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Run and tumble particle under resetting: a renewal approach
Martin R. Evans, Satya N. Majumdar
TL;DR
The paper studies how Poissonian resetting of position and velocity affects one-dimensional run-and-tumble particles, using renewal equations for stationary states and absorption. The stationary distribution is protocol-independent, while absorption outcomes depend on velocity resetting, with velocity randomization yielding shorter mean absorption times than position-only resetting.
Problem
The study asks how resetting affects run-and-tumble dynamics across diffusive and ballistic regimes, including resetting of both position and velocity rather than position alone.
Method
The authors analyze position and velocity resetting protocols using a renewal equation approach for stationary distributions and survival probabilities.
Results
The stationary state has a symmetric exponential-decay Laplace form independent of the velocity resetting protocol, whereas survival probability and mean first passage time depend on η.
Takeaways & Limitations
For fixed parameters, velocity randomization is more efficient than position-only resetting because it gives a smaller mean time to absorption, with an optimal resetting rate.
Takeaways & Limitations
The contour calculation leading to Eq. (20) is not detailed because the result is considered peripheral and well known.
Abstract
from arXiv · showhide
We consider a particle undergoing run and tumble dynamics, in which its velocity stochastically reverses, in one dimension. We study the addition of a Poissonian resetting process occurring with rate $r$. At a reset event the particle's position is returned to the resetting site $X_r$ and the particle's velocity is reversed with probability $η$. The case $η= 1/2$ corresponds to position resetting and velocity randomization whereas $η=0$ corresponds to position-only resetting. We show that, beginning from symmetric initial conditions, the stationary state does not depend on $η$ i.e. it is independent of the velocity resetting protocol. However, in the presence of an absorbing boundary at the origin, the survival probability and mean time to absorption do depend on the velocity resetting protocol. Using a renewal equation approach, we show that the the mean time to absorption is always less for velocity randomization than for position-only resetting.
1. Introduction
The paper studies resetting in one-dimensional run-and-tumble dynamics, extending resetting analysis beyond diffusion to a process interpolating between ballistic and diffusive motion. It uses renewal equations to compare velocity-resetting protocols and their effects on stationary and first-passage behavior.
- Run-and-tumble dynamics models correlated-noise motion and serves as the continuum limit of persistent random walks.
- Resetting extends beyond position space because both the particle’s position and velocity can be reset.The resetting position is fixed at X_r, while velocity-resetting protocols are varied.
- Renewal equations describe survival for velocity randomization and require joint survival-velocity equations for general velocity resetting.
- The stationary state is a symmetric Laplace distribution independent of the velocity-resetting protocol under symmetric initial conditions.
- Survival probability and mean first-passage behavior depend on the velocity-resetting parameter η in the presence of an absorbing boundary.
2. Run and tumble particle dynamics
This section formulates run-and-tumble motion through velocity-resolved probability equations and Laplace transforms. The resulting dynamics connects ballistic and diffusive limits, while the time-dependent distribution follows from inverse-transform analysis.
- The velocity-resolved density P_σ(x,t) describes the particle’s position and velocity, with switching between σ = ±1 occurring at rate γ.The velocity correlation time is 1/γ.
- Symmetric initial conditions place the particle at the origin with equal probabilities for the two initial velocities.
- The first-order master-equation system is converted into decoupled second-order equations away from x = 0.
- Solutions are selected to remain finite as x approaches ±∞, with coefficients fixed by returning to the first-order equations.
- The Laplace transform is inverted through a Bromwich contour and branch-cut integral to obtain the time-dependent distribution.The authors omit the lengthy algebra because the result is peripheral and well known.
3. Run and tumble particle under position resetting and velocity randomization
The particle is reset in position and velocity at Poissonian times, with velocity randomized equally between the two directions. The resulting stationary distribution is Laplace-shaped, with a decay length controlled by speed, switching, and resetting rates.
- Resetting protocol: Resetting occurs at rate r, returning the particle to the origin while choosing velocity σ = ±1 with probability 1/2.This protocol is termed position resetting and velocity randomization.
- Stationary state: The stationary distribution is a double exponential, also known as a Laplace distribution.It is obtained from the long-time limit of the renewal equation.
- Stationary state: The decay length increases with speed v0 but decreases with switching rate γ and resetting rate r.
- Limiting cases: As r → 0, the decay length diverges as r^-1/2, indicating the absence of a stationary state.
- Limiting cases: In the ballistic limit γ → 0, the decay length approaches v0.
- Limiting cases: The diffusive limit recovers the expression for diffusive resetting.
4. Survival Probability
The absorbing-boundary problem is formulated through renewal equations for survival with and without resetting. Backward equations and an absorbing boundary condition determine the survival probabilities used in the resetting solution.
- Problem setup: The origin is treated as an absorbing target, so touching the boundary corresponds to locating the target in a search process.
- Definitions: Qr(x0, t) denotes survival with resetting, while Q0(x0, t) denotes survival without resetting for equally likely initial velocities.
- Renewal formulation: The survival probability with resetting satisfies a renewal equation separating trajectories with no reset from those whose last reset occurred at time t − τ.
- Renewal formulation: For Poisson resetting at rate r, the Laplace-transformed renewal relation has general applicability when resetting returns the process to its initial conditions.
- Solution: Substituting the no-reset survival transform into the renewal relation yields the resetting survival result.
- Backward equations: The backward survival equations are solved on x0 ≥ 0 with initial survival equal to one and boundary condition Q0^−(0, t) = 0.The boundary condition reflects immediate absorption for a particle starting at the origin with negative velocity.
- Backward equations: The single absorbing condition Q0^−(0, t) = 0 is sufficient to provide a unique solution to the coupled equations.
5. Mean first passage time
The mean first passage time to the absorbing origin is obtained from the survival formulation and analyzed across resetting regimes. It diverges for both very weak and very strong resetting, producing a unique intermediate-rate minimum.
- Definition: The mean first passage time T(Xr) is the mean time to absorption at the origin.
- Limiting case: The diffusive limit gives λr → (r/D)^1/2 and recovers the known result for diffusive resetting.
- Resetting-rate dependence: T(Xr) diverges as r^-1/2 when r → 0 and exponentially in r when r → ∞.
- Resetting-rate dependence: The divergence at both resetting extremes implies a minimum at an intermediate resetting rate.
- Reduced variables: Reduced variables R and ξ represent ratios involving resetting, velocity switching, target distance, and mean travel distance between reversals.
- Optimization: At fixed ξ, minimizing with respect to R gives a unique minimum, illustrated by T(R, ξ = 1) versus R.
6. General velocity resetting
The paper introduces general velocity resetting, parameterized by η, and derives renewal equations for survival and mean absorption time. Under symmetric initial conditions, the stationary state is η-independent, but absorption observables depend on η; velocity randomization is more efficient than position-only resetting.
- 6. General velocity resetting: η = 1/2 gives velocity randomization, while η = 0 gives position-only resetting.At each reset, velocity reverses with probability η and remains unchanged with probability 1 − η.
- 6. General velocity resetting: For symmetric initial conditions, the stationary state is independent of the velocity resetting protocol.The velocity distribution remains symmetric, causing the η-dependent terms to cancel in the renewal equations.
- 6.1. Survival probability: In the presence of an absorbing boundary, the survival probability depends on η.The general survival calculation requires joint survival and final-velocity distributions, leading to a system of renewal equations.
- 6.1. Survival probability: The general renewal scheme yields a Laplace-transform expression for total survival probability and recovers the η = 1/2 result as a special case.For η = 1/2, the coefficients simplify and the earlier renewal expression is recovered.
- 6.3. Position-only resetting: For fixed reduced variables, the mean absorption time is greater for η = 0 than for η = 1/2, and each protocol has an optimal resetting rate.Both mean-time curves have a unique minimum as functions of R.
- 6.3. Position-only resetting: The optimized diffusive search time is 1.54414…, compared with 5.48571… for run-and-tumble dynamics with η = 1/2.The corresponding optimal reduced resetting rates are R*=2.53964… for diffusion and R*=0.55873… for run-and-tumble dynamics.
7. Conclusion
The paper finds that resetting run-and-tumble particles produces an η-independent stationary state under symmetric initial conditions, while absorption behavior depends on the velocity protocol. Position-only resetting has a greater mean absorption time, and the renewal approach suggests extensions to broader settings.
- 7. Conclusion: The stationary distribution is Laplace-shaped and has the same form as that of a diffusive particle under position resetting.Its width decreases with γ and increases with the velocity correlation time.
- 7. Conclusion: The survival probability and mean absorption time depend on the velocity resetting protocol.The paper derives explicit mean-time expressions for velocity randomization and position-only resetting.
- 7. Conclusion: For fixed other parameters, position-only resetting gives a greater mean absorption time than velocity randomization.In reduced variables, an optimal resetting rate minimizes the mean absorption time.
- 7. Conclusion: The renewal equation approach facilitates the calculations and may extend to higher-dimensional run-and-tumble resetting.The authors also identify resetting of other correlated-noise processes as a possible application.
- 7. Conclusion: Future experiments may approximate velocity resetting by manipulating bacterial swimming dynamics with light.This is presented as a speculative experimental direction.