velocity update allows separation. The system $z = (r, \sigma)$ is fully specified by the inter-particle separation vector $r = (r_i)_i$, where $r_i$ denotes the distance between particle $i$ and particle $(i+1)$, together with the velocity

<footnote>* leo.hahn@unine.ch</footnote>

<footnote>† arnaud.guillin@uca.fr</footnote>

<footnote>‡ manon.michel@uca.fr</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 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 $w_J$ (green, long dashes), bulk $w_B$ (blue, solid) and separated $w_S$ (orange, dash-dotted)) as a function of $\omega L$ for the 2-RTP and 3-RTP systems respectively. For the 3-RTP system, $w_S$ decomposes into its uniform $w_{eq}w_S$ (yellow, dashed) and exponential part $w_{rel}w_S$ (maroon, dotted); $w_S$ achieves a maximum of $\approx 0.582$ at $\omega L \approx 2.16$. For $\omega L \lesssim 0.759$, jammed states dominate, then for $0.759 \lesssim \omega L \lesssim 6.16$ separated ones do, after which the free states dominate. In the 2-RTP case, $w_J = w_B$ at $\omega L = 2$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 2: (a) Detailed-jamming for instantaneous tumbles (↔↔↔). (b) Global-jamming for finite tumbles (↔↔↔↔0↔↔): exiting with (↔↔↔) is impossible as $\mu_J(↔↔) = \mu_J(↔↔) = 0$.</figcaption></figure>

vector $\sigma = (\sigma_i)_i$. A periodic convention is used for the indices/particle labels, i.e. $x_i = x_{i\pm3}$. For two RTP, the steady state was solved for arbitrary tumbles [21] and is of the form $\pi = w_B\pi_B + \frac{w_J}{2}\sum_{k=1}^2 \pi_{J_k}$, where $\pi_B$ is supported on free states $B$, and $\pi_{J_k}$ on jammed states, $J_1, J_2$ (boundaries of $B$, see Fig. 1),

$$ \pi_B(z) = \frac{w_{eq}}{L} \mu_B(\sigma) + w_{rel}\gamma(z), \quad \pi_{J_k}(z) = \mu_J(\sigma_k, \sigma_{k+1}). $$

The distributions $\mu_B$ and $\mu_J$ are the respective tumble steady states, left invariant by the jump process for $\sigma$.

The crucial observation of [21] is that conservation of probability flows defines an active global balance in $B$. Detailed satisfaction of this balance requires $\pi(r, (\sigma_1, \sigma_2)) = \pi(r, (\sigma_2, \sigma_1))$, which $\mu_B$ automatically possesses due to particle indistinguishability. Thus, deviations from detailed symmetry in $\sigma$ arise from the imbalance between entering and exiting scenarios at jamming boundaries, which equivalently reflect the incompatibility between $\mu_B$ and $\mu_J$, as shown in Fig. 2. Accordingly, detailed-jamming systems necessarily have $w_{rel} = 0$, maintaining a uniform steady state $\propto \mu_B$ in $B$, identical to the passive equilibrium distribution. This occurs for instantaneous-tumble RTPs, where velocities $\sigma_i = \pm 1$ flip with rate $\omega > 0$. By contrast, global-jamming systems break this detailed symmetry. The system is shifted away from uniformity at the jamming boundary, resulting in a catenary relaxation term $w_{rel}\gamma$. This arises for

finite tumble RTPs, where $\sigma_i \in \{-1, 0, +1\}$ and $\gamma$ is a single-scale catenary relaxation.

This highlights the interplay between the richness of tumbling mechanism, symmetry preservation and boundary-induced relaxation. A natural next question is whether increasing the number of particles alone can generate non-equilibrium deviations away from uniformity. In particular, can three instantaneous RTPs already produce a non-uniform steady state—potentially stabilizing separated phases?

**Three-RTP steady state.** Increasing the particle number alone gives rise to qualitatively new phenomena: in the three-particle system, three-particle clusters and phase-separated states emerge, with two particles jammed and one remaining free. By deriving the exact invariant measure for a three-RTP system with instantaneous tumbles, we show that the symmetry picture remains relevant and demonstrate the stabilization of phase-separated configurations.

As illustrated in Fig. 1 and detailed in Appendix B, the state space $\Omega$ is split into: the bulk $B$ where all particles are free, the three edges $S_k$ where the system is separated into a cluster of two jamming particles and a free one and the three vertices $J_k$ where all particles are jamming into a single cluster. The steady state $\pi$ is a mixture over distributions $\pi_B, \pi_{S_k}, \pi_{J_k}$ respectively supported on the free, separated and jammed states,

$$ \pi = w_B \pi_B + \frac{1}{3} \sum_{k=1}^{3} [w_S \pi_{S_k} + w_J \pi_{J_k}]. \quad (1) $$

We show that, for $(r, \sigma) \in B$ (resp. $J_k, S_k$),

$$ \begin{align} \pi_B(r, \sigma) &= \frac{2}{L^2} \mu_B(\sigma), \quad \pi_{J_k}(r, \sigma) = \mu_J(\tau_k(\sigma)) \tag{2} \\ \pi_{S_k}(r, \sigma) &= \frac{w_{eq}}{L} \mu_S(\tau_k(\sigma)) + w_{rel}\gamma(r_{k+1}, \tau_k(\sigma)) \nonumber \end{align} $$

where, for any vector $x$, the $\tau_k(x_1, x_2, x_3) = (x_k, x_{k+1}, x_{k+2})$ are the index shifts. The distributions
$\mu_B$, $\mu_S$ and $\mu_J$ stand for the respective tumble steady state. There is no relaxation in the free states and the relaxation term $\gamma$ in the separated ones takes the form,

$$ \gamma(r, \sigma) = \frac{\nu_S(\sigma)}{N_\gamma} \mathrm{ch}(\lambda(\frac{L}{2} - r)) + \frac{\sigma_3 \nu_A(\sigma)}{N_\gamma} \mathrm{sh}(\lambda(\frac{L}{2} - r)), \quad (3) $$

where $\lambda = 2\sqrt{2}\omega$ and $N_\gamma = \frac{16\mathrm{sh}(\lambda L/2)}{\lambda}$. Appendix A summarizes the full formula. At the marginal level, the state space reduces to a simplex and $\pi$ sums up to a pure catenary relaxation along each separated-state edge,

$$ \pi(r) = w_B \frac{2 \mathbb{1}_B(r)}{L^2} + \frac{1}{3} \sum_{k=1}^{3} \left[ w_J \mathbb{1}_{J_k}(r) + w_S \mathbb{1}_{S_k}(r) \left( \frac{w_{eq}}{L} + w_{rel} \frac{\lambda \mathrm{ch}(\lambda(L/2 - r_{k+1}))}{2\mathrm{sh}(\lambda L/2)} \right) \right]. \quad (4) $$

Compared to the 2-RTP case, the 3-RTP steady state still reflects particle indistinguishability—appearing as a sum structure—and remains uniform over free states. The crucial novelty is the emergence of phase-separated configurations, where the steady state is not of product form, marking a stronger departure from equilibrium driven purely by the increase in particle number. In these states, relaxation proceeds in a catenary fashion under the constraint imposed by jamming boundaries. This reflects a fundamental change in symmetry: while instantaneous tumbles enforce detailed-jamming symmetry at the free/separated boundary, within separated states the jammed pair behaves as an effective single particle with richer tumble dynamics—akin to a finite tumble with

a non-motile state—thereby breaking detailed-jamming and producing global-jamming relaxation.

The central question of the three-RTP problem is the stabilization of the separated phase. To address this, we show how we obtained the explicit steady state using PDMP, including the weights (see (8), (10), (11)). For further details on the PDMP approach for RTPs, see [21, 22]. Here, the PDMP is characterized by its generator, for any state $z = (r, \sigma)$ and $f$ some test function,

$$ A f(z) = \underbrace{\langle \phi(z), \nabla_r f(z) \rangle}_{\text{Run}} + \omega \sum_j \underbrace{\frac{f(\iota_j(z)) - f(z)}{\text{Tumble}}}_{\text{Tumble}}, \quad (5) $$

where each $\iota_k$ flips the k-th particle velocity, $\sigma_k \to -\sigma_k$, translating the k-th particle tumble at rate $\omega$, and the flow $\phi$ encodes the dynamics of $r$ set by the velocities $\sigma$, with $\dot{r} = \phi(z) = (\phi_i(z))_i$, with $\phi_i(z)$ equal to,

$$ \left\{ \begin{array}{l} 0 \text{ if } (r_i, \sigma_{i+1} - \sigma_i) \in \{(0, -2), (0, 0), (L, 2)\}, \\ -\sigma_i \text{ if } (r_{i+1}, \sigma_{i+2} - \sigma_{i+1}) = (0, -2), r_i > 0, \\ \sigma_{i+1} \text{ if } (r_{i+2}, \sigma_i - \sigma_{i+2}) = (0, -2), r_i > 0, \\ \sigma_{i+1} - \sigma_i \text{ otherwise.} \end{array} \right. \quad (6) $$

A key feature of the dynamics is the emergence of velocities $\phi_i(z) = \pm 1$ in separated states down to $\phi = 0$ in jammed ones, which underlies the velocity slowdown as clustering and separation develop.

As detailed in Appendix B, the invariance condition $\int_{\Omega} A f d\pi = 0$ translates into the following conservation laws on probability flows, with contributions ordered as tumble, run, and boundary exchanges (the latter arising from integration by parts of the run term in (5)),

$$ \sum_j (\pi_B(\iota_j(z)) - \pi_B(z)) = \sum_j \frac{\phi_j(z)}{\omega} \partial_{r_j} \pi_B(z) + 0 \quad (C_B) $$

$$ \sum_j (\pi_{S_k}(\iota_j(z)) - \pi_{S_k}(z)) = \frac{\phi_{k+1}(z)}{\omega} (\partial_{r_{k+1}} \pi_{S_k}(z) - \partial_{r_{k+2}} \pi_{S_k}(z)) + \frac{3w_B \phi_k(z)}{w_S \omega} \mathbb{1}_{\partial_k B}(z) \lim_{\zeta \in B \to z} \pi_B(\zeta) \quad (C_{S_k}) $$

$$ \sum_j (\pi_{J_k}(\iota_j(z)) - \pi_{J_k}(z)) = 0 - \frac{w_S \phi_{k+2}(z)}{w_J \omega} (\mathbb{1}_{\partial_k S_k}(z) \lim_{\zeta \in S_k \to z} \pi_{S_k}(\zeta) + \mathbb{1}_{\partial_k S_{k+1}}(z) \lim_{\zeta \in S_{k+1} \to z} \pi_{S_{k+1}}(\zeta)), \quad (C_{J_k}) $$

where $\partial_k B$ and $\partial_k S_{k'}$ respectively gather the entering and exiting boundary points between free/separated and separated/jammed phases, i.e. the annihilation or creation of a jamming pair through particle transport.

Similarly to the 2-RTP case [21], the conservation of probability flows is ensured by the compensation between tumble and run terms and the exchanges on the boundary between the different phases. However, in addition to the free and jamming phases, the novel separated phase complexifies the condition hierarchy. Fortunately, we can extend the dynamical symmetry approach stemming from the active global balance [21] to restrict even further

those conditions and provide a complete solution:

(C$_B$) Since tumbles are instantaneous, the detailed dynamical symmetry, (e.g. $P(\oplus\ominus\ominus\ominus) = P(\ominus\ominus\ominus\ominus)$), can be satisfied at the free/separated boundary (see Appendix C). This leads to $\pi(r, \sigma) = \pi(r, -\sigma)$, $(r, \sigma) \in B$. Solving (C$_B$) under this symmetry necessarily cancels the derivative term and fixes the contribution $\pi_B$ in (1) to be the uniform solution of (C$_B$), set by the bulk tumble measure, $\mu_B(\sigma) = 1/8$.

(C$_{S_k}$) We refine the general expression of $\pi_{S_k}$ (2) by identifying $\gamma$ as the relaxation signature of broken dynamical symmetry at jamming. As shown in Appendix C, jam-
ming and unjamming states at the separated/jammed boundary cannot be balanced in a detailed-jamming manner (e.g. $P(\leftarrow\leftarrow\leftarrow) \neq P(\leftarrow\leftarrow\leftarrow)$) due to the incompatibility of the tumble mechanisms in the respective phases. To determine $\gamma$, we first exploit the uniform form of $\pi_B$ which generates constant source terms in $(C_{S_k})$. Their effect on the separated phase can be viewed as a modification of the tumble mechanism, since outflows from certain configurations are reinterpreted as inflows into others. As detailed in Appendix D, this leads directly to the pure catenary relaxation,

$$ \partial_{\bar{r}\bar{r}}^2 (\pi_{S_k}^{+} \pm \pi_{S_k}^{-}) = \lambda^2 L_{\pm} (\pi_{S_k}^{+} \pm \pi_{S_k}^{+}), \quad \lambda = 2\sqrt{2\omega}, \quad (7) $$

where $\pi_{S_k}^{\pm}(\bar{r}) = (\pi_{S_k}(r, \sigma_n^{\pm}))_{n=1}^2$, with $r_{k+1} = \bar{r}$ and $\sigma_n^{\pm}$ so that the flow $\phi_{k+1}(r, \sigma_n^{\pm}) = \pm n$. The matrices $L_{\pm}$ admit as eigenvalues 0 and 1, setting $\gamma$ to the form (3). Further constraints from the eigenvectors of $L_{\pm}$, symmetry around $L/2$ and first-order arguments lead to,

$$ \nu_S(\sigma) = \begin{cases} \sigma_1\sigma_3 & \nu_A(\sigma) = \begin{cases} \sigma_1\sigma_3\sqrt{2} & \text{if } \sigma_1 = \sigma_2 \\ 2\sqrt{2} & \text{otherwise} \end{cases} \end{cases}. \quad (8) $$

The separated tumble measure $\mu_S$ is the uniform solution of (7), but also of $(C_{S_k})$ once composed with $\tau_k$, setting the weight ratio between free and separated states,

$$ \mu_S(\sigma) = \frac{1 + \delta_{\sigma_1, -\sigma_2}}{8}, \quad \frac{w_B}{w_S w_{eq}} = \frac{\omega L}{6}. \quad (9) $$

Therefore, at the free-separated boundary, the exchanges does not involve the term $w_S w_{rel}\gamma$ and reduces to a mapping of the tumble term $w_S w_{eq}\mu_S$ onto the bulk source terms, consistent with their role as tumble modifiers.

Parallels can be drawn between $\pi_{S_k}$ and a generic 2-RTP system, but with crucial differences. A separated configuration maps onto an effective two-particle system, now distinguishable—the single particle and the jamming pair exhibit different tumbles—and further impacted by bulk source terms. A simple catenary relaxation term was expected from applying spectral argument [21] to the effective system but particle distinguishability manifests through the existence of an antisymmetric sh contribution even in fully aligned $\pm \pm \pm$ states.

($C_{J_k}$) As shown in Appendix E, the jammed constraints fix the jammed tumble measure $\mu_J$ and determine $w_{eq} = 1 - w_{rel} < 1$, as the detailed-jamming breaking generates the relaxation $w_{rel}\gamma$ in the separated phase,

$$ \mu_J(\sigma) = \frac{1+2\delta_{\sigma_1,-\sigma_3}}{8}, \quad w_{eq} = 1 - \frac{1}{1+\omega L \left[ \frac{2\sqrt{2}}{\tanh(\omega L\sqrt{2})} + \frac{10}{3} \right]}. \quad (10) $$

(w) As shown in Appendix F, the weights $w_B, w_S, w_J$ are set by expliciting the boundary terms in $(C_{S_k})$ (i.e. (9)), and in $(C_{J_k})$ and the normalisation condition,

$$ \begin{aligned} w_B &= \frac{(\omega L)^2 w_{eq}}{N_w}, & w_S &= \frac{6\omega L}{N_w}, & w_J &= \frac{(6+4\omega L)w_{eq}-4\omega L}{N_w} \\ N_w &= 2\omega L + w_{eq}[6+4\omega L+(\omega L)^2]. \end{aligned} \quad (11) $$

**Separated-phase stabilization.** We now discuss the central issue of the stabilization of the separated phase through the weight competition, as shown in Fig. 1. The explicit weights (10) and (11) reveal three distinct regimes, controlled solely by the activity $\omega L$. The separated uniform component $w_{eq}$ remains $\sim 1$ throughout, tending to $2/3$ as $\omega L \ll 1$ and to $1$ as $\omega L \gg 1$, making it a natural reference for reading the regimes directly from the weight hierarchy in (11). In the persistent regime $\omega L \ll 1$, the weights scale as $w_J \approx 1$ and $w_S \propto (\omega L)^2$. Therefore, the system predominantly forms a single three-particle cluster, spending almost all its time in fully jammed configurations. This reflects the fact that particles traverse the state space much faster than they tumble, yielding effective ballistic dynamics. In the opposite diffusive regime $\omega L \gg 1$, the weights scale as $w_B \approx 1$, $w_S \propto (\omega L)^{-1}$, and $w_J \propto (\omega L)^{-2}$. Jammed configurations become negligible, particles are almost always free, distributed quasi-uniformly over the torus. This is consistent with the now frequent velocity tumbles leading to a motion that is essentially diffusive.

These two asymptotic regimes already appear in 2-RTP systems, as shown for both the steady state and mixing time [21, 22]. The striking novelty in the 3-RTP case is the emergence of a separated regime at finite $\omega L = O(1)$, in which two particles are jammed while the third remains free, leading to a slowdown of the average velocity. Here, the separated phase dominates the invariant measure: the most probable configurations consist of two particles jamming while the third remains free. The sharp growth of $w_S$ along $\omega L$ and its slow subsequent decay highlight the robustness of this regime. The secondary role of the relaxation component in the invariant measure is now apparent. For all values of $\omega L$, the relaxation contribution $w_S w_{rel}$ always remains subdominant in comparison to the structural ones $w_B, w_S w_{eq}$ and $w_J$. Moreover, in the diffusive regime, relaxation—controlled by the scale $\lambda$—is essentially instantaneous and the weight $w_{rel}$ scales as $\sim (\omega L)^{-1}$. In the persistent regime, $w_{rel} \sim 1/3$ but $\gamma$ is effectively uniform. Thus, relaxation plays at most a secondary role relative to the dominant structural weights, primarily mediating the impedance between boundaries. Its limited impact, also observed in the general 2-RTP case [21, Section VI], indicates that relaxation—sometimes reinterpreted as an effective attraction—cannot be regarded as the primary driver of clustering. Even in N-particle systems, the formation and stability of clusters are therefore expected to be dictated by the dominant structural weights.

**Discussion.** The most compelling direction for future work is determining the fate of the separated regime in the N-particle case. Understanding the nature of clustering in this regime—i.e. whether a single macroscopic cluster forms, or the system fragments into multiple clusters, or displays fractal behavior—remains an important open question. This analysis of the exact microscopic steady state offers an alternative and complementary perspective to the effective-attraction picture often used in
coarse-grained theories [11, 24, 27]. It is particularly valuable for models with jamming in the absence of thermal noise, where exact hydrodynamic limits remain elusive [34, 35]. The key framework here is that of PDMPs, whose generator provides an efficient mean to handle the otherwise delicate boundary conditions in the Fokker-Planck formulation [21]. The invariance condition then translates into a hierarchy of PDEs, encoding the conservation of probability flows under deterministic RTP motion and stochastic velocity flips. For three RTPs, the competition displayed by $w_J$, $w_S$ and $w_B$ thus provides the first fully controlled microscopic picture of how activity alone can bias a many-body system toward finite, self-selected clusters.

While a full solution of the N-particle steady state lies beyond the scope of this work, the three-RTP case already yields insights that extrapolate naturally. First, the three-body steady state already contains the basic building blocks of the general problem: a bulk region where all particles move freely, (k-1)-dimensional faces associated with k clusters, and a hierarchy of probability flows between them. Second, the persistent and diffusive regimes extend directly: for $\omega L \gg 1$ the process should approach a uniform distribution on the simplex, whereas for $\omega L \ll 1$ the probability mass concentrates on subsets of jammed configurations (no free particles). Third, and more importantly, the hierarchical organization of structural weights in powers of $\omega L$ depending on the number of jamming particles is going to persist for general N, as this originates from the boundary exchange processes between run and tumble terms. The crucial open problem is to determine how these competing scales interact

and which phase dominates at finite $\omega$. This lies at the heart of whether separated phases survive in larger systems and, more broadly, whether MIPS can be rigorously established or ruled out.

Our analysis shows how delicate this balance is, with clustering outcomes hinging sensitively on activity strength. This motivates both further analytical progress and carefully targeted numerical studies, for which our results provide clear guidance on the parameter ranges to probe. At the same time, our work indicates a possible route to overcome the combinatorial complexity of the N-particle case: leveraging the recursive structure suggested by the effective-particle behavior of jammed clusters to build a tractable solution. In the detailed-jamming setting, the fact that the invariant measure remains uniform over free states provides a natural starting point for tackling the PDE hierarchy. Altogether, our approach lays the foundation for a new theoretical framework to probe phase separation in jamming systems, offering an alternative MIPS criterion directly connected to the microscopic scale rather than to coarse-grained or high-level observables.

## ACKNOWLEDGMENTS

All the authors acknowledge the support of the French ANR under the grant ANR-20-CE46-0007 (SuSa project). LH acknowledges support from the Swiss National Science Foundation (grant no. 200029-21991311). A.G. has benefited from the support of a government grant managed by the Agence Nationale de la Recherche under the France 2030 investment plan ANR-23-EXMA-0001.

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Appendix A: 3-RTP steady state

We summarize here the complete expression of the 3-RTP steady state, for $(r, \sigma) \in \Omega$,

$$ \pi(r, \sigma) = w_B \frac{1_B(r, \sigma)}{4L^2} + \frac{1}{3} \sum_{k=1}^{3} \left( w_S 1_{S_k}(r, \sigma) \left[ w_{\text{eq}} \frac{1 + \delta_{\sigma_1, -\sigma_2}}{8L} + w_{\text{rel}} \gamma(r_{k+1}, \tau_k(\sigma)) \right] + w_J 1_{J_k}(r, \sigma) \frac{1 + 2\delta_{\sigma_1, -\sigma_3}}{8} \right), \quad (A1) $$

with,

$$ \gamma(\bar{r}, \sigma) = \frac{\omega\sqrt{2}(2\delta_{\sigma_1, -\sigma_2} + \sigma_1\sigma_3\delta_{\sigma_1, \sigma_2})}{8\text{sh}(\omega L\sqrt{2})} \left[ (1 + \delta_{\sigma_1, -\sigma_2})\text{ch}\left(2\sqrt{2}\omega(\frac{L}{2} - \bar{r})\right) + \sigma_3\sqrt{2}\text{sh}\left(2\sqrt{2}\omega(\frac{L}{2} - \bar{r})\right) \right], $$

and,

$$ w_{\text{eq}} = \frac{\omega L \left[ \frac{2\sqrt{2}}{\text{th}(\omega L\sqrt{2})} + \frac{10}{3} \right]}{1 + \omega L \left[ \frac{2\sqrt{2}}{\text{th}(\omega L\sqrt{2})} + \frac{10}{3} \right]}, $$

$$ w_{\text{rel}} = \frac{1}{1 + \omega L \left[ \frac{2\sqrt{2}}{\text{th}(\omega L\sqrt{2})} + \frac{10}{3} \right]}, $$

$$ w_B = \frac{(\omega L)^2 w_{\text{eq}}}{2\omega L + w_{\text{eq}} [6 + 4\omega L + (\omega L)^2]}, $$

$$ w_S = \frac{6\omega L}{2\omega L + w_{\text{eq}} [6 + 4\omega L + (\omega L)^2]}, $$

$$ w_J = \frac{(6 + 4\omega L)w_{\text{eq}} - 4\omega L}{2\omega L + w_{\text{eq}} [6 + 4\omega L + (\omega L)^2]}. $$

The partition of the state space $\Omega$ into $\mathcal{B}$ (free states), $\cup_k S_k$ (separated states) and $\cup_k J_k$ (jammed states) is precisely defined in Appendix B 1.

Appendix B: Conservation constraint derivation

1. State space definition

In this section, we provide a detailed and precise definition of the state space, as this is essential for computing the integral in the invariance condition. The subtlety lies in the integration by parts, where the boundary terms must be specified with great care in order to determine unambiguously which states belong to which set. A state is assigned to a given set (separated or free) only if it does not immediately leave it under the dynamics (i.e., through the creation or annihilation of a jamming pair). This makes a careful understanding of the dynamics indispensable, and we therefore propose here an explicit construction of the state space, illustrated in Fig. 1.

The state space for $r$ is split into:

* the simplex $\Delta = \{r \in ]0, L^3; \sum_k r_k = L\}$

* the three edges $\partial_k \Delta = \{r; \tau_k(r) \in \{0\} \times ]0, L^2 \text{ and } \sum_k r_k = L\}$ with normal $n_k^S = -(\delta_{1,k}, \delta_{2,k}, \delta_{3,k})$

* the three vertices $\partial_k^2 \Delta = \{r; \tau_k(r) = (0, 0, L)\}$ with normal $n_k^J = (\delta_{1,(k+2)}, \delta_{2,(k+2)}, \delta_{3,(k+2)})$.

The state space for the velocities is $\Sigma = \{-1, 1\}^3$ and we define the subsets $\Sigma_{k^\pm} = \{\sigma; \pm(\sigma_{k+1} - \sigma_k) > 0\}$ and $\Sigma_{k^0} = \{\sigma; \sigma_{k+1} = \sigma_k\}$. Using the definition of the flow $\phi$ (6), the complete state space $\Omega$ for $(r, \sigma)$ identifies with $\mathcal{B} \cup (\cup_{k=1}^3 S_k) \cup (\cup_{k=1}^3 J_k)$ where:

* (Free) the bulk $\mathcal{B}$ identifies with the three RTP being free, i.e.

$$ \mathcal{B} = (\Delta \times \Sigma) \cup (\cup_k \partial_k \mathcal{B}^+), \quad (B1) $$

with the sets of entering states

$$ \partial_k \mathcal{B}^+ = \partial_k \Delta \times \Sigma_{k^+}. \quad (B2) $$

The set of exit states (not included in $\mathcal{B}$) are

$$ \partial_k \mathcal{B}^- = \partial_k \Delta \times \Sigma_{k^-}. \quad (B3) $$

The set $\cup_k \partial_k \mathcal{B}^+ \cup_k \partial_k \mathcal{B}^-$ amounts to $\partial \mathcal{B}$ as source term in $(C_{S_k})$.
• (Separated) the edge $S_k$ correspond to RTP $k$ and $k+1$ jamming and the $k+2$-one being free, i.e.

$$S_k = (\partial_k \Delta \times (\Sigma \setminus \Sigma_{k+})) \cup \partial_k S_k^+ \cup \partial_{k+2} S_k^+ \quad (B4)$$

with the set of entering states into $S_k$ from vertex $J_k$ and $J_{k+2}$,

$$\partial_k S_k^+ = \partial_k^2 \Delta \times (\Sigma_{(k+1)+} \setminus \Sigma_{k+}) \text{ and } \partial_{k+2} S_k^+ = \partial_{k+2}^2 \Delta \times (\Sigma_{(k+2)+} \setminus \Sigma_{k+}). \quad (B5)$$

It includes exit states from $\mathcal{B}$. The sets of exit states (not included in $S_k$) are

$$\partial_k S_k^- = \partial_k^2 \Delta \times (\Sigma_{(k+1)-} \setminus \Sigma_{k+}) \text{ and } \partial_{k+2} S_k^- = \partial_{k+2}^2 \Delta \times (\Sigma_{(k+2)-} \setminus \Sigma_{k+}). \quad (B6)$$

The set $\partial_k' S_k^+ \cup \partial_k' S_k^-$ amounts to $\partial_k S_k'$ as source term in $(C_{J_k})$.

• (Jamming) the vertex

$$J_k = \partial_k^2 \Delta \times (\Sigma \setminus (\Sigma_{k+} \cup \Sigma_{(k+1)+})) \quad (B7)$$

corresponds to the three RTP jammed in the order $k, k+1, k+2$. It includes exit states from $S_k$ and $S_{k+1}$.

## 2. Conservation constraints

Plugging the generator expression (5) into the invariance condition $\int_{\Omega} A f d\pi = 0$, we obtain for $f$ a continuous test function:

$$\int_{\mathcal{B}} \langle \phi(z), \nabla_r f(z) \rangle d\pi(z) + \sum_k \int_{S_k} \langle \phi(z), \nabla_r f(z) \rangle d\pi(z) + \int_{\mathcal{B} \cup (\cup_k S_k) \cup (\cup_k J_k)} \omega \sum_j (f(\iota_j(z)) - f(z)) d\pi(z) = 0 \quad (B8)$$

The first terms are treated by integration by parts, leading to,

$$\int_{\mathcal{B}} \langle \phi(z), \nabla_r f(z) \rangle d\pi(z) = - \int_{\mathcal{B}} \langle \phi(z), \nabla_r \pi(z) \rangle f(z) dz + \sum_k \int_{\partial_k \mathcal{B}^+ \cup \partial_k \mathcal{B}^-} \langle \phi(z), n_k^S \rangle \lim_{\zeta \in \mathcal{B} \to z} \pi(\zeta) f(z) dz \quad (B9)$$

$$\int_{S_k} \langle \phi(z), \nabla_r f(z) \rangle d\pi(z) = - \int_{S_k} \langle \phi(z), \nabla_r \pi(z) \rangle f(z) dz + \sum_k \int_{\partial_k S_k^+ \cup \partial_k S_k^-} \langle \phi(z), n_k^J \rangle \lim_{\zeta \in S_k \to z} \pi(\zeta) f(z) dz \\ + \sum_k \int_{\partial_{k+2} S_k^+ \cup \partial_{k+2} S_k^-} \langle \phi(z), n_{k+2}^J \rangle \lim_{\zeta \in S_k \to z} \pi(\zeta) f(z) dz \quad (B10)$$

Since $f$ is an arbitrary test function, the relation must hold for each value of $z$. By collecting all contributions associated with a given $z$ and replacing the normals by their expression and $\pi$ by its decomposition (1), we therefore arrive at the conservation conditions $(C_{\mathcal{B}}), (C_{S_k})$ and $(C_{J_k})$.

## Appendix C: Run-Tumble flow diagrams and jamming-unjamming (im)balance at phase boundaries

In this appendix we provide diagrams of the flow exchanges in the separated (Fig. 3) and jammed (Fig. 4) phases. They show that detailed jamming is trivially satisfied at the free-separated boundary ($P(\text{圈圈圈}) = P(\text{圈圈圈})$) but necessarily broken at the separated-jammed one. A direct inspection makes this evident: enforcing detailed balance on the jamming/unjamming flows is impossible, since the tumble steady states of the two phases do not match. Specifically, the separated tumble steady state requires

$$P(\text{圈圈圈}) = 2P(\text{圈圈圈})$$

while the jammed steady state enforces

$$P(\text{圈圈圈}) = 8P(\text{圈圈圈})$$

demonstrating the incompatibility and the breaking of the detailed-jamming symmetry. As a consequence, the solution in the separated phase can no longer reduce to a uniform form proportional to the tumble steady state; an additional relaxation contribution is required to reconcile the imbalance.
<figure><img src="image_5.png" /><figcaption>FIG. 3: 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).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 4: 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).</figcaption></figure>

## Appendix D: Relaxation equation in the separated phase

By symmetry, the distribution $\pi_{S_k}$ simplifies without loss of generality to its expression in (2). We determine $\gamma$ by considering the case over $S_1$. We introduce the vector $\pi^S(\bar{r}) = (\pi_{S_1}((0, \bar{r}, L - \bar{r}), \sigma)_{\sigma \in \Sigma})$, which contains all the densities on the edge $S_1 = \mathcal{J}_3\mathcal{J}_1$. The parameter $\bar{r}$ lives in $[0, L]$ and $\bar{r} = 0$ corresponds to $\mathcal{J}_1 = (0, 0, L)$, while $\bar{r} = L$ corresponds to $\mathcal{J}_3 = (0, L, 0)$. Expliciting the condition $(C_{S_k})$ following the diagram in Fig. 3 leads to,

$$ 
\begin{align*}
\partial_{\bar{r}} \pi^S_{+-} &= -3\omega \pi^S_{+-} + \omega \pi^S_{++} + \omega \pi^S_{--} + \omega \pi^S_{---} + K, && (C^S_{+-}) \\
\partial_{\bar{r}} \pi^S_{---} &= 3\omega \pi^S_{---} - \omega \pi^S_{--} - \omega \pi^S_{+-} - \omega \pi^S_{++} - K, && (C^S_{---}) \\
2\partial_{\bar{r}} \pi^S_{+-} &= -3\omega \pi^S_{+-} + \omega \pi^S_{--} - \omega \pi^S_{++}, && (C^S_{--}) \\
2\partial_{\bar{r}} \pi^S_{++} &= 3\omega \pi^S_{++} - \omega \pi^S_{--} - \omega \pi^S_{---}, && (C^S_{+-}) \\
K &= \omega \pi^S_{+++} + \omega \pi^S_{---}, && (C^S_{+-}) \\
K &= \omega \pi^S_{---} + \omega \pi^S_{+-}, && (C^S_{--}) \\
3\omega \pi^S_{+++} &= \omega \pi^S_{+-} + \omega \pi^S_{++}, && (C^S_{++}) \\
3\omega \pi^S_{---} &= \omega \pi^S_{--} + \omega \pi^S_{---}. && (C^S_{---})
\end{align*}
$$

where

$$ K = \frac{3w_B}{w_S} \frac{1}{2L^2} $$

stems for the source terms from the bulk (that are constant as we restrict possible solutions to being the uniform distribution $\frac{2}{L^2}\mu_B$ in $B$ and $\mu_B(\sigma) = 1/8$). Using $(C^S_{+-})$, $(C^S_{--})$, $(C^S_{++})$ and $(C^S_{---})$, we reduce the system of
equations to the following systems on states with nonzero flow $\phi$,

$$ \partial_{\bar{r}} \begin{pmatrix} \pi_{\pm--}^{S} \\ \pi_{--\pm}^{S} \\ \pi_{+-}^{S} \\ \pi_{++-}^{S} \end{pmatrix} = \frac{\omega}{3} \begin{pmatrix} -7 & 6 & 3 & 2 \\ 3/2 & -4 & 1/2 & 0 \\ -3 & -2 & 7 & -6 \\ -1/2 & 0 & -3/2 & 4 \end{pmatrix} \begin{pmatrix} \pi_{\pm--}^{S} \\ \pi_{--\pm}^{S} \\ \pi_{+-}^{S} \\ \pi_{++-}^{S} \end{pmatrix}. \quad (D1) $$

This resulting matrix coincides with the matrix B in [21] if using a mapping to an effective two-particle system with a tumble mechanism modified by the bulk source terms.

It leads to the following second-order relaxation equations that recover (7),

$$ \partial_{\bar{r}}^2 (\pi_{S_k}^{+} + \pi_{S_k}^{-}) = 8\omega^2 L_+(\pi_{S_k}^{+} + \pi_{S_k}^{-}), \quad \text{with } L_+ = \frac{1}{3} \begin{pmatrix} 2 & -4 \\ -1/2 & 1 \end{pmatrix} \quad (D2) $$

$$ \partial_{\bar{r}}^2 (\pi_{S_k}^{+} - \pi_{S_k}^{-}) = 8\omega^2 L_{-}(\pi_{S_k}^{+} - \pi_{S_k}^{-}), \quad \text{with } L_{-} = \frac{1}{3} \begin{pmatrix} 2 & -2 \\ -1 & 1 \end{pmatrix}, \quad (D3) $$

where $\pi_{S_1}^+ = (\pi_{\pm--}^S, \pi_{--\pm}^S)$, $\pi_{S_1}^- = (\pi_{+-}^{S}, \pi_{++-}^{S})$, and more generally, $\pi_{S_k}^{\pm}(\bar{r}) = (\pi_{S_k}(r, \sigma_n^{\pm}))_{n=1}^2$, with $r_{k+1} = \bar{r}$ and $\sigma_n^{\pm}$ so that the flow $\phi_{k+1}(r, \sigma_n^{\pm}) = \pm n$.

The matrix $L_+$ admits $\mu_S = (2, 1)$ for eigenvector of eigenvalue 0 and $\nu_S = (4, -1)$ for eigenvector of eigenvalue 1. The matrix $L_-$ admits $\mu_- = (1, 1)$ for eigenvector of eigenvalue 0 and $\nu_A = \sqrt{2}(2, -1)$ for eigenvector of eigenvalue 1. The fact that $\mu_S \not\propto \mu_-$ is consistent with the breaking of the detailed-jamming symmetry [21]. The solution necessarily decomposes into,

$$ \pi_{S_k}^{+}(\bar{r}) + \pi_{S_k}^{-}(\bar{r}) = w_{\text{eq}}\mu_S + w_{\text{rel}}\nu_S \left[ a_{+} \text{ch} \left( \lambda \left( \frac{L}{2} - \bar{r} \right) \right) + b_{+} \text{sh} \left( \lambda \left( \frac{L}{2} - \bar{r} \right) \right) \right] \quad (D4) $$

$$ \pi_{S_k}^{+}(\bar{r}) - \pi_{S_k}^{-}(\bar{r}) = w\mu_{-} + w_{\text{rel}}\nu_{A} \left[ a_{-} \text{ch} \left( \lambda \left( \frac{L}{2} - \bar{r} \right) \right) + b_{-} \text{sh} \left( \lambda \left( \frac{L}{2} - \bar{r} \right) \right) \right] \quad (D5) $$

By symmetry, $\pi_{S_k}^+(L/2) = \pi_{S_k}^-(L/2)$, therefore,

$$ w = 0, \quad a_{-} = 0. \quad (D6) $$

Plugging (D5) and (D4) into (D1) yields,

$$ a_{+} = b_{-}, \quad b_{+} = 0, \quad (D7) $$

and deriving $(C_{-+}^S)$ and $(C_{+-}^S)$ leads to $\partial\pi_{++}^S = -\partial\pi_{--}^S$ and $\partial\pi_{--}^S = -\partial\pi_{+-}^S$, recovering the expressions for $\nu_S$ and $\nu_A$ in (8). The vector $\mu_S$ identifies with the edge tumble measure $\mu_S$ in (9).

## Appendix E: Jammed constraints

By symmetry, the distribution $\pi_{J_k}$ simplifies without loss of generality to its expression in (2) and can be determined by considering the case over $J_1$. We introduce the vector $\mu = (\mu_J(\sigma))_{\sigma \in \Sigma}$, which contains all the densities on the edge $J_1$. Explicating the condition $(C_{J_k})$ according to the diagram in Fig. 4 leads to,

$$ \omega\mu_{++-} = 3\omega\mu_{+--}, \quad (C_{+--}^{J}) $$

$$ \omega\mu_{+--} = 3\omega\mu_{---}, \quad (C_{---}^{J}) $$

$$ \omega\mu_{---} + \omega\mu_{+--} = 3\omega\mu_{+--} - \frac{w_S}{w_J}(\pi_{+-}^S(0^+) + 2\pi_{--}^S(L^-)), \quad (C_{+-}^{J}) $$

$$ \omega\mu_{+++} + \omega\mu_{+--} = 3\omega\mu_{+--} - \frac{w_S}{w_J}(\pi_{+-}^S(L^-) + 2\pi_{++-}^S(0^+)), \quad (C_{+-}^{J}) $$

$$ \omega\mu_{---} = 2\frac{w_S}{w_J}\pi_{--+}^S(0^+), \quad (C_{--+}^{J}) $$

$$ \omega\mu_{+++} = 2\frac{w_S}{w_J}\pi_{++-}^S(L^-), \quad (C_{-++}^{J}) $$

$$ \omega\mu_{---} + \omega\mu_{+--} = \frac{w_S}{w_J}\pi_{+-}^S(L^-), \quad (C_{-+-}^{J}) $$

$$ \omega\mu_{+++} + \omega\mu_{+--} = \frac{w_S}{w_J}\pi_{+-}^S(0^+), \quad (C_{+-+}^{J}) $$
where the vector $\pi^S$ is defined in previous Appendix D.

Conditions $(C^{\mathcal{J}}_{++})$ and $(C^{\mathcal{J}}_{--})$ and symmetry (or equivalency the equality of $(C^{\mathcal{J}}_{--})/(C^{\mathcal{J}}_{++})$ or $(C^{\mathcal{J}}_{+-})/(C^{\mathcal{J}}_{+-})$) lead to,

$$ \mu_{\mathcal{J}}((\pm 1, \pm 1, \pm 1)) = \frac{1}{8}, \quad \mu_{\mathcal{J}}((1, \pm 1, -1)) = \frac{3}{8}. \qquad (E1) $$

Expliciting the vector $\mu$ and combining $(C^{\mathcal{J}}_{--})$ and $(C^{\mathcal{J}}_{+-})$ fix the ratio $w_{\text{eq}}$ and $w_{\text{rel}}$ and recovers (10),

$$ \begin{gathered} 8\pi^{S}_{--}(0^{+}) = \pi^{S}_{+-}(0^{+}) \\ \frac{w_{\text{eq}}}{L} - \frac{w_{\text{rel}}}{\mathcal{N}_{\gamma}}[8\text{ch}(\lambda\frac{L}{2}) + 8\sqrt{2}\text{sh}(\lambda\frac{L}{2})] = \frac{w_{\text{eq}}}{4L} + \frac{w_{\text{rel}}}{\mathcal{N}_{\gamma}}[4\text{ch}(\lambda\frac{L}{2}) + 2\sqrt{2}\text{sh}(\lambda\frac{L}{2})] \qquad (E2) \\ w_{\text{eq}} = w_{\text{rel}}\omega L \left[ \frac{2\sqrt{2}}{\text{th}(\omega L\sqrt{2})} + \frac{10}{3} \right] \end{gathered} $$

## Appendix F: Weight determination

We set the values of the weights $w_B$, $w_S$, $w_J$ through the conditions involving the boundary terms in $(C_{S_k})$ (e.g. $(C^{S}_{++})$),

$$ 3\frac{w_B}{2L^2} = \frac{\omega}{4L}w_S w_{\text{eq}} \rightarrow w_B = \frac{\omega L w_{\text{eq}}}{6}w_S, \qquad (F1) $$

and the boundary terms in $(C_{J_k})$ (e.g. $(C^{\mathcal{J}}_{--})$),

$$ w_S \left( \frac{w_{\text{eq}}}{2L} + w_{\text{rel}}\omega \frac{\sqrt{2}}{4\text{th}(\omega L\sqrt{2})} \right) = w_J \frac{5\omega}{8} \rightarrow w_{\text{eq}}(3 + 2\omega L) - 2\omega L = 3\omega L \frac{w_J}{w_S}, \qquad (F2) $$

and the normalisation condition,

$$ w_B + w_S + w_J = 1. \qquad (F3) $$

We obtain,

$$ \left( \frac{\omega L w_{\text{eq}}}{6} + 1 + \frac{w_{\text{eq}}(3 + 2\omega L) - 2\omega L}{3\omega L} \right) w_S = 1 \qquad (F4) $$

That fix the weights to,

$$ w_B = \frac{w_{\text{eq}}(\omega L)^2}{2\omega L + w_{\text{eq}}(6 + 4\omega L + (\omega L)^2)} \qquad (F5) $$

$$ w_S = \frac{6\omega L}{2\omega L + w_{\text{eq}}(6 + 4\omega L + (\omega L)^2)} \qquad (F6) $$

$$ w_J = \frac{w_{\text{eq}}(6 + 4\omega L) - 4\omega L}{2\omega L + w_{\text{eq}}(6 + 4\omega L + (\omega L)^2)} \qquad (F7) $$