[Paper Review] Target search by active particles

Urna Basu, Sanjib Sabhapandit+1 more|arXiv (Cornell University)|Nov 29, 2023

Diffusion and Search Dynamics4 citations

TL;DR

This paper investigates target search dynamics of active particles—run-and-tumble, active Brownian, and direction-reversing active Brownian particles—using overdamped Langevin equations with colored noise to model self-propulsion. It derives exact expressions for first-passage and survival probabilities, showing that resetting enables finite mean first-passage times and reveals a persistence exponent of α=1 for direction-reversing active Brownian particles in the long-time regime.

ABSTRACT

Active particles, which are self-propelled nonequilibrium systems, are modelled by overdamped Langevin equations with colored noise, emulating the self-propulsion. In this chapter, we present a review of the theoretical results for the target search problem of these particles. We focus on three most well-known models, namely, run-and-tumble particles, active Brownian particles, and direction reversing active Brownian particles, which differ in their self-propulsion dynamics. For each of these models, we discuss the first-passage and survival probabilities in the presence of an absorbing target. We also discuss how resetting helps the active particles find targets in a finite time.

Motivation & Objective

• To understand how active particles with persistent motion search for targets in stochastic environments.
• To analyze first-passage and survival probabilities for three key active particle models: run-and-tumble, active Brownian, and direction-reversing active Brownian particles.
• To investigate the impact of stochastic resetting on reducing mean first-passage time in non-Markovian active search processes.
• To derive exact analytical expressions for survival and first-passage time distributions using Laplace transforms and inverse transforms.
• To determine the persistence exponent α in the long-time limit and identify scaling behaviors in the survival probability.

Proposed method

• Model active particles via overdamped Langevin equations with colored noise to capture self-propulsion dynamics.
• Use Laplace transforms to solve the Fokker-Planck equation for the position distribution, enabling derivation of survival and first-passage time probabilities.
• Apply inverse Laplace transform techniques, including residue calculus on the complex s-plane, to obtain time-domain solutions.
• Identify singularities in the Laplace-transformed position distribution and compute residues to express the solution as a sum over exponential terms.
• Use special functions such as the regularized hypergeometric function and the Gamma function to express exact scaling forms.
• Derive the survival probability in scaling form, showing dependence on the ratio $ y_0 / (4 ilde{\Lambda}t) $, and extract asymptotic behavior for large times.

Experimental results

Research questions

• RQ1 How do first-passage and survival probabilities behave for run-and-tumble, active Brownian, and direction-reversing active Brownian particles in the presence of an absorbing target?
• RQ2 What is the effect of stochastic resetting on the mean first-passage time in active particle search processes?
• RQ3 What is the persistence exponent α in the long-time limit for direction-reversing active Brownian particles, and how does it compare to Brownian motion?
• RQ4 Can exact analytical expressions be derived for the first-passage time distribution and survival probability in non-Markovian active search models?
• RQ5 How does the survival probability scale with time and initial distance to the target in the long-time regime?

Key findings

• The survival probability for direction-reversing active Brownian particles decays as $ S_y(t; y_0) o rac{ ext{const}}{t} $ at large times, indicating a persistence exponent α = 1.
• The exact first-passage time distribution is derived as $ F_y(t; y_0) = rac{y_0 ilde{\gamma}^{3/2}}{v_0^3 t^2 ilde{D}_R^{1/2}} fig( rac{y_0}{v_0 t} ilde{\gamma}^{1/2} / (8 ilde{D}_R)^{1/2} ig) $, with $ f(z) $ defined via the Gamma function.
• The survival probability exhibits a scaling form $ S_y(t; y_0) = gig( y_0 / (4 ilde{\Lambda}t) ig) $, where $ g(z_0) = 2 ilde{\int}_0^{z_0} dz f(z) $, and $ f(z) $ is expressed in terms of the Gamma function.
• In the regime $ \gamma^{-1} \ll t \ll D_R^{-1} $, the survival probability decays as $ S_y(t; y_0) \sim \frac{\Gamma(1/4)^2}{2\pi^{3/2}} \sqrt{\frac{\gamma}{D_R}} \frac{y_0}{v_0 t} $, confirming the power-law decay with exponent α = 1.
• The mean first-passage time becomes finite under stochastic resetting, with an optimal resetting rate minimizing the average search time.
• The exact solution for the position distribution is expressed as $ P(y,t;y_0) = \frac{1}{4\tilde{\Lambda}t} \big[ f\big( \frac{y-y_0}{4\tilde{\Lambda}t} \big) - f\big( \frac{y+y_0}{4\tilde{\Lambda}t} \big) \big] $, where $ f(z) = \frac{1}{\sqrt{2\pi^3}} \Gamma(\frac{1}{4} + iz) \Gamma(\frac{1}{4} - iz) $.