Stochastic Position-Orientation Resetting

Updated 8 July 2026

Stochastic position-orientation resetting is a protocol where both spatial coordinates and orientations (and sometimes velocities) are renewed at random, typically via a Poisson process.
Different resetting schemes—complete, position-only, and orientation-only—alter active particle dynamics by preserving or erasing anisotropy and directional memory.
Renewal theory underpins these analyses, yielding exact moment hierarchies, localization laws, and non-Gaussian steady states with experimental implications in colloids, granular particles, and robotic systems.

Stochastic position-orientation resetting denotes restart protocols in which a particle is returned not only to a prescribed spatial point but also to a prescribed orientational state, typically at Poissonian times. In active and anisotropic systems this is qualitatively different from positional resetting alone, because orientation controls propulsion, translational anisotropy, or angular persistence. The resulting nonequilibrium dynamics has been analyzed for overdamped and inertial active Brownian particles, anisotropic Brownian particles, run-and-tumble particles, and chiral swimmers, with closely related work on position-only and orientation-only protocols used to isolate what is specific to joint resetting (Kumar et al., 2020, Chaki et al., 2024, Ghosh et al., 9 Jan 2025, Patel et al., 24 Feb 2026).

1. Protocol definitions and conceptual scope

The basic complete-reset rule in overdamped active matter is

$(\mathbf r,\mathbf u)\rightarrow (\mathbf r_0,\mathbf u_0),$

or equivalently $(x,y,\theta)\to(x_0,y_0,\theta_0)$ , with resets usually generated by a Poisson process of rate $r$ . Standard benchmark choices are $\mathbf r_0=\mathbf 0$ and $\theta_0=0$ , so that the heading is reset along the $x$ -axis. In anisotropic Brownian and anisotropic active Brownian models, the same formal distinction is made among complete resetting, position-only resetting, and orientation-only resetting. These are not merely bookkeeping variants: complete resetting can preserve anisotropy and directional bias indefinitely, whereas position-only resetting can erase late-time orientational memory from the stationary spatial state, and orientation-only resetting can sustain anisotropic transport without producing positional confinement (Chaki et al., 2024, Ghosh et al., 9 Jan 2025).

In inertial active matter the protocol must be specified more carefully. The inertial active Brownian particle studied in 2026 is often described as a position-orientation resetting problem, but the solved model is in fact a complete reset of position, velocity, and orientation,

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$

That distinction is essential because, unlike in the overdamped case, velocity is not instantaneously aligned with orientation; it relaxes only over the finite momentum-relaxation time $m/\gamma$ . The reported localization laws and steady-state moments in that model therefore belong specifically to complete state resetting with a “cold restart” of momentum (Patel et al., 24 Feb 2026).

A neighboring but distinct branch replaces joint position-orientation resetting by orientation-only or velocity-only resetting while leaving position unchanged. This is central in sorting problems, where resetting position would simply remix species, whereas resetting orientation or velocity can generate species-dependent drifts. Such protocols are related to the broader resetting literature but are not themselves examples of joint stochastic position-orientation resetting (Cleuren et al., 19 Mar 2026).

2. Canonical stochastic models

The most common setting is the two-dimensional active Brownian particle (ABP). In free space the overdamped dynamics is

$\dot x(t)=v_0\cos\theta(t),\qquad \dot y(t)=v_0\sin\theta(t),\qquad \dot\theta(t)=\sqrt{2D_R}\,\eta(t),$

while trapped variants add a harmonic drift $-\mu k\,\mathbf r$ . Joint resetting then repeatedly restores both the particle’s location and its propulsion direction, making the process exactly renewable in the full phase space when the reset state is fixed (Kumar et al., 2020, Shee, 2024).

Anisotropic Brownian and anisotropic active Brownian particles extend this by allowing unequal mobilities or diffusivities along body axes. In the overdamped anisotropic Brownian case the translational noise covariance depends explicitly on $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 0 through a mobility tensor $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 1, so orientational resetting feeds directly into spatial transport. In the anisotropic ABP, propulsion acts along the major body axis, and the coupling between translational anisotropy and rotational diffusion produces the familiar crossover from short-time anisotropic motion to long-time isotropic diffusion in the absence of resetting. Resetting protocols determine whether that isotropization is interrupted, erased, or continuously reimposed (Chaki et al., 2024, Ghosh et al., 9 Jan 2025).

Run-and-tumble particles (RTPs) supply a second major class. In two dimensions, one protocol resets the particle to the origin and randomizes the new orientation uniformly on $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 2, preserving isotropy while still performing a full position-orientation renewal. In one-dimensional half-line problems, the reset can restore the complete initial state $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 3, where $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 4 is the orientation or velocity state. This produces a sharp dependence of both steady states and first-passage observables on whether resetting relaunches the particle toward or away from a boundary (Santra et al., 2020, Bressloff et al., 2 Feb 2026).

Chiral and inertial ABPs add further dynamical structure. For a chiral swimmer,

$(x,y,\theta)\to(x_0,y_0,\theta_0)$ 5

joint position-orientation resetting interrupts circular motion and competes with both intrinsic spin $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 6 and rotational diffusion $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 7. For inertial ABPs, underdamped Langevin dynamics introduces the dimensionless inertia $(x,y,\theta)\to(x_0,y_0,\theta_0)$ 8, so resetting competes not only with active persistence but also with finite momentum relaxation (Shee, 17 Aug 2025, Patel et al., 24 Feb 2026).

3. Renewal theory and exact analytical machinery

The dominant analytical framework is renewal theory. For fixed-state Poissonian resetting, many observables satisfy the standard relation

$(x,y,\theta)\to(x_0,y_0,\theta_0)$ 9

where the right-hand side uses the no-reset dynamics. For steady-state moments this often reduces, via Laplace transforms and the Final-Value Theorem, to

$r$ 0

This is the key mechanism behind the exact closed-form moment hierarchies derived for trapped ABPs and inertial ABPs (Shee, 2024, Patel et al., 24 Feb 2026).

When all variables are reset, one can also write full phase-space renewal equations. For anisotropic Brownian particles, with $r$ 1, the propagator obeys

$r$ 2

Complete resetting corresponds to $r$ 3, whereas position-only and orientation-only resetting require more elaborate kernels because the non-reset variables retain memory across reset events (Chaki et al., 2024).

For first-passage observables the same structure appears in Laplace space. In the absorbing-wall ABP problem with resetting of both position and orientation, the survival probability satisfies

$r$ 4

so the reset problem is determined once the no-reset survival kernel is known. This exact renewal relation is formally close to passive resetting, but the renewed state is now a point in configuration space, not only in physical space, and the dependence on the initial orientation $r$ 5 becomes physically decisive (Baouche et al., 3 Apr 2025).

A recurring methodological point is that exact solvability depends strongly on what is reset. Complete resetting yields a genuine renewal of the full state. Position-only or orientation-only resetting generally renews only part of the dynamics, so one obtains exact formulas for selected moments, effective diffusion tensors, or stationary marginals rather than a closed stationary law for the whole process. This suggests that protocol specification is not auxiliary but structurally part of the model.

4. Stationary states, localization, and anisotropy

Whenever position is reset, a nonequilibrium stationary spatial state typically exists. For the free 2D ABP under joint position-orientation resetting, the stationary marginals are asymptotically exponential in the slow-reset regime $r$ 6,

$r$ 7

and similarly for $r$ 8. In the opposite regime $r$ 9, the stationary state becomes strongly anisotropic: the $\mathbf r_0=\mathbf 0$ 0-marginal remains finite at $\mathbf r_0=\mathbf 0$ 1, whereas the transverse $\mathbf r_0=\mathbf 0$ 2-marginal diverges algebraically,

$\mathbf r_0=\mathbf 0$ 3

revealing a directional singularity at the reset point (Kumar et al., 2020).

Position-only resetting can produce a stationary state with very different symmetry. For the same ABP, the large- $\mathbf r_0=\mathbf 0$ 4 stationary marginal becomes

$\mathbf r_0=\mathbf 0$ 5

so near the origin

$\mathbf r_0=\mathbf 0$ 6

The stationary state is isotropic because orientation is not reinitialized and is effectively averaged over at reset ages. The anisotropic Brownian particle shows the same principle more explicitly: under complete resetting the stationary variances retain memory of both intrinsic asymmetry and initial orientation, with anisotropy ratio $\mathbf r_0=\mathbf 0$ 7, whereas under position-only resetting

$\mathbf r_0=\mathbf 0$ 8

and the stationary law becomes exactly the standard isotropic resetting law $\mathbf r_0=\mathbf 0$ 9 with $\theta_0=0$ 0 (Kumar et al., 2020, Chaki et al., 2024).

Orientation-only resetting behaves differently because there is no spatial reinjection. For anisotropic Brownian particles there is no stationary spatial density; instead the late-time density is Gaussian with an effective diffusion tensor that contains off-diagonal terms. For the overdamped ABP under orientational resetting, the mean displacement grows linearly,

$\theta_0=0$ 1

so the long-time state is drift plus anisotropic diffusion rather than localization. The same conclusion appears in the anisotropic active Brownian particle: orientation resetting sustains asymmetry even at late times, but does not itself create positional confinement (Baouche et al., 2024, Ghosh et al., 9 Jan 2025).

These comparisons resolve a common misconception. Resetting orientation is not a small variant of position resetting, and position resetting is not a proxy for position-orientation resetting. Whether orientation is renewed determines if anisotropy is preserved, restored, or washed out; whether position is renewed determines whether a spatial steady state exists at all.

5. Higher-order structure: non-Gaussianity, chirality, inertia, and confinement

Second moments alone do not characterize reset-controlled active steady states. In several models the central diagnostic is the excess kurtosis,

$\theta_0=0$ 2

or its position- and velocity-space analogues. Positive values signal distributions that are more sharply peaked and heavy-tailed than a Gaussian with the same variance, while negative values indicate flatter, activity-dominated statistics (Shee, 2024, Shee, 17 Aug 2025, Patel et al., 24 Feb 2026).

For an overdamped ABP in a harmonic trap under complete resetting, the exact steady-state MSD is

$\theta_0=0$ 3

The fourth moment yields a steady-state phase structure organized by $\theta_0=0$ 4. The system admits resetting-dominated ( $\theta_0=0$ 5), activity-dominated ( $\theta_0=0$ 6), and Gaussian ( $\theta_0=0$ 7) regimes. As a function of trap strength, the paper reports a re-entrant sequence

$\theta_0=0$ 8

with critical values $\theta_0=0$ 9 and $x$ 0 for the example $x$ 1 (Shee, 2024).

Chirality enriches this picture further. For the chiral ABP under joint position-orientation resetting, the steady-state MSD is

$x$ 2

When resets are infrequent compared to chiral rotation, the active contribution is maximized at

$x$ 3

so rotational diffusion can increase the steady-state spread by breaking circular loops before resetting does. The paper uses $x$ 4 together with the oscillatory or monotonic character of the orientation autocorrelation to define three states: an activity-dominated chiral state, a resetting-dominated state with chirality, and a resetting-dominated state without effective chirality. The crossover line $x$ 5 separates oscillatory from monotonic orientational dynamics (Shee, 17 Aug 2025).

Inertia changes the asymptotics even more drastically. For the inertial ABP under complete resetting of position, velocity, and orientation, the exact steady-state MSD obeys

$x$ 6

at high reset rate, instead of the overdamped $x$ 7 suppression. The paper also finds strongly non-Gaussian steady states with a sharp central peak and heavy tails, and reports that the positional excess kurtosis approaches

$x$ 8

under strong resetting. The tail weight varies non-monotonically with reset rate, implying an optimal regime for rare long excursions generated by the competition between momentum relaxation and resetting (Patel et al., 24 Feb 2026).

A plausible implication is that stochastic position-orientation resetting should be viewed not as a single transport-control mechanism but as a family of nonequilibrium selectors: confinement, chirality interruption, inertial cold restarts, and trap relaxation each produce distinct fourth-order signatures even when the second moments look similar.

6. First-passage theory and optimal resetting

The first-passage problem shows most sharply why the orientational part of the reset matters. For a 2D ABP near an absorbing wall, resetting the full initial configuration $x$ 9 makes the mean first-passage time depend explicitly on whether the reset heading is favorable or unfavorable. In the low- $(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 0 theory the passive benchmark is

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 1

with optimal passive reset rate

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 2

The active correction is orientation-dependent already at first order, proportional to $(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 3. Consequently, resetting can either reduce or increase the minimal MFPT relative to passive diffusion: particles initially facing the wall can outperform the passive benchmark, while particles initially facing away can become slower, especially at short distances (Baouche et al., 3 Apr 2025).

The two-dimensional RTP with resetting to the origin plus orientation randomization provides a complementary benchmark. The process reaches a stationary state, and for first passage to an absorbing line through the reset point the MFPT is

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 4

This decreases monotonically with $(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 5. When reset point and target differ, however, numerical results show a non-monotonic MFPT and hence a nonzero optimal reset rate. The same paper emphasizes that changing the orientation part of the reset from randomization to a fixed direction breaks isotropy and qualitatively alters the stationary marginals (Santra et al., 2020).

One-dimensional sticky-boundary RTPs make the orientation dependence even more explicit. With bulk resetting to the complete state $(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 6, the adsorption MFPTs

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 7

depend strongly on whether resetting restores motion away from or toward the boundary. The paper finds that if the initial orientation is toward the boundary, desorption-induced resetting can reduce the MFPT more effectively than non-resetting desorption; when the initial orientation is away from the boundary, the opposite tendency appears. This is a direct first-passage manifestation of position-orientation resetting as directional restart, not only spatial restart (Bressloff et al., 2 Feb 2026).

The experimentally targeted platforms are broad. Exact-reset predictions have been framed as testable in inertial colloids, vibrated granular particles, optically controlled active particles, microrobotic and robotic agents, asymmetric colloids manipulated by optical tweezers and magnetic alignment, and chiral swimmers in targeted-delivery or search settings (Chaki et al., 2024, Shee, 2024, Shee, 17 Aug 2025, Patel et al., 24 Feb 2026).

A closely related control paradigm replaces joint position-orientation resetting by resetting only orientation or velocity while keeping position fixed. In overdamped colloids with translation-rotation coupling, periodic orientation resetting to $(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 8 produces the cycle-averaged drift

$(\mathbf r,\mathbf v,\theta)\rightarrow (\mathbf r_0,\mathbf v_0,\theta_0).$ 9

which depends on particle shape and enables sorting. This establishes a sharp practical distinction: position resetting is “evidently unsuitable” for sorting, whereas resetting non-positional degrees of freedom can generate directed transport and species separation (Cleuren et al., 19 Mar 2026).

Several protocol distinctions remain non-negotiable. Orientation-only resetting usually does not produce a stationary positional state, even though it can generate a steady drift or an anisotropic effective diffusion tensor (Baouche et al., 2024, Ghosh et al., 9 Jan 2025). Position-only resetting can suppress spatial wandering while erasing anisotropy from the stationary law (Chaki et al., 2024). Inertial “position-orientation resetting” results may depend quantitatively on simultaneous velocity resetting and therefore do not automatically extend to protocols that leave velocity unchanged (Patel et al., 24 Feb 2026).

The current literature therefore supports a precise usage. Stochastic position-orientation resetting is best understood as reset control in an extended state space, where the orientational variable is dynamically active rather than auxiliary. Its characteristic outcomes are protocol-sensitive renewal structures, stationary or non-stationary spatial laws depending on whether position is reset, and transport or search properties that can improve or degrade depending on the heading reinstated at each restart.