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Experimental realization of diffusion with stochastic resetting
Ofir Tal-Friedman, Arnab Pal, Amandeep Sekhon, Shlomi Reuveni, Yael Roichman
TL;DR
The paper addresses the absence of a well-controlled experimental platform for studying stochastic resetting. It realizes colloidal diffusion and resetting with holographic optical tweezers, corroborates theoretical steady-state results, and measures resetting costs in steady-state and first-passage settings. The results show that realistic resetting costs cannot be made arbitrarily small.
Problem
Controlled experimental evidence for diffusion with stochastic resetting was lacking despite extensive theoretical study.
Method
The paper realizes colloidal diffusion with stochastic resetting using holographic optical tweezers and analyzes steady-state and first-passage resetting costs.
Results
The experiments corroborate central theoretical results and show that resetting has a non-arbitrarily-small energetic cost in steady-state and first-passage settings.
Takeaways & Limitations
The experimental method provides a platform for studying stochastic resetting phenomena beyond diffusion.
Takeaways & Limitations
The optical trapping method used in the study is far from the most efficient.
Abstract
from arXiv · showhide
Stochastic resetting is prevalent in natural and man-made systems giving rise to a long series of non-equilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. We provide the first experimental corroboration of central theoretical results, and go on to measure the energetic cost of resetting in steady-state and first-passage scenarios. In both cases, we show that this cost cannot be made arbitrarily small due to fundamental constraints on realistic resetting protocols. The methods developed herein open the door to future experimental study of resetting phenomena beyond diffusion.
Introduction
The paper addresses the lack of controlled experimental studies of stochastic resetting by realizing colloidal diffusion with resetting using holographic optical tweezers. It examines realistic constant-velocity and constant-time returns, their energy costs, and first-passage behavior.
- Introduction: The study fills a gap in stochastic-resetting research by providing a controlled experimental realization of diffusive resetting.The setup uses a colloidal particle and holographic optical tweezers.
- Introduction: Resetting returns the particle to the origin either at constant velocity or within a constant time, with exponentially distributed intervals between events.The mean interval between resetting events is 1/r.
- Introduction: Each protocol step alternates diffusion, optical-trap return, and a short waiting period for improved localization at the origin.The particle diffuses to (x_i,y_i), is dragged back, and then remains briefly trapped.
- Introduction: The experimental platform measures long-time position distributions, energy spent per resetting event, and mean first-passage times to spatial targets.Energy costs are studied in both steady-state and first-passage settings.
Stochastic resetting with instantaneous returns
For instantaneous returns, stochastic resetting confines an otherwise unbounded diffusing particle and produces a steady-state position distribution. The experimentally measured distribution agrees well with the theoretical prediction.
- Stochastic resetting with instantaneous returns: Repeated instantaneous resetting confines free Brownian motion and produces a non-equilibrium steady-state position distribution.Without resetting, the Gaussian variance grows linearly with time and the particle is not spatially bound.
- Stochastic resetting with instantaneous returns: The experiment estimates the x-axis steady-state distribution after digitally removing return and waiting phases from trajectories.This isolates the diffusive portions used for comparison with theory.
- Stochastic resetting with instantaneous returns: The experimentally measured x-axis distribution conforms well with the theoretical prediction.The comparison is shown in Fig. 2a.
- Stochastic resetting with instantaneous returns: The radial steady-state distribution also shows excellent agreement with theory.The radial coordinate is the distance R = √(x^2+y^2) from the origin.
Stochastic resetting with non-instantaneous returns
The paper tests physically realistic returns using optical tweezers at constant velocity or constant duration. Both theoretical radial steady-state distributions agree very well with experiments, while return dynamics affect the distribution outside the instantaneous limit.
- Stochastic resetting with non-instantaneous returns: Realistic resetting protocols include finite-duration returns at constant radial velocity and returns completed within a constant time.These protocols account for physical return processes absent from instantaneous-resetting models.
- Stochastic resetting with non-instantaneous returns: Bessel functions appear in the constant-velocity theory because the process and resetting protocol have rotational symmetry.The corresponding conditional densities describe diffusive and return phases.
- Stochastic resetting with non-instantaneous returns: The constant-velocity radial steady-state distribution agrees very well with experimental data.Experiments at v = 0.8µm/s are compared with Eq. (1), using independently measured D = 0.18±0.02µm^2/s and set r = 0.05s^-1 without fitting.
- Stochastic resetting with non-instantaneous returns: The finite-return equations interpolate between instantaneous returns and infinitely slow returns, with short times and high velocities behaving similarly.Marked differences emerge in the slow-return extreme because return statistics dominate the radial distribution.
Energy cost per resetting
The experiments quantify the energetic cost of stochastic resetting in steady state. Although faster returns reduce energy expenditure, optical-trap constraints impose a nonzero minimum cost.
- Energy measurement: The optical trap captures the diffusing particle and returns it to the origin, with energy per event given by E = Pτ(R).Here P is fixed laser power and τ(R) is the time required to trap and return a particle initially at distance R.
- Energy measurement: The resetting distance fluctuates between events, making the energy spent per resetting event random.Its distribution is derived from the resetting-time density and diffusion propagator, and agrees with experimental data.
- Velocity dependence: ⟨E⟩ ∝ v^-1, so increasing the return velocity lowers the mean energy cost per resetting event.This reduction holds until the return velocity approaches the maximum compatible with trapping the particle.
- Velocity dependence: The trap must satisfy k ≥ 2γv, because its stiffness must oppose drag and keep the particle confined.With stiffness proportional to laser power, this condition sets a maximum allowed return velocity.
- Minimum cost: The mean energy cannot be lowered indefinitely: a maximal feasible velocity produces a minimal energy cost per resetting event.Figure 4d identifies this optimum where the trap is just strong enough to prevent escape.
Energy cost per first-passage
The first-passage experiments test how stochastic resetting changes target-hitting times and quantify the associated energy expenditure. The results agree with theory while showing that realistic return protocols impose a lower energy bound.
- Experimental setup: The first-passage setup uses a virtual absorbing wall at x = L and holographic optical tweezers to return the particle to the origin stochastically.Experiments use both constant-time and constant-velocity return protocols.
- First-passage times: The theoretical mean-FPT expression agrees with experimental data and accurately predicts the resetting rate minimizing the mean first-passage time.The comparison includes non-instantaneous and instantaneous returns.
- Protocol constraints: For constant-time returns, the average return velocity can exceed vmax at low resetting rates, causing particles to escape and breaking the protocol.Thus, the apparent vanishing energy cost as r → 0 is not physically realizable across that range.
- Energy cost: For constant-velocity returns, every r > 0 gives ⟨EFP⟩ > PL/v, so the energy per first-passage event cannot fall below direct transport to L.The bound applies for v < vmax.
- Energy cost: Setting v = vmax yields ⟨EFP⟩ > 2Lγ/C, independent of laser power and return velocity.This establishes a protocol-independent lower bound within the stated trapping model.
Discussion and future outlook
The study establishes holographic optical tweezers as a controllable platform for experimentally realizing stochastic resetting, testing theory, and measuring resetting energetics. It also identifies extensions beyond current theories while highlighting the poor energy efficiency of the present optical-trapping method.
- Experimental platform: The platform experimentally realizes diffusion with stochastic resetting and allows many parameters to be controlled.The setup is presented as a unique and versatile method for studying stochastic motions and resetting protocols.
- Experimental platform: The experiments corroborate existing theoretical predictions and motivate studies of novel, more realistic resetting aspects.The authors frame the experimental results as a basis for extending research beyond established theoretical treatments.
- Energetic cost: The energetic cost of resetting is characterized in both steady-state and first-passage settings, revealing lower bounds on energy expenditure.Analytical expressions are combined with the physics of holographic optical tweezers to identify these bounds.
- Future outlook: The setup can be adapted to arbitrary dimensions, resetting time distributions, return protocols, and multibody systems with strong interactions.These extensions are described as going beyond the reach of existing theories and are identified as future experimental directions.
- Energetic cost: Developing energy-efficient resetting methods remains a future challenge because the optical trapping method is far from the most efficient way to apply force.The experiments used 1W of laser output to create a trap with k = 30pN/µm for a silica bead of radius a = 0.75µm.
- Energetic cost: The work required to reset position is orders of magnitude smaller than the energy actually spent when resetting uses holographic optical tweezers.For the reported experiment, the average energy expenditure was ⟨E⟩=3.68±0.05J, while the frictional work is much smaller.
Supporting Information Available
The paper’s supporting information is available free of charge and contains additional derivations, methods, and results.
- The supporting files are available free of charge.
- The supplementary information provides details of the theoretical derivations.
- The supplementary information includes experimental methods and other results.