Stochastic Resetting vs. Thermal Equilibration: Faster Relaxation, Different Destination

TL;DR

The paper compares relaxation to a non-equilibrium steady state under stochastic resetting with thermal relaxation for a Brownian particle in a harmonic potential. By introducing the dimensionless parameter $S=\gamma r/\kappa$, the authors show resetting always accelerates relaxation to a NESS and that the speedup follows a universal master curve as a function of $S$, while the steady state itself differs from Boltzmann equilibrium. They demonstrate that the full relaxation is governed by the slowest moment, with the second-moment timescale under resetting being $(S+2)^{-1}$ (in dimensionless units) compared to the thermal value $1/2$, leading to a speedup of $(S+2)/2$. A trade-off exists between speedup and spatial exploration, and resetting to positions far from the potential minimum can still accelerate relaxation, highlighting potential applications in fast state-to-state control and optimization of search processes in micro- and nanoscale systems. The results lay groundwork for optimized control strategies in experiments and simulations involving trapped Brownian particles and related stochastic systems.

Abstract

Stochastic resetting is known for its ability to accelerate search processes and induce non-equilibrium steady states. Here, we compare the relaxation times and resulting steady states of resetting and thermal relaxation for Brownian motion in a harmonic potential. We show that resetting always converges faster than thermal equilibration, but to a different steady-state. The acceleration and the shape of the steady-state are governed by a single dimensionless parameter that depends on the resetting rate, the viscosity, and the stiffness of the potential. We observe a trade-off between relaxation speed and the extent of spatial exploration as a function of this dimensionless parameter. Moreover, resetting relaxes faster even when resetting to positions arbitrarily far from the potential minimum.

Figures (5)

• Figure 1: KLD as a function of time for thermal equilibration (blue) and SR (red, $r=1.8 \,\mathrm{kHz} = 7.5 \omega_0$) using trap stiffness $\kappa = 2 \, \mathrm{pN /\mu m}$. The relaxation times (yellow circles) are determined when the KLD first drops below a threshold (dashed yellow line). This allows us to measure $\mathcal{T}_{\rm eq} =5.49 \, \mathrm{ms} = 1.31\omega_0^{-1}$ and $\mathcal{T}_{\rm SR} =1.14 \, \mathrm{ms}= 0.27\omega_0^{-1}$. The inset graphically shows the evolution of the probability distribution for both cases.
• Figure 2: (a) The speedup induced by SR, $\mathcal{T}_{\rm eq}/\mathcal{T}_{\rm SR}$, against $r$ for $\kappa= 1 \, \mathrm{pN/\mu m}$. Red circles for $r\neq0$, blue triangle underlines the thermal case $r=0$ as shown in the legend of (b). (b) Relaxation times against $\kappa$ for $r=1.12 \, \mathrm{kHz}$. The duration of relaxation under SR is smaller than the corresponding thermal process for all $\kappa$. This effect diminishes at large $\kappa$. (c) Speedup $\mathcal{T}_{\rm eq}/\mathcal{T}_{\rm SR}$, measured for various systems with different values of $r$ and $\kappa$, collapses onto a single master curve when plotted against the dimensionless parameter $S$, demonstrating that relaxation dynamics are governed solely by $S$. The dashed black line is the ratio between the timescales of the second moments, showing that the speedup in relaxation of the full distribution can be characterized by the relaxation of the moments.
• Figure 3: Speedup vs. spatial exploration shows a direct inverse relation, black dashed line follows the analytical expression $y=x^{-1}$, here $Speedup=(\sigma_{SR}^2/\sigma_{eq}^2)^{-1}$. Inset, red - speedup $\mathcal{T}_{\rm eq}/\mathcal{T}_{\rm SR}$, blue - relative spatial exploration $\sigma_{\rm SR}^2/\sigma_{\rm eq}^2$, dashed lines represent the corresponding analytical expressions red - $(S+2)/2$, blue - $2/(S+2)$.
• Figure 4: The NESS $P_{\rm s}(x)$of an SR process for various resetting positions from right to left: centered process $X_r^{(1)}=0$ (blue), $X_r^{(2)}\approx 2.25 \sigma$ (yellow), $X_r^{(3)}\approx 4.5\sigma$ (red). Increasing the distance between the resetting position and the center of the potential leads to skewed distributions, but also to larger spatial extension.
• Figure 5: The speedup is presented as a function of $S$ for various initial positions. Dashed lines (dashed black and solid blue) represent the ratio of the first and second moments' timescales, respectively. The speedup follows either the first or second moment ratios, depending on initial conditions, with some transition between these two regimes.