Jerky Active Particles: Intermittent Dynamics

• Jerky active particles are a class of active matter systems defined by intermittent, burst-like motion governed by higher-order temporal dynamics.
• Models range from minimal Poisson-kick formulations yielding non-Gaussian displacements to explicit third-order Langevin equations with superballistic spreading.
• They have applications in understanding confined transport, collective nonequilibrium dynamics, and enhanced diffusivity due to transient bursts.

Searching arXiv for recent and foundational papers on jerky active particles and related active-matter frameworks. Jerky active particles are active-matter systems whose motion is intermittent, bursty, or explicitly governed by higher-order temporal dynamics, rather than by smooth overdamped self-propulsion alone. Across the literature, the term encompasses several related but distinct constructions: coarse-grained active-matter theories in which intermittent motion emerges from noise, hydrodynamic instabilities, curvature-induced currents, clustering, and jamming (Ramaswamy, 2010); minimal stochastic models in which Brownian tracers are driven by Poisson-distributed active kicks (Park et al., 2021); continuum field theories with third-order-in-time polarization dynamics arising from translational and orientational memory (Vrugt et al., 2021); and single-particle Langevin models in which a third-order derivative in time, the jerk, appears explicitly in the equation of motion and produces anomalous spreading and confinement breakdown (Löwen, 11 Jul 2025). Related manifestations occur in chiral active motion (Jose et al., 25 Aug 2025), confined single-file transport (Locatelli et al., 2014), disordered obstacle environments (Reichhardt et al., 4 Mar 2026), active baths (Hossein et al., 2017), frictional active Brownian particles (Nie et al., 2019), kinetic Monte Carlo active dynamics with discrete jumps (Klamser et al., 2021), and confined active assemblies that exhibit caging, yielding, and segregation (Yang et al., 2014). Taken together, these studies show that jerkiness in active matter can denote either a microscopic temporal structure of rare kicks and abrupt reorientations, or a macroscopic consequence of collective nonequilibrium feedback that converts smooth local rules into stop-and-go trajectories, giant fluctuations, and burst-like rearrangements (Ramaswamy, 2010).

1. Conceptual scope and definitions

In the broad active-matter framework, active particles are entities that consume energy locally, convert it into systematic motion or stress, and do so without being driven by externally imposed forces or flows (Ramaswamy, 2010). Within that class, jerky behavior appears when trajectories are punctuated by abrupt changes in speed, direction, or acceleration, or when the effective coarse-grained dynamics itself becomes higher than first order in time.

One line of work defines jerky active particles through explicit third-order temporal dynamics. In the single-particle model of "Gigantic dynamical spreading and anomalous diffusion of jerky active particles" (Löwen, 11 Jul 2025), a linear jerk equation of motion combines a third-order derivative in time with Stokes friction, a spring force, and active Ornstein–Uhlenbeck propulsion. In the chiral extension, jerky chiral active Brownian particles are defined by the translational dynamics

$\lambda \,\dddot{\vec r}(t) + m \,\ddot{\vec r}(t) + \gamma \,\dot{\vec r}(t) = \gamma v_0 \,\hat n(t) + \sqrt{2D\,\gamma^2}\,\boldsymbol{\eta}(t),$

supplemented by chiral rotational dynamics for the propulsion direction (Jose et al., 25 Aug 2025). In this usage, “jerky” is literal: the time derivative of acceleration is a dynamical variable.

A second line of work uses “jerky” more phenomenologically to describe intermittent paths generated by discrete active events. In "Rapid-Prototyping a Brownian Particle in an Active Bath" (Park et al., 2021), jerkiness is implemented as a compound Poisson process of finite-duration kicks. The particle obeys

$$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$

where the active force arises from random trap-center displacements with Poisson waiting times and exponential relaxation. Sparse kicks produce jumpy trajectories with non-Gaussian displacement statistics (Park et al., 2021).

A third line treats jerkiness as an emergent outcome of collective active-matter dynamics. The review "The Mechanics and Statistics of Active Matter" emphasizes that in polar flocks, active nematics, and suspensions, intermittent motion follows from noise-driven reorientation, sound-like density-orientation modes, curvature-induced currents, giant number fluctuations, hydrodynamic instabilities, active turbulence, clustering, and jamming (Ramaswamy, 2010). In that broader sense, jerky active particles need not possess an explicit jerk term; rather, their trajectories become bursty because the active environment and collective fields fluctuate strongly in space and time.

This suggests two analytically distinct but physically connected notions: “jerky dynamics” as a higher-order temporal law, and “jerky motion” as intermittent nonequilibrium transport.

2. Single-particle stochastic formulations

A minimal stochastic realization of jerky active motion is the Brownian tracer in an active bath driven by Poissonian kicks (Park et al., 2021). The tracer is confined by a harmonic potential

$U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$

and is subject to thermal white noise

$$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$

Activity is implemented by shifting the trap center at random Poisson times $t_i$, with amplitudes $d_i$ drawn from a Gaussian distribution of variance $\chi^2$, each shift decaying exponentially with time constant $\tau_c$: $$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$ The waiting times are exponentially distributed,

$P(\Delta t)=\frac{1}{\tau_p}e^{-\Delta t/\tau_p},$

so the kick rate is $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$0 (Park et al., 2021).

The active-force autocorrelation is

$$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$1

which resembles Ornstein–Uhlenbeck noise at the level of two-point correlations, but the force remains generally non-Gaussian because it is generated by discrete pulses (Park et al., 2021). This distinction is central. When $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$2, kicks are rare and short-lived, so the force is mostly near zero with sharp bursts; when $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$3, many kicks overlap and the force becomes effectively Gaussian by the central limit theorem (Park et al., 2021). The jerky regime is therefore the Poissonian, non-Gaussian-force regime with $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$4.

The stationary position distribution can become non-Gaussian, and the non-Gaussianity is measured by

$$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$5

For $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$6, the position distribution has a Gaussian core and exponential tails, while even in parameter regimes where the stationary position distribution looks nearly Gaussian, the van Hove self-correlation function $$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$7 remains non-Gaussian at short lag times (Park et al., 2021). This is a precise statistical signature of jerkiness: rare strong bursts dominate short-time displacement tails.

In the untrapped case, the mean-squared displacement is

$$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$8

with

$$\gamma \dot{x}(t) = -k\,x(t) + \xi_{\text{th}}(t) + \xi_{\text{act}}(t),$$9

Thus the long-time effective diffusion is $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$0, while the short-time statistics remain strongly intermittent if kicks are sparse (Park et al., 2021). The same model reproduces tracer diffusion in quasi-2D algal baths, including the crossover from highly non-Gaussian displacement distributions at low swimmer concentration to Gaussian ones at high concentration, and the scaling $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$1 when $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$2 (Park et al., 2021).

A distinct stochastic framework arises in active kinetic Monte Carlo models, where trajectories are intrinsically discrete and therefore jerky in configuration space (Klamser et al., 2021). There, finite jumps can mimic active dynamics, but a purely active KMC scheme has an ill-defined continuous-time limit in which core active-matter effects such as MIPS and ratchets vanish and pressure diverges. Mixing passive with active steps regularizes the limit and yields well-defined continuous dynamics connected to AOUP, ABP, and RTP models (Klamser et al., 2021). This establishes that not all jerky discretizations are faithful coarse-grainings of active transport.

3. Explicit jerk equations and anomalous transport

The most literal theory of jerky active particles is the third-order Langevin model introduced in (Löwen, 11 Jul 2025). In one dimension, after a feedback-based construction and a coordinate shift, the dynamics is

$U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$3

where $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$4 is an active Ornstein–Uhlenbeck process,

$U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$5

with

$U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$6

and correlation

$U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$7

Here $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$8 is the jerk coefficient, $U(x)=\frac12 kx^2,\qquad F_{\text{trap}}=-kx,$9 an effective mass, $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$0 the friction coefficient, and $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$1 the spring constant (Löwen, 11 Jul 2025).

The model admits exact Green-function solutions for the mean-squared displacement and related observables. Its main result is gigantic superballistic spreading in weakly damped, weakly confined regimes. For the pure jerk case $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$2, the Green function is $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$3, and the MSD behaves as

$$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$4

with exponents $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$5 and $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$6 arising from the competition between jerk and persistent activity (Löwen, 11 Jul 2025). Including inertia but no friction or trap reduces the long-time scaling to $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$7, while still leaving the short-time $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$8 regime intact (Löwen, 11 Jul 2025). In the damped case without a trap, multiple anomalous exponents $$\langle \xi_{\text{th}}(t)\rangle=0,\qquad \langle \xi_{\text{th}}(t)\xi_{\text{th}}(t')\rangle=2\gamma k_B T\,\delta(t-t').$$9 can appear as successive crossovers, depending on the ordering of the timescales

$t_i$0

This hierarchy is one of the clearest mathematical manifestations of jerk-induced transport regimes (Löwen, 11 Jul 2025).

Confinement produces another distinctive effect. In a harmonic trap, the localization length

$t_i$1

defines an order parameter $t_i$2 for a localization–delocalization transition. The paper shows that a jerky active particle can escape harmonic confinement through a sharp transition, which can be first or second order as a function of jerkiness (Löwen, 11 Jul 2025). The transition is controlled by dimensionless combinations of the jerk, mass, friction, and trap strength, and is accompanied by an enormous increase in the kinetic temperature relative to the AOUP bath temperature (Löwen, 11 Jul 2025). This marks a qualitative departure from standard active Ornstein–Uhlenbeck particles.

The chiral extension (Jose et al., 25 Aug 2025) adds an orientational dynamics

$t_i$3

to the translational jerk equation in two dimensions. In this case, jerk induces oscillatory corrections to ordinary circular swimming and leads to mean trajectories that interpolate between damped or exploding Lissajous patterns and the spira mirabilis familiar from conventional chiral active Brownian particles (Jose et al., 25 Aug 2025). The MSD retains the short-time anomalous scaling

$t_i$4

and crosses over at long times to ordinary diffusive behavior with the standard chiral-ABP effective diffusion coefficient

$t_i$5

showing that jerk primarily reshapes transient transport and trajectory geometry rather than asymptotic diffusivity (Jose et al., 25 Aug 2025).

4. Continuum and hydrodynamic theories of intermittent active motion

Jerky active behavior also appears at the continuum level without an explicit third derivative. In polar flocking on substrates, the Toner–Tu description uses density $t_i$6 and polarization/velocity $t_i$7, with a schematic equation

$t_i$8

coupled to

$t_i$9

Here self-advection, alignment, density coupling, and additive Gaussian noise together generate intermittent reorientation and burst-like acceleration events in particle trajectories (Ramaswamy, 2010). In the ordered phase, density-orientation coupling produces sound-like modes with dispersion

$d_i$0

so local reorientation events can propagate and induce correlated bursts over long distances (Ramaswamy, 2010).

Active nematics on substrates provide a different route to jerky motion. There, orientational curvature directly induces currents: $d_i$1 Because orientational fluctuations are large Goldstone modes, the resulting density fluctuations satisfy

$d_i$2

which are the giant number fluctuations characteristic of active nematics (Ramaswamy, 2010). A particle entering or leaving the dense bands and voids generated by these currents experiences stop-and-go motion, intermittent caging, and rapid ejection, even though the underlying coarse-grained equations are smooth (Ramaswamy, 2010).

In active suspensions with fluid flow, the active stress

$d_i$3

enters a generalized Navier–Stokes equation,

$d_i$4

coupled to orientational dynamics

$d_i$5

The feedback between active stresses and orientational distortions destabilizes ordered states, producing spatiotemporally irregular flows and “turbulence at zero Reynolds number” (Ramaswamy, 2010). Individual tracers or swimmers in such flows undergo nearly straight segments within coherent vortices or jets, punctuated by rapid turns when those structures merge or decay (Ramaswamy, 2010). This is another canonical mechanism for jerky active trajectories.

A still more formal continuum route to jerk appears in the active phase-field crystal theory with translational and orientational memory (Vrugt et al., 2021). There, the density field $d_i$6 and polarization field $d_i$7 obey memory-integral equations with two distinct relaxation rates, $d_i$8 and $d_i$9. Eliminating the memory kernels yields a second-order density dynamics and a third-order polarization dynamics. The key equation is

$\chi^2$0

which the authors identify as a spatiotemporal jerky dynamics (Vrugt et al., 2021). In this framework, the linear stability of the liquid state depends on the damping coefficients, unlike in passive phase-field crystal theory, and sound can propagate through two distinct mechanisms, one pressure-mediated and one polarization-mediated (Vrugt et al., 2021). This suggests that jerkiness at the field level can qualitatively alter pattern selection and wave attenuation.

5. Disorder, confinement, friction, and caging as sources of jerkiness

Jerky motion is often most visible when active particles move in geometrically constrained or crowded environments. In a random Lorentz gas, an active Brownian particle moving through a random array of impenetrable obstacles obeys

$\chi^2$1

with activity measured by

$\chi^2$2

Near the obstacle percolation density $\chi^2$3, both active and Brownian particles exhibit subdiffusive scaling

$\chi^2$4

set by the geometry of the percolating void network (Reichhardt et al., 4 Mar 2026). Yet active particles reach this regime more rapidly and, at high activity, exhibit lower long-time diffusivity than Brownian particles because persistence enhances self-trapping in dead ends and concave obstacle pockets (Reichhardt et al., 4 Mar 2026). The resulting trajectories show long plateaus while trapped, followed by jumps along channels when rotational diffusion finally permits escape (Reichhardt et al., 4 Mar 2026).

Single-file confinement offers a different mechanism. In narrow channels where particles cannot pass one another, active Brownian particles projected onto the channel axis satisfy

$\chi^2$5

with

$\chi^2$6

The key control parameters are

$\chi^2$7

which separate translational active particles (TAPs) from rotational active particles (RAPs) (Locatelli et al., 2014). TAPs form “active clusters” that merge and split, and in biased channels these clusters can release particles “one after the other in a rapid sequence” as they collectively orient against the bias (Locatelli et al., 2014). The motion is therefore stick–slip-like: long jammed intervals followed by correlated bursts of escape (Locatelli et al., 2014).

Frictional active Brownian particles provide another microscopically distinct route to jerkiness (Nie et al., 2019). In that model, explicit Coulomb friction at interparticle contacts adds tangential forces and torques. The overdamped translational and rotational velocities are

$\chi^2$8

and the tangential contact force satisfies $\chi^2$9 (Nie et al., 2019). Friction suppresses sliding resolution of collisions and instead causes extended blocked contacts and collision-induced reorientations. In the dilute regime, the effective rotational diffusion obeys

$\tau_c$0

so collisions themselves become a source of abrupt angular changes (Nie et al., 2019). At the collective level, friction stabilizes long-lived rotating clusters and drives the low-density MIPS spinodal to zero at large $\tau_c$1, indicating that interaction-induced jerkiness can qualitatively reshape phase behavior (Nie et al., 2019).

Confinement and crowding also induce jerky behavior in soft repulsive disks. In a square box, overdamped self-propelled disks with harmonic repulsions obey

$\tau_c$2

with wall confinement and rotational noise $\tau_c$3 (Yang et al., 2014). At low density and low $\tau_c$4, particles spontaneously accumulate at walls; at higher packing fractions, the system develops a finite critical speed for aggregation near $\tau_c$5, the jamming point for athermal monodisperse disks (Yang et al., 2014). The paper reports force chains, rattling within cages, pressure anomalies, and barrier-crossing segregation in bidisperse mixtures, all of which imply stop-and-go trajectories and rare rearrangement events (Yang et al., 2014).

6. Active baths, active strings, and other environments that induce bursty trajectories

A particle need not be intrinsically active to become jerky. In a quasi-2D active filament bath, passive particles intermittently bind to self-propelled filaments and inherit their active drift (Hossein et al., 2017). The particle velocity is effectively

$\tau_c$6

where $\tau_c$7 is a telegraph process describing bound versus unbound states, and $\tau_c$8 has a finite persistence time $\tau_c$9 (Hossein et al., 2017). Trajectories alternate between caged diffusive phases and advective bursts, leading to multimodal displacement distributions, superdiffusive intermediate-time MSDs, and an effective diffusion coefficient

$$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$0

The first term is an active diffusion generated by intermittent hitch-hiking on the active medium (Hossein et al., 2017).

In active dipolar systems, long-range anisotropic interactions create another environment that promotes jerky transport. Brownian dynamics of dipolar active Brownian particles in 3D produce string fluids and percolated active gels, depending on the dipolar coupling

$$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$1

and active force $$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$2 (Kelidou et al., 2024). Strong dipolar coupling and moderate activity yield an active gel: a percolated network of active chains whose overall structure remains interconnected, but whose bond lifetime is reduced by self-propulsion (Kelidou et al., 2024). The bond time autocorrelation decays much faster than in the passive gel, while translational and rotational diffusion are enhanced relative to passive counterparts (Kelidou et al., 2024). This suggests a trajectory pattern of long constrained intervals within the network punctuated by bond-breaking and network-rearrangement events.

Hamiltonian active particles with an internal energy depot offer still another perspective (Eichmann et al., 2021). The coupled motional and depot Hamiltonian generates active and inactive phases separated by a separatrix in an effective reduced phase space, while environmental noise appears as random momentum kicks and viscous drag (Eichmann et al., 2021). The model displays long episodes of approximately constant-speed motion against an external force, interrupted by stochastic exits from the active phase or re-entry through depot refilling (Eichmann et al., 2021). Although the paper does not use the term “jerky,” its dynamics of separatrix crossing, noise-induced kicks, and depot recharge is naturally intermittent (Eichmann et al., 2021).

7. Statistical signatures, misconceptions, and limitations

Jerky active motion is not defined by a single observable. Instead, different models exhibit different statistical diagnostics. In the Poisson-kick tracer model, non-Gaussian van Hove functions and exponential tails in displacement PDFs are direct markers of intermittent kicks (Park et al., 2021). In explicit jerk equations, superballistic short-time MSD scaling such as $$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$3 or $$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$4 is the primary signature (Löwen, 11 Jul 2025, Jose et al., 25 Aug 2025). In hydrodynamic active matter, giant number fluctuations,

$$x_c(t)=\sum_i d_i e^{-(t-t_i)/\tau_c}\Theta(t-t_i),\qquad \xi_{\text{act}}(t)=k x_c(t).$$5

sound-like propagating fluctuations, or long-range stress and velocity correlations signal collective intermittency rather than microscopic jumps (Ramaswamy, 2010). In disordered media, subdiffusive plateaus and abrupt hops along channels characterize self-trapping and release (Reichhardt et al., 4 Mar 2026). In single-file channels, bursty escape sequences and cluster splitting/merging are more relevant than one-body displacement moments (Locatelli et al., 2014).

A common misconception is that jerkiness must imply inertia. The literature does not support that equivalence. The Poisson-kick model is overdamped yet strongly jerky (Park et al., 2021); active nematics and bacterial turbulence produce bursty motion through collective field instabilities without a third time derivative (Ramaswamy, 2010); and frictional ABPs generate abrupt reorientations through contact mechanics in an overdamped setting (Nie et al., 2019). Conversely, explicit jerk terms in the Langevin equation do not automatically imply visible stop-and-go trajectories unless the relevant timescales place the system in an oscillatory or superballistic regime (Löwen, 11 Jul 2025, Jose et al., 25 Aug 2025).

Another misconception is that activity always enhances transport. In obstacle arrays, strong activity can reduce long-time diffusion below the Brownian value because persistence causes self-trapping (Reichhardt et al., 4 Mar 2026). In confined or jammed systems, a finite activity may be required just to yield the cage structure (Yang et al., 2014). In frictional ABPs, increased activity plus friction can stabilize dense clusters rather than fluidizing them (Nie et al., 2019). This suggests that the relation between activity and jerkiness is not monotonic: more persistent self-propulsion can either create bursts or lock particles into longer waiting states.

The main limitations also differ across approaches. Continuum theories such as Toner–Tu or active nematic hydrodynamics coarse-grain away discrete biochemical cycles, motor stepping, or run-and-tumble states, folding them into effective noises and stresses (Ramaswamy, 2010). The Poisson-kick active bath model is intentionally agnostic about microscopic interactions, which aids phenomenology but limits mechanistic interpretation (Park et al., 2021). Explicit jerk models are analytically tractable but mostly single-particle and linear; many-body extensions remain open (Löwen, 11 Jul 2025, Jose et al., 25 Aug 2025). Single-file and obstacle models capture crowding and disorder but simplify hydrodynamics and often work in low dimensions (Locatelli et al., 2014, Reichhardt et al., 4 Mar 2026). These differences matter because the origin of jerkiness can be internal, environmental, or collective, and a model that captures one mechanism may miss another.

A plausible implication is that “jerky active particles” should be understood as a family of nonequilibrium transport processes rather than a single universality class. What unifies them is not a unique equation, but the coexistence of persistence and abrupt change: finite-time runs or trapped intervals, punctuated by strong reorientation, release, or acceleration events. That unification is visible across discrete kick models (Park et al., 2021), third-order jerk equations (Löwen, 11 Jul 2025), memory-driven field theories (Vrugt et al., 2021), and collective active-matter hydrodynamics (Ramaswamy, 2010).