Run-and-Tumble Particle Dynamics

• Run-and-Tumble Particles (RTPs) are active particles that alternate between constant-speed ballistic runs and stochastic tumbles, forming a core model for non-equilibrium systems.
• Mathematical frameworks such as stochastic differential equations, Chapman–Kolmogorov equations, and PDMPs rigorously capture RTP dynamics and transitions from ballistic to diffusive regimes.
• Applications of RTP models span bacterial motion, Janus particles, molecular motors, and phase separation, providing actionable insights into transport phenomena and critical behaviors.

A run-and-tumble particle (RTP) is an active particle whose dynamics alternate between ballistic “runs” and stochastic “tumbles” that reshuffle its direction, orientation, or propulsion velocity. In one dimension, the canonical RTP switches between right- and left-moving states; in higher dimensions it moves at constant speed along a direction that is renewed at tumble events. RTPs are a foundational model for self-propelled particles such as bacteria or Janus particles, and they also arise in descriptions of molecular motors, microtubules, and cytonemes (Hahn et al., 2023, Bressloff, 2021). Across single-particle, interacting, and boundary-driven settings, RTPs provide an analytically tractable nonequilibrium framework in which ballistic persistence, stochastic switching, resetting, jamming, and phase separation can all be studied explicitly (Malakar et al., 2017, Guillin et al., 2024).

1. Canonical definitions and mathematical representations

In the standard one-dimensional formulation, the RTP position obeys

$\frac{dx}{dt}=v\,\sigma(t)+\sqrt{2D}\,\eta(t),$

where $\sigma(t)=\pm 1$ is a dichotomous variable that flips at rate $\gamma$, $v$ is the propulsion speed, and $D$ is an additional diffusion constant when translational noise is included (Malakar et al., 2017). A related formulation treats the velocity state as a two-state Markov chain $n(t)\in\{0,1\}$, giving

$\frac{dX}{dt}=v[1-2n(t)],$

with space-dependent switching rates $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$ in inhomogeneous environments (Bressloff, 2021).

A broader class of RTPs redraws the propulsion velocity at each tumble. In one dimension, one such model assumes i.i.d. Gaussian velocities $v_i$ and exponentially distributed run times $\tau_i$, so that after $\sigma(t)=\pm 1$0 runs the displacement is

$\sigma(t)=\pm 1$1

with the $\sigma(t)=\pm 1$2 i.i.d. and non-Gaussian (Mori et al., 2021). In two dimensions, the kinematics are often written as

$\sigma(t)=\pm 1$3

where $\sigma(t)=\pm 1$4 is either reset from a continuous uniform distribution or drawn from a discrete set of orientations at tumbling events (Santra et al., 2020, Santra et al., 2020).

Several mathematical formalisms coexist. The Chapman–Kolmogorov equations govern the joint probability densities of position and internal state in many one-dimensional models (Malakar et al., 2017, Bressloff, 2024). For continuous-space interacting systems, piecewise deterministic Markov processes (PDMPs) provide an exact description in which deterministic transport between tumbles is combined with Poissonian state changes and boundary events (Hahn et al., 2023, Guillin et al., 2024). A complementary pathwise formulation represents tumbles and resettings by independent Poisson processes and uses a generalized Itô formula for jump-diffusion functionals $\sigma(t)=\pm 1$5, thereby linking trajectory-level dynamics directly to the underlying Chapman–Kolmogorov equation (Bressloff, 2024).

2. Free motion, asymptotics, and universal projected statistics

For a single RTP on the infinite line, the position distribution approaches a Gaussian at long times, with effective diffusion constant $\sigma(t)=\pm 1$6, but its finite-time structure can be qualitatively different from Brownian motion. In particular, for sufficiently small $\sigma(t)=\pm 1$7, the one-dimensional distribution becomes multimodal at intermediate times before crossing over to the long-time Gaussian regime (Malakar et al., 2017). In two dimensions, the long-time distribution admits a perturbative expansion in powers of $\sigma(t)=\pm 1$8; the leading term is Gaussian with $\sigma(t)=\pm 1$9, while higher corrections satisfy inhomogeneous diffusion equations whose source terms are recursively determined by lower orders (Santra et al., 2022).

The short-time, ballistic regime retains explicit signatures of the tumble rule. For two-dimensional RTPs with discrete orientations, the support of the position distribution reflects the underlying polygonal kinematics, whereas for continuous orientations it is concentrated on a circle of radius $\gamma$0. Exact marginal distributions are known for the $\gamma$1, $\gamma$2, and continuous-orientation cases, and all exhibit a crossover from ballistic to diffusive behavior. At long times the central part becomes Gaussian, but atypical fluctuations remain governed by explicit large-deviation functions, for example

$\gamma$3

for the $\gamma$4 and continuous-orientation models (Santra et al., 2020).

A strikingly robust feature concerns the $\gamma$5-component alone. For a $\gamma$6-dimensional RTP with exponentially distributed run times, the probability $\gamma$7 that the $\gamma$8-component does not change sign up to time $\gamma$9 is independent of $v$0 for any finite $v$1. The exact result,

$v$2

follows from a mapping to a one-dimensional symmetric random walk and the Sparre Andersen theorem. The same universality extends to arbitrary speed distributions $v$3, and related observables such as the time of the maximum and record statistics are likewise dimension-independent within this class (Mori et al., 2020, Mori et al., 2020).

Geometric observables can also be computed exactly. For a planar RTP, the mean perimeter of the convex hull is known in both the fixed-number-of-tumbles and fixed-time ensembles. In the fixed-time ensemble it behaves as $v$4 for $v$5, while for $v$6 it crosses over to

$v$7

consistent with the Brownian limit with effective diffusion $v$8 (Hartmann et al., 2019).

3. Inhomogeneous media, external potentials, and first-passage structure

Spatially varying tumble rates generate effective trapping and bias. In one important class of one-dimensional inhomogeneous RTPs, the flip rates $v$9 and $D$0 scale as $D$1 with different coefficients on the two sides of the origin. When $D$2, the particle admits a steady-state distribution on the infinite line whose exact form depends on $D$3; when $D$4, the long-time dynamics is subdiffusive with $D$5; and when $D$6, the density develops two peaks drifting away from the origin. Detailed balance holds only in the unbiased case $D$7 (Singh et al., 2020).

Space-dependent switching can also be reinterpreted as an effective trapping potential. For a one-dimensional RTP with rates $D$8, asymptotic weak-noise analysis yields mean first-passage times of the activated form

$D$9

where $n(t)\in\{0,1\}$0 is the quasipotential. This framework captures escape from symmetric traps and cubic-like barrier problems and shows that the diffusion approximation fails exponentially in the weak-noise regime (Bressloff, 2021).

For an RTP in an arbitrary one-dimensional external potential $n(t)\in\{0,1\}$1, backward Fokker–Planck analysis leads to a closed second-order ODE for the mean first-passage time to an absorbing target at the origin,

$n(t)\in\{0,1\}$2

with $n(t)\in\{0,1\}$3. Depending on the turning-point structure of $n(t)\in\{0,1\}$4, four distinct “phases” arise, including regimes with finite MFPT, mandatory drift toward the target, and regimes in which the origin is inaccessible and the MFPT diverges. Explicit formulae were derived for double-well and logarithmic potentials, together with an active generalization of Kramers’ law involving

$n(t)\in\{0,1\}$5

so that $n(t)\in\{0,1\}$6 in the weak-noise limit (Guéneau et al., 2024).

4. Resetting, boundaries, and nonequilibrium stationary states

Boundaries generate steady states with no passive analogue. In a finite one-dimensional interval, the stationary RTP density is peaked at the reflecting boundaries rather than uniform. In the limit $n(t)\in\{0,1\}$7, the steady state consists of a uniform bulk contribution plus $n(t)\in\{0,1\}$8-function peaks at the walls, and the approach to stationarity can be analyzed through a spectral expansion whose slowest mode controls relaxation (Malakar et al., 2017).

In two dimensions, stochastic resetting can act simultaneously on position and orientation. When an RTP is reset at rate $n(t)\in\{0,1\}$9 to a fixed point and its orientation is randomized uniformly, the stationary radial density remains finite at the origin,

$\frac{dX}{dt}=v[1-2n(t)],$0

whereas the stationary $\frac{dX}{dt}=v[1-2n(t)],$1-marginal diverges logarithmically,

$\frac{dX}{dt}=v[1-2n(t)],$2

Both marginals have the same exponential decay far from the origin. The relaxation to stationarity shows a dynamical transition with a front moving outward at constant speed, and the resetting rate can minimize the mean first-passage time (Santra et al., 2020).

Sticky and partially absorbing boundaries can be incorporated directly into a pathwise RTP calculus. In one dimension, a sticky wall at $\frac{dX}{dt}=v[1-2n(t)],$3 may be represented by an auxiliary bound state $\frac{dX}{dt}=v[1-2n(t)],$4, with boundary conditions such as

$\frac{dX}{dt}=v[1-2n(t)],$5

and partial absorption is introduced by replacing $\frac{dX}{dt}=v[1-2n(t)],$6. The same formalism can be combined with resetting and occupation-time-dependent absorption (Bressloff, 2024).

Renewal theory yields a complementary exact treatment of a half-line RTP with both stochastic resetting and a sticky target at $\frac{dX}{dt}=v[1-2n(t)],$7. In that setting, the nonequilibrium stationary state and MFPT depend strongly on the initial orientation and on whether desorption restarts the particle from the boundary or resets it to the initial state $\frac{dX}{dt}=v[1-2n(t)],$8. The MFPT retains the usual unimodal dependence on the bulk resetting rate, but the protocol that optimizes absorption can reverse when desorption kinetics are matched to bulk tumbling or resetting rates (Bressloff et al., 2 Feb 2026).

An additional boundary-driven extension replaces the target by a fertile site. There, RTPs crossing the origin generate offspring until they next flip direction. The density of right-movers at the origin grows exponentially, $\frac{dX}{dt}=v[1-2n(t)],$9, where $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$0 is determined by the fertility rate and fertility function alone. After normalization by $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$1, the total density converges to a stationary profile with a strict local minimum at the fertile site, satisfying $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$2 (Grange et al., 2020).

5. Interactions, jamming, and condensation mechanisms

Two-body RTP interactions already display genuinely nonequilibrium stationary structure. For two general RTPs on a one-dimensional periodic domain with hardcore jamming, the steady state can be constructed exactly in the PDMP framework from probability-flow conservation laws. A central organizing principle is “active global balance,” and the invariant measure falls into two universality classes: the detailed-jamming class, in which jamming flows are balanced state by state and the bulk is uniform, and the global-jamming class, in which only global balance survives and the bulk develops exponential growth or decay terms, sometimes modulated by polynomial factors. In both cases, the invariant measure carries positive Dirac mass at the jammed configuration $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$3, and in the global-jamming class the bulk obeys a catenary-like constraint (Hahn et al., 2023).

A rigorous discrete-to-continuous limit has been established for a pair of on-lattice and continuous RTPs with jamming interactions. As the lattice spacing vanishes, the discrete dynamics converges to a continuous PDMP on the torus, and the invariant measures converge as well. The continuous steady state exhibits finite mass at the jammed configurations and exponential decay away from them, which indicates effective attraction and is consistent with motility-induced phase separation. Coupling methods further yield non-asymptotic mixing-time bounds distinguishing a persistent regime, $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$4, from a diffusive regime, $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$5 (Guillin et al., 2024).

At the single-particle level, a different form of condensation appears in the displacement statistics of an RTP whose velocity is redrawn from a Gaussian distribution at each tumble. Because the single-run displacement has a stretched-exponential tail,

$0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$6

the total displacement after $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$7 runs develops an anomalous large-deviation regime at $0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$8,

$0 \xrightleftharpoons[\alpha(x)]{\beta(x)} 1$9

The rate function $v_i$0 is continuous but has a discontinuous derivative at $v_i$1, signaling a first-order condensation transition: beyond $v_i$2, a single large run contributes a finite fraction of the total displacement. The participation ratio

$v_i$3

acts as an order parameter and jumps from $v_i$4 below $v_i$5 to a nonzero value above it, with $v_i$6 (Mori et al., 2021).

6. Phase separation, criticality, and thermodynamic interpretations

On two-dimensional lattices, interacting RTPs display re-entrant geometric and thermodynamic transitions. In one model, particles hop with unit rate along their orientation, hop to the other nearest neighbours with rate $v_i$7, and tumble with rate $v_i$8. For small $v_i$9, increasing $\tau_i$0 first induces a phase-separated state and then destroys it, yielding a re-entrant site-percolation transition. The order parameter $\tau_i$1, susceptibility, and Binder cumulant show that the critical exponents vary continuously along the critical line in the $\tau_i$2 plane, while the scaling function $\tau_i$3 remains invariant and coincides with that of the Ising-model site-percolation transition. The critical exponents of the underlying motility-induced phase separation are related to the percolation exponents by the same multiplicative factors that connect magnetic and percolation exponents in the Ising model (Saha et al., 2024).

When explicit attractive interactions $\tau_i$4 are added, RTPs can show a second form of re-entrance. At high motility, increasing $\tau_i$5 can suppress motility-induced phase separation, restore a homogeneous phase, and then re-establish phase separation at still larger attraction. Dense-cluster percolation again tracks the transition, and although the exponents $\tau_i$6, $\tau_i$7, and $\tau_i$8 vary continuously along the critical line, some scaling functions remain identical to those of equilibrium interacting percolation, leading to the characterization of the transition as exhibiting Ising-like super universality (Bhowmick et al., 5 Jun 2025).

Recent thermodynamic work has recast confined RTP dynamics in terms of entropy exchange with a heat bath. For a one-dimensional RTP in an external potential with continuous velocity redraws, the steady-state Fokker–Planck equation yields an explicit local entropy flux

$\tau_i$9

from which one infers an effective temperature field $\sigma(t)=\pm 1$00 by matching to an overdamped Clausius representation,

$\sigma(t)=\pm 1$01

This “Inverse Clausius Thermodynamics” framework also introduces a bath-mediated approximation for the local entropy production rate and is particularly suited to RTPs whose post-tumble velocities are redrawn from a continuous distribution rather than restricted to discrete states (Farago, 10 Sep 2025).