Three-State Run-and-Tumble Dynamics

• Three-state run-and-tumble dynamics is an active-matter framework that extends binary models by incorporating a sedentary or intermediate state, enhancing transport and reorientation analysis.
• It unifies explicit Markov chain formulations with emergent sector decompositions, linking one-particle theories, finite-duration tumbles, and density-dependent effects.
• The approach reveals unique phenomena such as central attractors in harmonic traps, exact transport coefficients, and scale-dependent collision impacts in complex systems.

Three-state run-and-tumble dynamics denotes a class of active-matter descriptions in which motility cannot be reduced to a binary run/tumble telegraph process. In the recent literature, the term covers both models with three explicit internal motility states—most commonly $\{-v,0,+v\}$ or right-moving/sedentary/left-moving kinetics—and systems whose underlying continuous or many-body dynamics admit a robust tripartite coarse-graining into coherent running, an intermediate escape or inactive sector, and reorientation. This broader usage connects exact one-particle theories, transport–proliferation kinetics, finite-duration tumbles, anisotropic turn processes, and emergent switching induced by coupling, steric interactions, or jamming (Basu et al., 2019, Breoni et al., 2022, Romo-Cruz et al., 13 Aug 2025, Paramanick et al., 3 Feb 2025, Hahn et al., 10 Sep 2025).

1. Conceptual scope and taxonomy

The most literal meaning of a three-state run-and-tumble process is a Markovian internal dynamics with three discrete motility states. In one dimension, the canonical state set is $\{-v,0,+v\}$, where the zero-velocity sector is either a genuine rest state, a tumbling state, or a sedentary proliferative state. This structure appears explicitly in the harmonic-trap model with internal states $\sigma_x(t)\in\{+1,0,-1\}$, in the non-reciprocal trichotomous-velocity model with six independent transition rates $\vartheta_n^m$, and in the cell-cycle model with densities $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$ (Basu et al., 2019, Romo-Cruz et al., 13 Aug 2025, Breoni et al., 2022).

A second use of the term is coarse-grained rather than literal. Several systems remain formally two-state at the microscopic level but display a third dynamical sector once finite tumble duration, hidden sequential switching, collision-modified motion, or metastable escape dynamics are resolved. The interacting bacterial-bath study keeps only running and tumbling microscopically but motivates a third collision-modified motility sector through scale-dependent effective parameters. The path-integral treatment of E. coli motion allows multi-step hidden transitions and gamma-distributed waiting times, which can be embedded in a finite-state Markov chain. The anisotropic run-and-tumble-turn model is continuous in angle but naturally decomposes into run, turn, and renewed run. These are not explicit three-state master equations in the papers’ formal sense, but they support a technically meaningful generalized three-state interpretation (Paoluzzi et al., 2013, Renadheer et al., 2018, Loewe et al., 2023).

A third usage is emergent and geometric. In coupled robots, jammed RTPs, and hydrodynamic Chlamydomonas models, the underlying variables are continuous or many-body, yet the observed dynamics partitions into three sectors such as coherent run, escape or separated configurations, and tumble or full jamming. This suggests that “three-state” can refer not only to internal-state cardinality but also to the minimal number of dynamically distinct sectors needed to organize the invariant measure or trajectory ensemble (Paramanick et al., 3 Feb 2025, Hahn et al., 10 Sep 2025, Bennett et al., 2012).

2. Explicit three-state formulations

Three representative formulations organize the explicit internal-state literature.

Formulation	State set	Distinctive role of the third state
Trapped RTP	$\{+v_0,0,-v_0\}$	Rest state creates a central attractor in a harmonic trap
Cell-cycle RT	$(\rho_+,\rho_0,\rho_-)$	Sedentary state is the unique site of reproduction
Non-reciprocal RT	$\{-v,0,+v\}$	Zero-velocity state enables irreversible internal cycles

In the harmonic-trap model, the overdamped dynamics is

$\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$

with $\sigma_x(t)\in\{+1,0,-1\}$. The allowed internal transitions are only

$\{-v,0,+v\}$0

with $\{-v,0,+v\}$1 and $\{-v,0,+v\}$2 at rate $\{-v,0,+v\}$3, and $\{-v,0,+v\}$4 at rate $\{-v,0,+v\}$5. The resulting stationary position density $\{-v,0,+v\}$6 has finite support $\{-v,0,+v\}$7, and the central novelty relative to the standard two-state RTP is that the zero-velocity state produces a third deterministic attractor at $\{-v,0,+v\}$8 (Basu et al., 2019).

In the cell-cycle model, the states are number densities rather than single-particle labels: $\{-v,0,+v\}$9 The mobile states advect with speeds $\sigma_x(t)\in\{+1,0,-1\}$0 and $\sigma_x(t)\in\{+1,0,-1\}$1, while $\sigma_x(t)\in\{+1,0,-1\}$2 is sedentary. The kinetic processes are exchange of running direction at rate $\sigma_x(t)\in\{+1,0,-1\}$3, settling at rate $\sigma_x(t)\in\{+1,0,-1\}$4, doubling of a sedentary cell into one right- and one left-mover at rate $\sigma_x(t)\in\{+1,0,-1\}$5, and death of motile cells at rate $\sigma_x(t)\in\{+1,0,-1\}$6. What distinguishes this model is not only the presence of a third state, but the functional division of roles: transport is carried by $\sigma_x(t)\in\{+1,0,-1\}$7, whereas proliferation occurs only in $\sigma_x(t)\in\{+1,0,-1\}$8 (Breoni et al., 2022).

In the non-reciprocal trichotomous-velocity model, the particle obeys

$\sigma_x(t)\in\{+1,0,-1\}$9

with $\vartheta_n^m$0 and Gaussian white noise $\vartheta_n^m$1. The six transition rates $\vartheta_n^m$2 need not satisfy reciprocity, so $\vartheta_n^m$3, $\vartheta_n^m$4, and $\vartheta_n^m$5. This explicitly breaks detailed balance in the internal dynamics and produces transport behaviors that do not occur in reciprocal models (Romo-Cruz et al., 13 Aug 2025).

3. Exact stationary structure and transport laws

The simplest exact structural result is the harmonic-trap shape transition controlled by

$\vartheta_n^m$6

For $\vartheta_n^m$7, the stationary density is double-concave and diverges algebraically both at the origin and at the boundaries with exponent $\vartheta_n^m$8. For $\vartheta_n^m$9, it becomes convex, has a finite single peak at the origin, and vanishes at the boundaries. At $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$0, it shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries. The zero-velocity state is decisive here: unlike the two-state RTP, it creates a central singularity because $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$1 has a stable point at $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$2 (Basu et al., 2019).

The non-reciprocal three-state model admits exact long-time transport coefficients. The stationary internal probabilities $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$3 determine the effective drift

$\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$4

and the asymptotic dynamics reduces to an advection–diffusion equation with

$\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$5

A central result is that broken detailed balance does not automatically imply long-time drift: there exists a manifold in rate space where $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$6, equivalently $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$7, and the long-time motion is purely diffusive despite internal irreversibility. The same model shows universal short-time ballistic growth, $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$8, while the kurtosis approaches $\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)$9 asymptotically (Romo-Cruz et al., 13 Aug 2025).

The cell-cycle model alters transport more strongly at early times than at late times. For an initially settled cell, $\{+v_0,0,-v_0\}$0, the full-population mean displacement starts as $\{+v_0,0,-v_0\}$1,

$\{+v_0,0,-v_0\}$2

and the full-population MSD also starts as $\{+v_0,0,-v_0\}$3,

$\{+v_0,0,-v_0\}$4

By contrast, the settled population shows the distinctive early-time super-ballistic scaling $\{+v_0,0,-v_0\}$5: $\{+v_0,0,-v_0\}$6 This follows from the sequential kinetics of doubling, motion, and re-settling. At long times, if $\{+v_0,0,-v_0\}$7, the full-population MSD is diffusive; if $\{+v_0,0,-v_0\}$8, it crosses over to ballistic $\{+v_0,0,-v_0\}$9 growth (Breoni et al., 2022).

4. Finite-duration tumbles, hidden phases, and generalized three-state embeddings

A major route from two-state to three-state RT dynamics is the resolution of finite-duration or hidden intermediate phases. In the bacterial-bath study, the microscopic dynamics uses only a binary state variable $(\rho_+,\rho_0,\rho_-)$0: $(\rho_+,\rho_0,\rho_-)$1 is a running state with propulsion $(\rho_+,\rho_0,\rho_-)$2, and $(\rho_+,\rho_0,\rho_-)$3 is a tumbling state with propulsion turned off and a random torque $(\rho_+,\rho_0,\rho_-)$4. Runs are interrupted by tumble events occurring as a Poisson process with rate $(\rho_+,\rho_0,\rho_-)$5, and each tumble lasts a constant time $(\rho_+,\rho_0,\rho_-)$6. The exact free-theory propagator is then used to fit the interacting bath, yielding effective parameters $(\rho_+,\rho_0,\rho_-)$7, $(\rho_+,\rho_0,\rho_-)$8, and $(\rho_+,\rho_0,\rho_-)$9 (Paoluzzi et al., 2013).

The same study motivates a generalized third state through interactions. Collisions increase $\{-v,0,+v\}$0, decrease $\{-v,0,+v\}$1, and reduce $\{-v,0,+v\}$2, with $\{-v,0,+v\}$3 as density increases. The speed distribution $\{-v,0,+v\}$4 evolves from bimodal at $\{-v,0,+v\}$5 to continuous support over $\{-v,0,+v\}$6. This suggests that, on coarse-grained scales, the clean run/tumble decomposition is replaced by run, tumble, and a collision-modified interruption state. The paper does not formulate that state explicitly, but the empirical signal for it is the broad low-speed sector and the $\{-v,0,+v\}$7-dependent parameter renormalization (Paoluzzi et al., 2013).

The path-integral formulation of bacterial motion generalizes the same idea at the trajectory level. It introduces arbitrary run and tumble survival functions $\{-v,0,+v\}$8 and $\{-v,0,+v\}$9, a tumble-dependent angular kernel $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$0, finite tumble durations, directional persistence, and gamma-distributed run intervals

$\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$1

where $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$2 is the number of hidden steps in a single $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$3 switch. For $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$4, this is exactly the sum of two exponential stages, so a three-state Markov embedding becomes natural. The paper does not write such a generator explicitly, but it supplies the formal machinery needed for it (Renadheer et al., 2018).

The anisotropic run-and-tumble-turn model provides a different generalization. Here the particle angle obeys a continuous SDE in a four-well angular potential, with preferred directions $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$5. Rare barrier crossings produce $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$6 turns, with effective one-sided rate

$\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$7

total turning rate $\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$8, and persistence scales

$\dot x(t)=-\mu x(t)+v_0\,\sigma_x(t),$9

The paper does not formulate a discrete three-state chain, but it makes explicit that the long-time statistics collapse when rescaled by $\sigma_x(t)\in\{+1,0,-1\}$0 and $\sigma_x(t)\in\{+1,0,-1\}$1, and that the hydrodynamic limit becomes diffusive before it becomes isotropic. This is a physically grounded run–turn–run structure rather than a literal three-state master equation (Loewe et al., 2023).

5. Emergent three-sector dynamics from coupling and geometry

In coupled self-propelled robots, three-state organization appears even though the individual units are not run-and-tumble particles themselves. Each free unit is an overdamped active Brownian robot; RT-like switching emerges only after mechanical coupling by a rigid rod attached at off-center pivots. The operational analysis uses a two-state vocabulary—run versus tumble—based on the pair-angle $\sigma_x(t)\in\{+1,0,-1\}$2 and centroid speed $\sigma_x(t)\in\{+1,0,-1\}$3, with an arbitrary threshold $\sigma_x(t)\in\{+1,0,-1\}$4 for run/tumble segmentation. Yet the phase-space description in $\sigma_x(t)\in\{+1,0,-1\}$5 reveals richer structure: stable fixed points or fixed lines for running, noise-driven escape from them, and large-$\sigma_x(t)\in\{+1,0,-1\}$6 reorientation events. For $\sigma_x(t)\in\{+1,0,-1\}$7, the running set is semi-stable in $\sigma_x(t)\in\{+1,0,-1\}$8 and neutral in $\sigma_x(t)\in\{+1,0,-1\}$9, so the system can slide along a marginal manifold before escaping. This strongly supports a tripartite reinterpretation into coherent run, escape or misaligned metastable state, and tumble or reorientation, even though the paper does not formulate a discrete three-state stochastic process (Paramanick et al., 3 Feb 2025).

A formally exact many-body realization of emergent three-sector structure is provided by the three-particle jamming problem on a ring. The microscopic particles still have only two velocity states $\{-v,0,+v\}$00, but the exact steady state decomposes into three geometric sectors: $\{-v,0,+v\}$01 The invariant measure is

$\{-v,0,+v\}$02

with explicit structural weights $\{-v,0,+v\}$03, $\{-v,0,+v\}$04, and $\{-v,0,+v\}$05. The resulting regime sequence as $\{-v,0,+v\}$06 increases is jammed $\{-v,0,+v\}$07 separated $\{-v,0,+v\}$08 free, and the separated sector dominates for $\{-v,0,+v\}$09. Here “three-state” refers not to internal variables but to the exact decomposition of the invariant measure into three support sectors (Hahn et al., 10 Sep 2025).

The hydrodynamic Chlamydomonas model is similar in spirit but less explicit. Its fundamental variables are the cell position, orientation, and the two flagellar phases, with noise entering the phase-dependent driving forces. The paper identifies run intervals as occupancy of synchronized or oscillatory phase-difference branches and tumbles as phase slips accompanied by sharp jumps in orientation. It does not formulate a three-state master equation, but it does display multiple deterministic oscillatory regimes near $\{-v,0,+v\}$10 and $\{-v,0,+v\}$11. A plausible implication is that synchronized run, out-of-phase run, and tumble can serve as an inferred three-state coarse-graining (Bennett et al., 2012).

6. Interactions, continuum limits, and collective organization

Three-state structure becomes especially consequential when coupled to interactions or population kinetics. In the cell-cycle model, moving cells are attracted toward regions with high settled-cell density through $\{-v,0,+v\}$12, while settled cells experience self-repulsion through $\{-v,0,+v\}$13. Linear stability then yields a finite-wavelength pattern-forming mechanism: attraction creates aggregation, repulsion regularizes it, and numerical simulations produce stationary waves when $\{-v,0,+v\}$14 and traveling waves when $\{-v,0,+v\}$15. A notable observation is that, in the final state, all three densities move together with a common wave speed (Breoni et al., 2022).

The interacting bacterial bath addresses collective effects differently. Rather than introducing a third microscopic state, it shows that steric interactions can be absorbed, over a range of densities and at fixed observation scale $\{-v,0,+v\}$16, into effective parameters of the free finite-$\{-v,0,+v\}$17 theory. The mapping is explicitly scale-dependent: $\{-v,0,+v\}$18, $\{-v,0,+v\}$19, and $\{-v,0,+v\}$20 depend on both $\{-v,0,+v\}$21 and $\{-v,0,+v\}$22. This implies that a single $\{-v,0,+v\}$23-independent Markovian state model is insufficient at high enough density or long enough scale, and it provides a quantitative reason why a coarse-grained collision state or a memory kernel becomes attractive (Paoluzzi et al., 2013).

The lattice RT framework supplies the corresponding microscopic template for explicit multi-species extensions. In its basic one-dimensional form the model is two-state, with right- and left-moving particles hopping and tumbling instantaneously. But the formulation is explicitly built from multiple species and transmutation, so a third non-propagating state $\{-v,0,+v\}$24 can be added naturally. The continuum run-and-tumble limit emerges only when the persistence length is large compared with the lattice spacing, $\{-v,0,+v\}$25, and the same logic would govern a finite-duration three-state extension. The paper also shows that kernel isotropy is crucial: isotropic kernels support a free-energy mapping, whereas anisotropic kernels produce strong run-length dependence and continuum failure (Thompson et al., 2010).

7. Limitations, ambiguities, and open directions

Despite the breadth of the recent literature, no single canonical three-state RT theory has emerged. Some papers study genuine three-state internal Markov chains; others study formally two-state systems whose effective description becomes tripartite only after coarse graining. This ambiguity is conceptually productive but technically important. In the coupled-robot work, for example, the authors do not formulate a discrete-state master equation with three states, do not measure separate rates such as $\{-v,0,+v\}$26, $\{-v,0,+v\}$27, $\{-v,0,+v\}$28, and do not derive a closed Kramers-like formula for $\{-v,0,+v\}$29. The intermediate state is visible only implicitly through continuous angular variables and threshold crossings (Paramanick et al., 3 Feb 2025).

A related limitation is scale dependence. In the interacting bacterial bath, the effective mapping onto free RT dynamics works only up to moderate density and is explicitly $\{-v,0,+v\}$30-dependent. At high density and low $\{-v,0,+v\}$31, the fit quality deteriorates, indicating dynamical correlations beyond what any effective free-particle description can capture. This means that some apparent three-state phenomenology is really a proxy for non-Markovianity or unresolved collective modes rather than evidence for a universal third microscopic state (Paoluzzi et al., 2013).

Many-body exact theory remains at an early stage. The three-body jamming solution resolves the invariant measure exactly and suggests a hierarchy of jammed, separated, and free sectors, but the general $\{-v,0,+v\}$32-body problem is open. The paper explicitly raises the question of whether a separated regime survives in the large-$\{-v,0,+v\}$33 limit, whether one macroscopic cluster dominates or multiple clusters coexist, and whether the separated regime occupies a finite interval of activity or collapses to a single point (Hahn et al., 10 Sep 2025).

The most systematic route toward unification is likely to combine the formal tools already available. The path-integral framework accommodates arbitrary waiting times, finite tumble durations, directional persistence, and hidden multi-step switching; the lattice transmutation framework provides an exact microscopic scaffold for adding a finite-residence $\{-v,0,+v\}$34 state; and the recent explicit three-state models provide exact observables against which any generalized theory must be benchmarked. A plausible implication is that future work will distinguish more sharply between three-state internal dynamics, three-stage cycle kinetics, and three-sector coarse-grained organization, rather than treating them as interchangeable labels (Renadheer et al., 2018, Thompson et al., 2010, Basu et al., 2019, Breoni et al., 2022, Romo-Cruz et al., 13 Aug 2025).