Run-and-Tumble Particles (RTPs) Overview

• Run-and-tumble particles (RTPs) are self-propelled active particles that alternate between persistent runs and random tumbles, establishing nonequilibrium steady states.
• They are modeled using kinetic equations in both continuum and lattice frameworks to investigate phenomena such as motility-induced phase separation, jamming, and anomalous diffusion.
• Coarse-graining methods reveal a hydrodynamic equivalence with active Brownian particles while highlighting distinct transport, confinement, and clustering behaviors under strong persistence.

Searching arXiv for recent and foundational papers on run-and-tumble particles to ground the article in published work. Run-and-tumble particles (RTPs) are active particles that self-propel at fixed speed along an internal orientation during persistent runs and then undergo stochastic reorientation events, or tumbles, that reset or randomize that orientation. In the minimal kinetic description, an RTP moves with propulsion velocity $v{\bf u}$, with ${\bf u}$ reoriented by a Poisson process of rate $\alpha$, or by analogous lattice updates in discrete-state models (Cates et al., 2012). Across continuum, lattice, interacting, and confined settings, RTPs serve as a canonical nonequilibrium model for persistent transport, motility-induced phase separation (MIPS), anomalous collective diffusion, boundary accumulation, jamming, and rare-event kinetics. A recurrent theme in the literature is that RTPs are microscopically distinct from active Brownian particles (ABPs), yet often share the same leading coarse-grained hydrodynamics in isotropic torque-free settings, while departing sharply under strong confinement, anisotropy, or explicit internal-state structure (Solon et al., 2015).

1. Definition, microscopic dynamics, and canonical formulations

In the standard continuum formulation, the one-particle probability density $\psi({\bf r},{\bf u},t)$ obeys a kinetic equation with self-propulsive advection, optional translational diffusion $D_t$, optional rotational diffusion $D_r$, and tumble loss/gain terms. For a pure RTP one sets $D_r=0$, leaving tumbles at rate $\alpha$ as the sole orientational decorrelation mechanism (Cates et al., 2012). In one dimension, the basic microscopic law is $dX/d\tau=v\sigma(\tau)$, where $\sigma(\tau)\in\{-1,+1\}$ flips as a Poisson process, while in two dimensions the orientation is commonly parameterized by an angle ${\bf u}$0 with ${\bf u}$1 and tumbles reset ${\bf u}$2 uniformly on ${\bf u}$3 (Grange et al., 2020, Santra et al., 2022).

Lattice RTPs replace continuous orientations by discrete nearest-neighbor headings. On a two-dimensional square lattice, a representative model assigns each particle an internal orientation ${\bf u}$4, enforces hard-core exclusion ${\bf u}$5, and combines oriented hopping, lateral hopping, and tumbling (Saha et al., 2024). In that model, a particle hops along its current orientation with unit rate, to each of the other nearest neighbors with rate ${\bf u}$6, and tumbles with rate ${\bf u}$7. The parameter ${\bf u}$8 acts as positional diffusion, and the paper emphasizes that this ingredient is essential for stable MIPS in that lattice family (Saha et al., 2024).

A useful generalization is the one-dimensional PDMP framework, where the state is ${\bf u}$9, deterministic motion is $\alpha$0, and the internal state jumps with rates $\alpha$1. This formulation covers RTPs in external potentials, finite-state internal dynamics, and resetting, and makes the invariant-measure problem amenable to local regularity analysis (Hahn, 2 Jun 2025). Another exact continuous-space PDMP treatment studies two interacting RTPs on a one-dimensional ring by reducing the problem to the interparticle distance $\alpha$2 plus an internal pair state, with jamming appearing as a boundary condition in configuration space (Hahn et al., 2023).

These model classes share the same physical core: bounded-speed self-propulsion, orientational persistence, stochastic reorientation, and nonequilibrium steady states even in simple confining geometries. What varies across formulations is the representation of orientation, the role of exclusion and interaction, and whether one studies single-particle transport, few-body jamming, or collective phases.

2. Coarse-graining, hydrodynamics, and relation to active Brownian particles

The central coarse-grained result for isotropic RTPs is that orientational moments relax on a finite persistence time and can be adiabatically eliminated at long times and large scales. Decomposing $\alpha$3 into scalar density $\alpha$4, polarization $\alpha$5, and higher harmonics yields, to leading order in gradients,

$\alpha$6

and therefore a diffusion-drift density equation

$\alpha$7

with

$\alpha$8

For pure RTPs, $\alpha$9, so the orientational relaxation rate is simply $\psi({\bf r},{\bf u},t)$0 [(Cates et al., 2012); (Solon et al., 2015)].

This leads to the well-known ABP/RTP mapping

$\psi({\bf r},{\bf u},t)$1

valid for isotropic motility parameters that depend on position or density but not on $\psi({\bf r},{\bf u},t)$2, and at leading order in the gradient expansion. Under that mapping, ABPs and RTPs share the same large-scale diffusion-drift equation, the same fluctuating hydrodynamics, and the same local MIPS criterion (Cates et al., 2012). The equivalence is therefore hydrodynamic rather than microscopic: the full angular hierarchies differ term by term, but the leading conserved density dynamics depends only on the combined orientational relaxation rate $\psi({\bf r},{\bf u},t)$3 (Solon et al., 2015).

The same coarse-graining yields a fluctuating density equation of the form

$\psi({\bf r},{\bf u},t)$4

with white noise $\psi({\bf r},{\bf u},t)$5 (Cates et al., 2012). When $\psi({\bf r},{\bf u},t)$6 is density dependent and local, $\psi({\bf r},{\bf u},t)$7, this admits an equilibrium-like free-energy representation at leading order, with

$\psi({\bf r},{\bf u},t)$8

and spinodal instability criterion

$\psi({\bf r},{\bf u},t)$9

which is the standard local MIPS condition in both RTP and ABP hydrodynamics (Solon et al., 2015).

The same papers are careful about where equivalence fails. Strong external potentials, confinement with size comparable to the run length, higher-order gradient terms, orientational torques, and explicitly $D_t$0-dependent motilities all expose the discrete tumble dynamics and spoil the mapping [(Cates et al., 2012); (Solon et al., 2015)]. This distinction between microscopic nonequivalence and hydrodynamic equivalence organizes much of the RTP literature.

3. Single-particle transport, confinement, and stationary structure

In weak external potentials, RTPs admit an effective-equilibrium regime. For two-dimensional sedimentation under $D_t$1, the exact distal steady state is exponential,

$D_t$2

with anisotropic angular distribution

$D_t$3

Expanding for $D_t$4 gives an effective temperature $D_t$5, but the paper emphasizes that beyond weak forcing no single effective temperature survives and RTP sedimentation is genuinely nonequilibrium (Solon et al., 2015).

Strong confinement reveals more distinctively RTP physics. In a harmonic trap, ABPs and RTPs share the same weak-trap Gaussian profile, but for strong persistence $D_t$6, RTPs travel across the trap along straight chords after tumbles at the boundary, leading to bulk density scaling $D_t$7, much larger than for ABPs in the same regime (Solon et al., 2015). In nested two-dimensional mazes, geometry alone separates RTPs from ABPs: ABPs escape outward from the center faster, whereas RTPs reach the center from the rim more easily because a tumble can instantly reverse direction and enter an inner opening. In the circular Matryoshka-like maze, ABPs accumulate in the outermost region at large persistence, while RTPs occupy all zones with nearly equal probability (Khatami et al., 2016).

Several exact one-dimensional and two-dimensional stationary-state problems further illustrate how persistence reshapes invariant measures. For a 1D RTP in an inhomogeneous medium with telegraphic noise $D_t$8 and position-dependent flip rates

$D_t$9

the particle has a normalizable steady state on the infinite line when $D_r$0, with exact density

$D_r$1

This gives exponential confinement for $D_r$2 and Gaussian confinement for $D_r$3, generated purely by inhomogeneous tumbling rather than an external potential (Singh et al., 2020).

Under stochastic resetting in two dimensions, a Poisson reset at rate $D_r$4 to the origin with orientation randomization creates a nonequilibrium stationary state with qualitatively different radial and Cartesian marginals. The radial stationary density tends to a constant, $D_r$5 as $D_r$6, whereas the stationary $D_r$7-marginal diverges logarithmically,

$D_r$8

and both decay at large distance with the same exponent $D_r$9 (Santra et al., 2020). In one-dimensional PDMP language, the regularity of stationary RTP measures under confinement can be characterized without an explicit closed-form density: for a wide class of 1D RTPs in a potential, the invariant density is continuous at high tumble rates and develops divergences at force-balance points at low tumble rates, a phenomenon termed shape transition (Hahn, 2 Jun 2025).

These results collectively show that RTP stationary structure is governed not only by external potentials, but by a broader competition among bounded propulsion, tumbling, inhomogeneous switching, and boundary conditions. A plausible implication is that singular accumulation at force-balance points is a generic nonequilibrium signature of persistent bounded-speed motion, rather than a peculiarity of any single model.

4. Interactions, MIPS, clustering, and criticality

For interacting RTPs, the central coarse-grained mechanism behind MIPS is density-suppressed motility. In the hydrodynamic theory with local $D_r=0$0, particles accumulate where they move more slowly; if $D_r=0$1, this positive feedback destabilizes the homogeneous state and drives phase separation even without attractive forces [(Cates et al., 2012); (Solon et al., 2015)]. Microscopic simulations with programmed $D_r=0$2 and with explicit WCA repulsions show that RTPs and ABPs have closely matching phase diagrams and coexistence densities under the mapping $D_r=0$3, supporting the practical usefulness of the hydrodynamic equivalence for MIPS (Solon et al., 2015).

A more microscopic collective-diffusion picture emerges in interacting lattice RTPs with hard-core exclusion. On a $D_r=0$4-dimensional periodic lattice, the bulk diffusion coefficient $D_r=0$5 obeys a scaling form

$D_r=0$6

controlled by the competition between persistence length $D_r=0$7 and mean gap $D_r=0$8 (Chakraborty et al., 2022). In the strong-persistence regime, the diffusivity varies as $D_r=0$9 over a broad density range, with $\alpha$0 crossing over from $\alpha$1 at low density to $\alpha$2 at high density, leading to nonlinear diffusion and anomalous density spreading (Chakraborty et al., 2022).

On a two-dimensional square lattice with oriented hopping and positional diffusion $\alpha$3, the percolation transition of occupied clusters provides a geometric route to characterize MIPS. At small tumble rate $\alpha$4, increasing $\alpha$5 produces a re-entrant sequence

$\alpha$6

because moderate positional diffusion stabilizes macroscopic clustering while too much diffusion destroys it (Saha et al., 2024). At $\alpha$7, Binder-cumulant crossings locate two transitions at $\alpha$8 and $\alpha$9; at $dX/d\tau=v\sigma(\tau)$0, the corresponding values are $dX/d\tau=v\sigma(\tau)$1 and $dX/d\tau=v\sigma(\tau)$2 (Saha et al., 2024).

The critical behavior is unusual. Along the critical line in the $dX/d\tau=v\sigma(\tau)$3 plane, the exponent ratios vary continuously, for example from $dX/d\tau=v\sigma(\tau)$4 at $dX/d\tau=v\sigma(\tau)$5 toward the two-dimensional $dX/d\tau=v\sigma(\tau)$6-percolation values near small $dX/d\tau=v\sigma(\tau)$7, while the scaling function $dX/d\tau=v\sigma(\tau)$8 remains invariant and matches the Ising-cluster percolation function (Saha et al., 2024). The same group extended this picture to interacting RTPs with explicit nearest-neighbor attraction $dX/d\tau=v\sigma(\tau)$9: at fixed $\sigma(\tau)\in\{-1,+1\}$0 and $\sigma(\tau)\in\{-1,+1\}$1, moderate attraction can suppress a motility-induced phase-separated state, while stronger attraction restores it, producing a reentrant transition in the $\sigma(\tau)\in\{-1,+1\}$2 plane (Bhowmick et al., 5 Jun 2025). The critical line again shows continuously varying exponents, but Binder-cumulant curves plotted against $\sigma(\tau)\in\{-1,+1\}$3 collapse onto the same function as equilibrium interacting percolation, motivating the language of Ising-like super universality (Bhowmick et al., 5 Jun 2025).

The geometric viewpoint is important here. These papers distinguish the percolation transition, defined by the emergence of a macroscopic connected dense cluster, from the MIPS transition, defined by density segregation into high- and low-density regions. Their claim is that the two occur at the same critical line, but with exponents related analogously to Ising versus Ising-cluster percolation (Saha et al., 2024, Bhowmick et al., 5 Jun 2025). This suggests that RTP phase separation can be simultaneously understood as a nonequilibrium thermodynamic instability and as a geometric connectivity transition.

5. Crowding, jamming, collective transport, and current fluctuations

Crowding changes the meaning of persistence. For a single RTP moving in a lattice of hard-core obstacles, the long-time diffusivity can become nonmonotonic in the tumbling probability $\sigma(\tau)\in\{-1,+1\}$4. In the discrete-time model of “Optimized Diffusion of Run-and-Tumble Particles in Crowded Environments,” the mean free run time between obstacle encounters is

$\sigma(\tau)\in\{-1,+1\}$5

for fixed obstacles, independent of $\sigma(\tau)\in\{-1,+1\}$6, by a generalized Kac theorem (Bertrand et al., 2017). The mean trapping time, however, diverges as $\sigma(\tau)\in\{-1,+1\}$7, so maximal persistence is no longer optimal. For static obstacles, the diffusivity has an optimum at $\sigma(\tau)\in\{-1,+1\}$8, with $\sigma(\tau)\in\{-1,+1\}$9, whereas obstacle mobility softens or suppresses this optimum (Bertrand et al., 2017).

At the pair level, jamming can be solved exactly in continuous space. For two RTPs on a one-dimensional ring, the stationary state falls into two universality classes determined by whether probability flows entering and leaving jamming configurations respect a detailed internal-state symmetry (Hahn et al., 2023). In the detailed-jamming class, the invariant measure is a uniform bulk density plus a Dirac mass at the jammed boundary ${\bf u}$00; in the global-jamming class, the jammed boundary injects an internal-state skewness that relaxes through the bulk via a matrix ODE, producing exponential or catenary-like stationary profiles (Hahn et al., 2023). The paper identifies active global balance as the nonequilibrium analog of ordinary global balance, and interprets jamming as a boundary source in configuration space rather than merely a contact event (Hahn et al., 2023).

Collective current fluctuations in interacting one-dimensional RTP systems exhibit another layer of hydrodynamic universality. For hardcore RTPs on a ring and for a long-ranged lattice-gas surrogate, the bond-current variance ${\bf u}$01 shows three regimes: model-dependent short-time behavior, a universal intermediate-time subdiffusive growth ${\bf u}$02, and a long-time diffusive growth ${\bf u}$03 (Chakraborty et al., 2023). After scaling by the bulk diffusion coefficient ${\bf u}$04, mobility ${\bf u}$05, and system size ${\bf u}$06,

${\bf u}$07

both models collapse onto the same scaling function ${\bf u}$08, with asymptotics ${\bf u}$09 for ${\bf u}$10 and ${\bf u}$11 for ${\bf u}$12 (Chakraborty et al., 2023). The same work finds exponentially decaying spatial current correlations with correlation length ${\bf u}$13 at large persistence, and a negative temporal current tail ${\bf u}$14 underlying the ${\bf u}$15 law (Chakraborty et al., 2023).

These transport results correct a common misconception that persistence always improves exploration. In free space, decreasing the tumble rate typically increases the diffusivity; in crowded media or under exclusion, persistence also enhances trapping, jamming, or boundary sticking, so optimal transport often occurs at intermediate reorientation rates (Bertrand et al., 2017, Chakraborty et al., 2023).

6. Exact asymptotics, branching variants, and rare trajectories

A large body of RTP work concerns exact or asymptotic results for propagators and stationary measures. For a free two-dimensional RTP with tumble rate ${\bf u}$16, the late-time propagator can be expanded as a series in ${\bf u}$17; the leading term is Gaussian with diffusivity

${\bf u}$18

while each subleading correction satisfies an inhomogeneous diffusion equation whose source depends on lower-order terms (Santra et al., 2022). The position density in the diffusive sector takes the form

${\bf u}$19

with explicit polynomials ${\bf u}$20 computed recursively (Santra et al., 2022). This makes precise how persistence leaves non-Gaussian fingerprints on top of the asymptotic diffusion law.

Branching or source-driven variants produce different long-time structures. In “Run-and-tumble particles on a line with a fertile site,” crossing the origin activates offspring production during the current run, with creation rate modulated by a fertility kernel ${\bf u}$21 (Grange et al., 2020). The density at the fertile site grows exponentially,

${\bf u}$22

with ${\bf u}$23 determined solely by the fertility kernel and fertility rate through the renewal equation ${\bf u}$24, while the normalized density converges to a stationary profile with a local minimum at the source,

${\bf u}$25

(Grange et al., 2020). This is a solvable branching extension of 1D RTP dynamics rather than a conservative transport model.

Rare-event sampling for RTPs has also become technically accessible. “Transition-path sampling for Run-and-Tumble particles” constructs a trajectory-space Monte Carlo scheme for an irreversible RTP model by defining a backward dynamics with a well-defined path probability and deriving the appropriate Metropolis acceptance rule (Kiechl et al., 2024). Applied to barrier crossing in a two-dimensional double-well potential, the method shows that intermittent run-and-tumble dynamics can yield shorter transition-path times than both passive diffusion and purely active motion, because tumbles help the particle escape long wall-surfing trajectories and favor more direct reactive channels (Kiechl et al., 2024).

Finally, mean-field interacting RTPs with nonconvex attraction-repulsion can have a far richer stationary structure than equilibrium Brownian systems. For one-dimensional RTPs interacting via

${\bf u}$26

the large-${\bf u}$27 stationary density has compact support and undergoes a transition from connected to disconnected support at a renormalized ${\bf u}$28, with multivalued self-consistency below a critical tumbling rate ${\bf u}$29 and even a continuous family of asymmetric stationary states in the disconnected-support phase (Touzo et al., 8 Oct 2025). This is a particularly sharp example of non-uniqueness generated by bounded active noise.

Run-and-tumble particles thus occupy a distinctive place in nonequilibrium statistical mechanics. They are simple enough to admit exact propagators, solvable PDMP reductions, and controlled hydrodynamics, yet rich enough to display confinement without potentials, jamming-induced singularities, MIPS, re-entrant criticality, anomalous collective diffusion, and trajectory-level optimization. A plausible summary of the literature is that RTPs are not merely a minimal active-particle toy model: they are a unifying framework in which persistence, bounded propulsion, and stochastic reorientation can be studied across microscopic, hydrodynamic, geometric, and thermodynamic levels [(Cates et al., 2012); (Solon et al., 2015)].