Search by Return: Stochastic Resetting in Fluctuating Harmonic Potentials

TL;DR

This work analyzes stochastic resetting in a fluctuating harmonic trap, enforcing return termination at the origin to fix the starting point for subsequent outward searches. By decomposing each reset cycle into outgoing and return phases, the authors derive MFPT expressions for four protocols, including a novel SR-C scheme in which only return motion drives the search. A key result is that the return-only protocol yields a closed-form MFPT and can outperform standard SR in the strong confinement or small-reset-time regimes, illustrating a genuine ‘search by return’ paradigm. The study also quantifies energy and entropy costs, showing that the information-driven control shortens return times without extra mechanical energy, situating the approach within Maxwell-demon–type frameworks. Overall, the paper provides a rigorous, analytically tractable framework for designing and evaluating SR protocols that exploit controlled return dynamics in fluctuating traps, with clear implications for experimental implementations using optical or feedback traps.

Abstract

We study a class of stochastic resetting (SR) processes in which a diffusing particle alternates between free motion and confinement by an externally controlled potential. When the particle is recaptured, it undergoes a return trajectory that drives it toward a designated reset point. In standard SR, such returns are treated as instantaneous, but in realistic setups they have finite duration and introduce imprecision in the starting points of subsequent search attempts. We analyze a fluctuating harmonic potential in which return trajectories are forcibly terminated the moment the particle reaches the origin, ensuring that all outward (diffusive) trajectories begin from the same point. This is implemented through instantaneous positional information: a feedback signal that shortens the return phase without incurring additional mechanical energetic cost. We examine several search protocols built on this controlled return mechanism and determine their mean first-passage times (MFPTs). Of particular interest is a protocol in which outward diffusion is eliminated entirely and the return motion itself becomes the search mechanism. This "search by return" perspective reverses the conventional logic of SR and yields a closed-form MFPT that highlights the efficiency of using return dynamics as the primary search strategy.

Figures (12)

• Figure 1: Steady-state distributions as a function of dimensionless distance $z = x/\sqrt{D\tau_K}$ for various values of $\alpha$. The left column shows the contributions from the two states: black lines correspond to the "off" state distribution $f\, n_{\text{off}}$ as given in Eq. (\ref{['eq:Laplace']}), and red lines correspond to the "on" state distribution $(1-f)\, n_{\text{on}}$ given in Eq. (\ref{['eq:rho-on']}). The right column shows the total normalized distribution $n = f\, n_{\text{off}} + (1-f)\, n_{\text{on}}$, where red circles denote simulation data, confirming analytical predictions.
• Figure 2: Fraction of particles in the "off" state, $f$, as a function of $\alpha = \tau_K/\tau$, where $\tau_K = 1/(\mu K)$. Red circles are simulation results.
• Figure 3: First-passage-time distributions as a function of dimensionless time $u = t/\tau_K$. The left panel shows $R_{\text{on}}(s\,|\,z_0)$ for a particle initially located at $z_0 = 5$, while the right panel displays the total return-time distribution $r_{\text{on}}(u)$ for different values of $\alpha$. The exact expressions are given in Eq. (\ref{['eq:R-on']}) for $R_{\text{on}}(s,z_0)$ and in Eq. (\ref{['eq:app1F']}) for $r_{\text{on}}(u)$.
• Figure 4: Mean return time $\tau_{\text{on}}$ as a function of $\tau$. The dashed line marks the passive equilibration time $\tau_{\mathrm{eq}}\!\approx\!3\tau_K$. $\tau_{\text{on}}$ remains below $\tau_{\mathrm{eq}}$ except for very large ratios $\tau/\tau_K$.
• Figure 5: MFPT as a function of $\tau$ for the SR-A and SR-B protocols, corresponding to Eq. (\ref{['eq:Tsr-1B']}) and Eq. (\ref{['eq:tau-sr-3']}), compared with the simulation data points. The results are for $b=1/\sqrt{0.3}$, where $b=L/\sqrt{D\tau_K}$.