Adding noise and scaling forces to speed up the Langevin clock

Abstract

Using experiments on a colloidal particle trapped in an optical tweezer, we confirm a recent proposal to increase the effective mobility or clock rate of systems described by Langevin dynamics, by simultaneously scaling deterministic forces and adding external noise. A corollary, which we also confirm experimentally, is that a system driven out of equilibrium by a time-dependent protocol can remain closer to thermal equilibrium. As an application, we demonstrate more precise recovery of free-energy differences from nonequilibrium work relations. Langevin clock rescaling provides a general strategy for accelerating calculations in the emerging field of thermodynamic computing, which uses stochastic devices governed by Langevin dynamics to do low-energy calculations.

Figures (11)

• Figure 1: Schematic of speed-up concept. (a) In a harmonic potential $U(x)$, a particle has a Gaussian position distribution $p(x)$ (blue shading) that is compiled from individual trajectories of particle position, $x(t)$ (orange trace). (b, c) From left to right, the potential $U(x)$ is scaled by a factor $\lambda$. From top to bottom, the temperature $T$ is scaled by $\lambda$. Scaling either $U(x)$ or $T$ alters the position distribution. (d) Simultaneously scaling $U(x)$ and $T$ leaves $p(x)$ unchanged, while speeding up dynamics.
• Figure 2: Experimental demonstration of the Langevin clock rescaling. (a) Time evolution of the measured particle position distribution $p(x,t)$ (markers) under the trap motion $c(t)$. The shaded distributions represent the equilibrium distribution corresponding to the trap position. The solid lines are predictions for our nonequilibrium transformation whose mean is given by Eq. \ref{['eqn:x(t)']}. At $\lambda=1$, the distribution stays farther from the trap center at the end of the protocol. But at higher $\lambda$, the distribution stays closer to the equilibrium distribution. (b) The trap center $c(t)$ moves at constant velocity over a distance $5\sigma_x$, within one relaxation time $t_\textrm{f}=\tau_\textrm{r}$. At $\lambda=1$, the ensemble-averaged position of the particle $\langle x_n\rangle$ remains farther from the trap position and does not reach a steady state. At $\lambda=10.8$, the mean position $\langle x_n\rangle$ relaxes faster and reaches a steady state much closer to the trap position. The solid curves are the predictions $\langle \tilde{x}(\tilde{t})\rangle$ based on Eq. \ref{['eqn:x(t)']}.
• Figure 3: Experimentally rescaling the Langevin clock alters work statistics and improves free-energy estimates. (a) The scaled reduced work distributions $p(W)$ for different scaling factor $\lambda$. At larger $\lambda$, distributions are narrower, and closer to $\Delta F=0$ (denoted by the dashed line). The solid lines are analytical predictions. (b) Forward (blue) and time-reversed (yellow) work distributions $p_\textrm{F}(W_\lambda),p_\textrm{R}(-W_\lambda)$. The Crooks free-energy estimator is obtained from their intersection. At higher $\lambda$, the distributions have more overlap and intersect closer to their mean values. (c) Increasing the scaling factor $\lambda$ improves the estimation of free-energy change $\Delta\hat{F}_\textrm{C}$, as obtained using the Crooks relation (Eq. \ref{['eq:Crooks_rel']}). The error-bars represent standard deviation obtained from Monte-Carlo simulation. The dashed line indicates equilibrium free-energy change $\Delta F=0$.
• Figure S1: Schematic diagram of the experimental apparatus. The 532 nm green laser is focused by microscope objective MO1 and traps a silica sphere. The forward scattered light from the 660 nm red laser is imaged on a quadrant photodiode (QPD) for position detection. Blue light from a 470nm LED is used to image the trapped sphere on the camera. An FPGA records the position data from the QPD and controls the deflection of the green laser using the AOD.
• Figure S2: Experimental flowchart for the spatial-temporal scaling protocol. (a) We first determine the baseline trap stiffness ($\kappa_1$) and diffusion constant at $\lambda = 1$. We then increase the trap laser power via an acousto-optic deflector (AOD) and (b) quantify the realized scaling factor $\lambda = \kappa_\lambda/\kappa_1$ by measuring the scaled stiffness $\kappa_\lambda$. (c) Using this $\lambda$, we calculate the requisite variance for the trap-center fluctuations using Eq. 7 to appropriately scale the effective temperature. (d) After scaled stationary conditions are reached, the nonequilibrium transformation is executed by translating the trap at a constant velocity.