[Paper Review] Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: Experiments, theory and numerical tests

F. Faisant, Benjamin Besga+3 more|arXiv (Cornell University)|Jun 16, 2021

Diffusion and Search DynamicsBiochemistry, Genetics and Molecular Biology56 references89 citations

TL;DR

This paper presents a comprehensive theoretical, experimental, and numerical study of optimal mean first-passage time (MFPT) for a Brownian searcher with resetting in one and two dimensions, where the reset position is drawn from a Gaussian distribution of finite width σ. It reveals a critical phase transition in MFPT at a threshold ratio b = L/σ, with a metastable minimum in MFPT emerging for b > bc(a), demonstrating that finite reset position variance induces non-trivial optimal search dynamics beyond the standard resetting paradigm.

ABSTRACT

We study experimentally, numerically and theoretically the optimal mean time needed by a Brownian particle, freely diffusing either in one or two dimensions, to reach, within a tolerance radius $R_{ ext tol}$, a target at a distance $L$ from an initial position in the presence of resetting. The reset position is Gaussian distributed with width $\sigma$. We derived and tested two resetting protocols, one with a periodic and one with random (Poissonian) resetting times. We computed and measured the full first-passage probability distribution that displays spectacular spikes immediately after each resetting time for close targets. We study the optimal mean first-passage time as a function of the resetting period/rate for different target distances (values of the ratios $b=L/\sigma$) and target size ($a=R_ ext{tol}/L$). We find an interesting phase transition at a critical value of $b$, both in one and two dimensions. The details of the calculations as well as experimental setup and limitations are discussed.

Motivation & Objective

• To investigate how finite reset position variance σ affects optimal search efficiency in one and two dimensions.
• To extend the resetting paradigm beyond fixed initial positions to include spatially distributed reset points.
• To experimentally verify theoretical predictions of MFPT optimization in 2D using optical tweezers.
• To analyze the emergence of phase transitions and metastable minima in MFPT as a function of the dimensionless ratio b = L/σ.
• To provide a complete theoretical framework, numerical simulations, and experimental validation for periodic and Poissonian resetting protocols.

Proposed method

• Theoretical analysis uses the survival probability and first-passage time distribution for Brownian motion in arbitrary dimensions with reset to a Gaussian-distributed initial position.
• The mean first-passage time (MFPT) is computed via Laplace transforms and numerical integration of modified Bessel functions and error functions.
• Numerical simulations solve the 1D and 2D Langevin equations with time-discretized noise and reset events at periodic or Poisson-distributed intervals.
• Experimental validation uses optical tweezers to trap silica microspheres in water, with position tracking via camera and quadrant photodiode.
• The system is reset to a Gaussian-distributed initial position by reactivating the optical trap, and first-passage times are measured when the particle crosses a target line (1D) or enters a tolerance radius (2D).
• Theoretical predictions are compared with experimental and numerical data across varying parameters a = Rtol/L and b = L/σ.

Experimental results

Research questions

• RQ1 How does a finite reset position variance σ affect the optimal mean first-passage time (MFPT) in one and two dimensions?
• RQ2 Does a phase transition in MFPT emerge as a function of the dimensionless ratio b = L/σ, and if so, what is its critical value?
• RQ3 Can the theoretical predictions for MFPT with Gaussian-distributed reset positions be experimentally verified in 2D?
• RQ4 How do periodic and Poissonian resetting protocols compare in terms of MFPT optimization and metastable behavior?
• RQ5 What is the impact of initial position anisotropy (σx ≠ σy) on MFPT, and how does it affect experimental agreement with theory?

Key findings

• A critical value bc(a) ≈ 5 exists for a = 0.5, below which no metastable minimum in MFPT appears; for b > bc(a), a metastable minimum at c∗1(a,b) and a maximum at c∗2(a,b) emerge in the MFPT vs. resetting period curve.
• For finite b (i.e., σ > 0), the scaled MFPT Wperiodic(a,b,c) decays monotonically to zero as c → ∞, in contrast to the b → ∞ (σ = 0) case where it diverges.
• The MFPT exhibits a non-monotonic dependence on resetting rate, with a unique optimal resetting period c∗1(a,b) for b > bc(a), indicating a true minimum in search time.
• Experimental data in 2D with periodic resetting show excellent agreement with theory when using σ = σmean = √(σ²x,exp + σ²y,exp)/2, confirming the model's predictive power.
• Anisotropy in the initial position distribution (σx ≠ σy) causes deviations of up to ±20% in MFPT, especially for high resetting frequency (large c) and a → 1.
• The full first-passage probability density F(t) displays sharp spikes immediately after each resetting time, particularly for close targets (small a), indicating a burst-like search pattern.