Experimentally achieving minimal dissipation via thermodynamically optimal transport

TL;DR

This work experimentally validates thermodynamically optimal transport by using scanning optical tweezers to drive Brownian microparticles along geodesics in distribution space, achieving minimal dissipation in finite time. It shows that the finite-time excess dissipation beyond the Landauer bound is precisely determined by the Wasserstein distance between the initial and final distributions, as captured by $w_d^{min}= γ D(p_0,p_τ)^2/τ$, and demonstrates this both for Gaussian translation/compression and for information erasure under controlled accuracy. The authors recover the time-dependent potential from distribution dynamics via the Fokker–Planck equation and quantify dissipation from distributions alone, without trajectories, illustrating a distribution-centric view of stochastic thermodynamics. They also reveal a universal energy–speed–accuracy trade-off, revealable through an experimental OT protocol, and contrast optimal transport with optimal control, highlighting the practical relevance for designing high-speed, low-energy information-processing systems.

Abstract

Optimal transport theory, originally developed in the 18th century for civil engineering, has since become a powerful optimization framework across disciplines, from generative AI to cell biology. In physics, it has recently been shown to set fundamental bounds on thermodynamic dissipation in finite-time processes. This extends beyond the conventional second law, which guarantees zero dissipation only in the quasi-static limit and cannot characterize the inevitable dissipation in finite-time processes. Here, we experimentally realize thermodynamically optimal transport using optically trapped microparticles, achieving minimal dissipation within a finite time. As an application to information processing, we implement the optimal finite-time protocol for information erasure, confirming that the excess dissipation beyond the Landauer bound is exactly determined by the Wasserstein distance - a fundamental geometric quantity in optimal transport theory. Furthermore, our experiment achieves the bound governing the trade-off between speed, dissipation, and accuracy in information erasure. To enable precise control of microparticles, we develop scanning optical tweezers capable of generating arbitrary potential profiles. Our work establishes an experimental approach for optimizing stochastic thermodynamic processes. Since minimizing dissipation directly reduces energy consumption, these results provide guiding principles for designing high-speed, low-energy information processing.

Figures (16)

• Figure 1: Optimal transport theory applied to thermal microparticles. (a) Transport of a sand pile in one dimension. The protocol with minimizes the cost, defined based on the transport distance, is achieved when the positional order is maintained so that the sand grains at the leftmost location are transported to the leftmost location, and so on. Optimal transport is a transport protocol that minimizes the cost. When the cost is defined based on the transport distance, the optimal transport of moving a sand pile in one dimension is achieved when the positional order is maintained so that the sand grains at the leftmost location are transported to the leftmost location, and so on. (b) Stochastic thermodynamics describes the thermodynamics in thermal fluctuating systems Seifert2012Ciliberto2017Pigolotti-Peliti. We think of transporting a probability distribution $p_0(x)$ at $t=0$ to $p_\tau(x)$ at $t=\tau$. Exemplified transport processes are shown on the right, which are the targets of this paper. (c) Geometric space with the Wasserstein distance (left). With the optimal transport in one-dimensional systems, each segment in the distribution is linearly transported without changing the positional order (right). (d) We experimentally implement the optimal transport by using a microscopic particle with a diameter of 0.5µ m trapped by a potential with a dynamically changing profile. We developed scanning optical tweezers that generate an arbitrary potential profile under the constraints determined by the device (SI Section \ref{['SI:Scanning optical tweezers']}). Right: examples of transport. Distribution of the particle positions (color) and potentials reconstructed from the experiments (solid curves).
• Figure 1: Optical tweezers system. Laser: SpectraPhysics V-106C-4000 (4W at maximum, 1064nm). Electric optical deflector (EOD) and amplifier: ConOptics 412-2Axis system. Attenuator: ThorLabs VA5-1064/M. Beam trap: ThorLabs BT610/M. CMOS camera: Basler ace acA1300-200. Microscope: Evident IX73. Objective lens: Evident UPlanSApo (100$\times$, NA1.40) customized for infrared. Multi-function IO board: NI PCIe-6374. L and M denote lens and mirror, respectively. M$\times$2 means that there are two overlapping mirrors that reflect light in the direction perpendicular to the paper and in the direction towards the right of the paper.
• Figure 2: Optimal transport in finite time with translation and compression protocol. (a -- c) Time evolution of probability distributions and potentials (a), mean $\mu$ (b), and width $d$ (c). The optimal protocol varies the potential profile so that $\mu_t$ and $d_t$ linearly vary. The naive protocol linearly varies the position and stiffness of the potential. The gearshift protocol combines two optimal protocols with different durations (fractions are 2/3 and 1/2) and speeds (ratio of 1 to 2). (a) Experimentally obtained distributions with Gaussian fittings and potentials for $\tau=50ms$. Open and closed circles indicate the centers of distribution and potential, respectively. The dotted curves in optimal and gearshift protocols are the potentials before the jumps of the potential position. (d) Trajectories in the ($\mu$, $d$) space, which implements the Wasserstein distance for Gaussian dynamics, for the same data in (a -- c). The optimal protocol is characterized by a uniform-speed transport on a geodesic (gray straight line) connecting the initial and final distributions. (e) The work $W$ vs the protocol speed $1/\tau$. $W$ was calculated based on Eq. \ref{['eq:W, F']}. Gray closed symbol corresponds to an experimental run consisting of more than 3,000 repetitions for $\tau\le200ms$ and 1,500 repetitions for $\tau=500ms$ for a particle. We performed four runs with four independent particles under each condition to measure the mean values (colored open symbols). Error bars indicate the standard error of the mean (s.e.m., four samples). The black open circle indicates the mean values of $\Delta F$ calculated from the initial and final distributions. The blue solid line indicates the theoretical minimum evaluated using the mean $\Delta F$ (0.680 $\pm$ 0.007, mean $\pm$ s.e.m. of all data of all protocols, 48 samples) as the intercept and the mean of $\tau w_\mathrm{d}^\mathrm{min}$ with $w_\mathrm{d}^\mathrm{min}$ calculated by Eq. \ref{['eq:Wdis:min']} as the slope. Some runs show $W$ values lower than this average theoretical minimum (also in Figs. \ref{['fig:landauer']}d and \ref{['fig:tradeoff']}), since the minimum $\Delta F+w_\mathrm{d}^\mathrm{min}$ differs from particle to particle even in the same condition due to the particle-dependent variation in $\gamma$ (Extended Data Fig. \ref{['exfig:power spectrum']}). We confirmed that each run satisfies the bound except for a few outliers due to statistical errors (Extended Data Fig. \ref{['exfig:Wdis:each']}). The colored thin solid lines connect experimental data of naive and gearshift protocols, which are extrapolated to the circle by dotted lines. (f) Evaluation of $w_\mathrm{d}$ from distributions without knowing individual trajectories (Eq. \ref{['eq:Wdis:W2']}). A typical example of gearshift protocol is shown. Inset: schematic of the segmentation. (g) Comparison of evaluation of $w_\mathrm{d}$ from recovered potentials (Eq. \ref{['eq:W, F']}) and from distributions (Eq. \ref{['eq:Wdis:W2']}).
• Figure 2: Exemplified scan pattern for the translation and compression protocol with $\tau=50ms$. The vertical axis is the voltage input for the EOD device (the same magnitude of voltages are applied to the two EOD devices), which diagonally translates the laser in the x-y plane to extend the scan range. The duration between red lines corresponds to a cycle, which was repeated more than 3,000 times for $\tau\le200ms$ and more than 1,500 times for $\tau=500ms$ for each particle. (b) is the magnification of (a). In this protocol, we generated Gaussian-profile potentials with a width of approximately 500nm. This width is sufficiently large so that the particle effectively feels a harmonic potential (see the first row in Extended Data Fig. \ref{['exfig:potential']}).
• Figure 3: Optimal information erasure in finite time. (a) Kymograph of the probability distributions constructed from 5,585 repetitions of information erasure with exemplified trajectories (solid). The cyan dashed curves indicate the tertile and mean of the distribution. (b) The distribution $p_t(x)$ and the recovered potential $V_t(x)$ under the optimal protocol. The optimal potential dynamics changed instantaneously at $t=0$ and $t=\tau$, similarly to the translation-compression setup. Each distribution is calculated from 31 successive video frames and spatially smoothened by being convolved with a Gaussian-shape window. (c) Accuracy of information erasure $\eta_\tau$ evaluated as the fraction of 0 at $t=\tau$. The inset is the bit erasure calculated as $\Delta H\times \log_2 e$ plotted against $1/\tau$. $\alpha$ is a parameter to control the accuracy and is the height ratio of the two peaks in the final target distribution. With $\alpha=0.5$, the potential is unchanged during the transport. (d) Work. Solid lines correspond to the theoretical minimum for work $\Delta F+w_\mathrm{d}^\mathrm{min}$, where we use the mean $\tau w_\mathrm{d}^\mathrm{min}$ for each $\alpha$ as the slope and the mean $\Delta F$ for each $\alpha$ as the intercept. Number of samples (particles) is three for each point in (c) and (d). Gray closed symbols correspond to each run of more than 5,000 repetitions. Colored open symbols are the mean of each condition. Error bars indicate s.e.m. (three samples for each).