---
title: Three-State Run-and-Tumble Dynamics
url: https://www.emergentmind.com/topics/three-state-run-and-tumble-dynamics
type: topic
---

# Three-State Run-and-Tumble Dynamics

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 [1910.10083][2206.00992][2508.10213][2502.01257][2509.08945].

## 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 \(\boldsymbol\rho=(\rho_+,\rho_0,\rho_-)\) [1910.10083][2508.10213][2206.00992].

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 [1305.6475][1801.03721][2305.03663].

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 [2502.01257][2509.08945][1211.3272].

## 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
\[
+1 \leftrightarrow 0 \leftrightarrow -1,
\]
with \(+1\to0\) and \(-1\to0\) at rate \(\gamma\), and \(0\to\pm1\) at rate \(\gamma/2\). The resulting stationary position density \(P(x)\) has finite support \(|x|\le v_0/\mu\), and the central novelty relative to the standard two-state RTP is that the zero-velocity state produces a third deterministic attractor at \(x=0\) [1910.10083].

In the cell-cycle model, the states are number densities rather than single-particle labels:
\[
\boldsymbol\rho(x,t)=(\rho_+,\rho_0,\rho_-).
\]
The mobile states advect with speeds \(v_+\) and \(-v_-\), while \(\rho_0\) is sedentary. The kinetic processes are exchange of running direction at rate \(\lambda_e\), settling at rate \(\lambda_s\), doubling of a sedentary cell into one right- and one left-mover at rate \(\lambda_d\), and death of motile cells at rate \(\mu\). What distinguishes this model is not only the presence of a third state, but the functional division of roles: transport is carried by \(\rho_\pm\), whereas proliferation occurs only in \(\rho_0\) [2206.00992].

In the non-reciprocal trichotomous-velocity model, the particle obeys
\[
\frac{d}{dt}x(t)=v_{\rm sp}(t)+\xi(t),
\]
with \(v_{\rm sp}(t)\in\{-v,0,+v\}\) and Gaussian white noise \(\xi(t)\). The six transition rates \(\vartheta_n^m\) need not satisfy reciprocity, so \(\vartheta_-^+\neq\vartheta_+^-\), \(\vartheta_-^0\neq\vartheta_0^-\), and \(\vartheta_0^+\neq\vartheta_+^0\). This explicitly breaks detailed balance in the internal dynamics and produces transport behaviors that do not occur in reciprocal models [2508.10213].

## 3. Exact stationary structure and transport laws

The simplest exact structural result is the harmonic-trap shape transition controlled by
\[
\beta=\frac{\gamma}{\mu}.
\]
For \(\beta<1\), the stationary density is double-concave and diverges algebraically both at the origin and at the boundaries with exponent \(\beta-1\). For \(\beta>1\), it becomes convex, has a finite single peak at the origin, and vanishes at the boundaries. At \(\beta=1\), 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 \(\dot x=-\mu x\) has a stable point at \(x=0\) [1910.10083].

The non-reciprocal three-state model admits exact long-time transport coefficients. The stationary internal probabilities \(p_n^{\rm st}\) determine the effective drift
\[
v_{\rm eff}=v(p_+^{\rm st}-p_-^{\rm st}),
\]
and the asymptotic dynamics reduces to an advection–diffusion equation with
\[
D_{\rm eff}=v^2\frac{\vartheta_0^-+\vartheta_0^+}{\lambda}.
\]
A central result is that broken detailed balance does not automatically imply long-time drift: there exists a manifold in rate space where \(p_+^{\rm st}=p_-^{\rm st}\), equivalently \(v_{\rm drift}=0\), and the long-time motion is purely diffusive despite internal irreversibility. The same model shows universal short-time ballistic growth, \(\langle x^2(t)\rangle_{\rm act}\approx \frac{2}{3}v^2 t^2\), while the kurtosis approaches \(3\) asymptotically [2508.10213].

The cell-cycle model alters transport more strongly at early times than at late times. For an initially settled cell, \(\boldsymbol\rho(x,0)=(0,\delta(x),0)\), the full-population mean displacement starts as \(O(t^2)\),
\[
\langle x(t)-x_0\rangle=\lambda_d v_d t^2+\cdots,
\]
and the full-population MSD also starts as \(O(t^2)\),
\[
\langle (x(t)-x_0)^2\rangle = 2D\lambda_d t^2+\cdots.
\]
By contrast, the settled population shows the distinctive early-time super-ballistic scaling \(\sim t^3\):
\[
\langle x(t)-x_0\rangle_0 \sim \frac13 \lambda_s\lambda_d v_d\, t^3,\qquad
\langle (x(t)-x_0)^2\rangle_0 \sim \frac23 D\lambda_s\lambda_d\, t^3.
\]
This follows from the sequential kinetics of doubling, motion, and re-settling. At long times, if \(v_+=v_-\), the full-population MSD is diffusive; if \(v_+\neq v_-\), it crosses over to ballistic \(t^2\) growth [2206.00992].

## 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 \(\sigma_i\in\{0,1\}\): \(\sigma_i=0\) is a running state with propulsion \(f_0\hat{\mathbf e}_i\), and \(\sigma_i=1\) is a tumbling state with propulsion turned off and a random torque \(\mathbf t_r\). Runs are interrupted by tumble events occurring as a Poisson process with rate \(\lambda\), and each tumble lasts a constant time \(\tau\). The exact free-theory propagator is then used to fit the interacting bath, yielding effective parameters \(v_{\rm eff}(q,\rho)\), \(\lambda_{\rm eff}(q,\rho)\), and \(\tau_{\rm eff}(q,\rho)\) [1305.6475].

The same study motivates a generalized third state through interactions. Collisions increase \(\lambda_{\rm eff}\), decrease \(v_{\rm eff}\), and reduce \(\tau_{\rm eff}\), with \(\tau_{\rm eff}(q,\rho)\to0\) as density increases. The speed distribution \(P(v)\) evolves from bimodal at \(\rho=0\) to continuous support over \(0<v<v_{\max}\). 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 \(q\)-dependent parameter renormalization [1305.6475].

The path-integral formulation of bacterial motion generalizes the same idea at the trajectory level. It introduces arbitrary run and tumble survival functions \(f(T)\) and \(g(\tau)\), a tumble-dependent angular kernel \(G_\tau(\theta\mid\theta_0)\), finite tumble durations, directional persistence, and gamma-distributed run intervals
\[
\xi_n(t)=\frac{R^n}{\Gamma(n)}t^{n-1}e^{-Rt}H(t),
\]
where \(n\) is the number of hidden steps in a single \(CW \leftrightarrow CCW\) switch. For \(n=2\), 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 [1801.03721].

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 \(\phi=0,\pi/2,\pi,3\pi/2\). Rare barrier crossings produce \(\pm 90^\circ\) turns, with effective one-sided rate
\[
\lambda=\tau^{-1},
\]
total turning rate \(2\lambda\), and persistence scales
\[
T=(2\lambda)^{-1},\qquad \ell=\frac{v_0}{2\lambda}.
\]
The paper does not formulate a discrete three-state chain, but it makes explicit that the long-time statistics collapse when rescaled by \(T\) and \(\ell\), 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 [2305.03663].

## 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 \(\beta(t)\) and centroid speed \(V(t)\), with an arbitrary threshold \(\beta=60^\circ\) for run/tumble segmentation. Yet the phase-space description in \((\theta_+,\theta_-)\) reveals richer structure: stable fixed points or fixed lines for running, noise-driven escape from them, and large-\(\beta\) reorientation events. For \(\alpha=90^\circ\), the running set is semi-stable in \(\theta_-\) and neutral in \(\theta_+\), 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 [2502.01257].

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 \(\sigma_i\in\{-1,+1\}\), but the exact steady state decomposes into three geometric sectors:
\[
\text{bulk/free},\qquad \text{separated edge},\qquad \text{fully jammed vertex}.
\]
The invariant measure is
\[
\pi = w_{\mathcal B}\pi_{\mathcal B} +\frac13\sum_{k=1}^3 \Big[ w_{\mathcal S}\pi_{\mathcal S_k} + w_{\mathcal J}\pi_{\mathcal J_k} \Big],
\]
with explicit structural weights \(w_{\mathcal B}\), \(w_{\mathcal S}\), and \(w_{\mathcal J}\). The resulting regime sequence as \(\omega L\) increases is jammed \(\to\) separated \(\to\) free, and the separated sector dominates for \(0.759\lesssim \omega L\lesssim 6.16\). Here “three-state” refers not to internal variables but to the exact decomposition of the invariant measure into three support sectors [2509.08945].

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 \(2\pi n\) and \((2n+1)\pi\). A plausible implication is that synchronized run, out-of-phase run, and tumble can serve as an inferred three-state coarse-graining [1211.3272].

## 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 \(-\kappa \rho_0\), while settled cells experience self-repulsion through \(+\kappa_0\rho_0\). Linear stability then yields a finite-wavelength pattern-forming mechanism: attraction creates aggregation, repulsion regularizes it, and numerical simulations produce stationary waves when \(v_+=v_-\) and traveling waves when \(v_+\neq v_-\). A notable observation is that, in the final state, all three densities move together with a common wave speed [2206.00992].

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 \(q\), into effective parameters of the free finite-\(\tau\) theory. The mapping is explicitly scale-dependent: \(\lambda_{\rm eff}\), \(v_{\rm eff}\), and \(\tau_{\rm eff}\) depend on both \(\rho\) and \(q\). This implies that a single \(q\)-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 [1305.6475].

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 \(0\) can be added naturally. The continuum run-and-tumble limit emerges only when the persistence length is large compared with the lattice spacing, \(d/\alpha\gg1\), 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 [1012.0786].

## 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 \(k_{R\to I}\), \(k_{I\to T}\), \(k_{T\to R}\), and do not derive a closed Kramers-like formula for \(\lambda\). The intermediate state is visible only implicitly through continuous angular variables and threshold crossings [2502.01257].

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 \(q\)-dependent. At high density and low \(q\), 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 [1305.6475].

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 \(N\)-body problem is open. The paper explicitly raises the question of whether a separated regime survives in the large-\(N\) 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 [2509.08945].

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 \(0\) 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 [1801.03721][1012.0786][1910.10083][2206.00992][2508.10213].

Source: https://www.emergentmind.com/topics/three-state-run-and-tumble-dynamics