Researchers Extract Work From a Fluid's "Memory," Reaching About 3× the Markovian Limit in One Case

When tiny particles in water are observed under a microscope, they keep moving irregularly because of collisions with water molecules. This phenomenon, called Brownian motion, is thermal fluctuation showing up as the motion of microscopic particles. In 1867, the physicist James Clerk Maxwell conceived of a hypothetical being that monitors the thermal motion of molecules and sorts them according to their speed. This thought experiment later came to be known as "Maxwell's demon."

In this thought experiment, using information about molecular motion appears to make it possible to create a temperature difference without adding any external work. That raised the question of whether it contradicts the second law of thermodynamics. Modern information thermodynamics, however, resolves the problem by considering the whole process: acquiring information through measurement, feedback control that uses it, and even erasing the memory that records it. Information processing is also a physical process, and when the thermodynamic balance of the whole system, including the apparatus, is taken into account, the second law is not violated.

This line of thinking developed into research on "information heat engines," microscopic systems that extract work. They measure something such as a particle's position and apply external control according to the result, using thermal fluctuations to obtain work. However, most previous experimental studies assumed a simplified condition in which the environment surrounding the particle retains no memory of the past.

A research team at the University of Konstanz and the Max Planck Institute for Dynamics and Self-Organization in Germany has succeeded in an experiment that goes beyond this assumption. The study was carried out by Lokesh Muruga, Felix Ginot, Sarah A. M. Loos, and Clemens Bechinger, and was published on October 16, 2026, in the journal Nature Physics as "The extraction of work from hidden degrees of freedom" (DOI: 10.1038/s41567-026-03442-w). A preprint was posted to arXiv on March 6, 2026, but this article relies primarily on the peer-reviewed paper.

What the team showed is that, using the "memory" left in the microstructure of a fluid that cannot be observed directly, it is possible to extract more work than the theoretical limit that assumes a memoryless environment. This neither violates the second law of thermodynamics nor creates energy from nothing. The researchers confirmed that the excess free energy of the composite system, which arises from fluctuations in thermal equilibrium and becomes usable depending on measurement outcomes, can be recovered through the memory remaining in the environment.

The effect of this memory changes greatly with the time interval between measurements. The effect weakens if the interval is too short or too long, and is most pronounced in a particular time window. The team introduced an order parameter $\Phi$ as an indicator of non-Markovianity and confirmed that the influence of memory is greatest when the measurement interval is about 1 second.

Treating the invisible micelle network as a "hidden degree of freedom"

In physics, the property that, once the state at a given moment is known, there is no need to know the past history to predict subsequent behavior is called the "Markov property." In a simple liquid such as water, correlations from molecular collisions are lost in an extremely short time, so on sufficiently long time scales the motion of a colloidal particle can be approximated well as a Markov process. In this model, the information and energy the environment receives through interaction with the particle dissipate quickly, and the particle's past motion does not affect its future behavior.

By contrast, the property that past history affects present motion is called "non-Markovianity." In fluids with complex structure, such as polymer solutions or the inside of living cells, when a particle moves, the surrounding molecular structure also deforms and takes a certain time to return to its original state. While the deformation remains, information about how the particle moved in the past is retained in the environment. As a result, the environment behaves not merely as a sink for heat but as a medium that remembers the particle's past motion.

To investigate this non-Markovianity, the team used colloidal silica particles 2.7 \mu\mathrm in diameter. The particles were dispersed in a viscoelastic micellar solution made by mixing cetylpyridinium chloride monohydrate and sodium salicylate at equimolar concentrations of 8 mM each. Unlike ordinary water, this solution has both elasticity, which resists deformation, and viscosity, which lets it flow.

At room temperature, the surfactant molecules in the solution form long, thread-like micelles that entangle with one another to create a dynamic network structure. The structural relaxation time of this micellar network is about 6 seconds, and deformation caused by the particle's motion relaxes gradually on this time scale. In other words, even when the particle moves, the surrounding network does not return to its original state immediately, and information about past deformation remains for some time.

In the experiment, the silica particle was trapped with "optical tweezers" that use focused laser light and confined in a harmonic potential, a field in which energy increases in proportion to the square of the distance from the center. The particle's position was recorded 100 times per second with a spatial resolution of \pm 5\mathrm. This made it possible to track in detail the tiny displacements of the particle caused by thermal fluctuations and how they change over time.

Importantly, the team directly observed only the position of the silica particle; it did not individually observe the deformation of the surrounding micellar network. The entanglement of the micelles and their internal deformation state were treated as "hidden degrees of freedom." By measuring the particle's position at different times and analyzing its subsequent motion, the team tried to read out indirectly the influence of the memory left in the invisible environment.

In conventional simple heat engine models, the unobservable degrees of freedom of the environment have mainly been treated as a destination into which energy and information dissipate. In this study, however, the team examined whether the temporal correlations held by these degrees of freedom could also be used to extract work. The focus of the experiment was whether that memory could be used without directly observing or controlling the internal structure of the environment.

A double measurement at 1-second intervals captures the fluid's lingering memory

To read out the environment's memory, the team adopted a "double-measurement protocol" in which the particle's position is measured twice with a time lag. The position is measured first, and after a set interval $\Delta t$ it is measured again. In each measurement, the team determined whether the particle was in the left or right region relative to a preset position and recorded the result as a binary 0 or 1.

The combinations of the two measurement results are four: $|00⟩$, $|01⟩$, $|10⟩$, and $|11⟩$. The left digit represents the result of the first measurement and the right digit that of the second. For example, $|10⟩$ means that the particle was on the right side of the threshold (state 1) in the first measurement and had moved to the left side (state 0) by the second measurement, after time $\Delta t$ had elapsed.

When the measurement interval $\Delta t$ was set to 1 second, a characteristic phenomenon appeared. Even among particles classified into the same state 0 in the second measurement, the subsequent relaxation behavior was clearly different depending on whether the first measurement result was 0 or 1. The team used trajectory data of 300 seconds and averaged about 1,000 measurement results for each condition to obtain relaxation curves.

In a Markovian environment, if the particle's current state is the same, its earlier position history does not change its future motion. In this viscoelastic fluid, however, the particle's past motion deforms the micellar network, and the effect returns to the particle with a delay. Therefore, even if the particle is in the same region in the second measurement, its subsequent relaxation changes depending on how it moved in the past.

The most striking case was the $|10⟩$ state. Under this condition, a non-monotonic relaxation was observed in which the particle's average potential energy temporarily increased and then decreased toward equilibrium. In ordinary relaxation, energy decreases toward the equilibrium value, but here there was a period in which it increased along the way. The team interprets this as experimental evidence of an "energy backflow" in which energy stored in the surrounding micellar network returns to the particle.

To check this interpretation, the team also performed a control experiment in which the measurement interval was extended to 200 seconds. When the interval greatly exceeded the network's structural relaxation time of about 6 seconds, the difference in relaxation behavior due to the first measurement result became virtually undetectable in the experiment. The surrounding network is thought to have relaxed enough that information about the moment of the first measurement no longer influenced the particle's subsequent motion. In the single-measurement limit, where the interval approaches zero, almost no additional effect from past history appeared either.

To quantify the influence of this memory, the team introduced the order parameter $\Phi$. Simply comparing position distributions after the two measurements cannot distinguish the effect of differing particle states right after measurement from the effect of the environment's memory. The team therefore constructed a control distribution that ignores past history but matches the particle's position distribution immediately after measurement, and compared how information decays afterward. In this way, they extracted a history dependence that cannot be explained by differences in initial conditions.

$\Phi$ is an indicator of how information about the particle's position decays compared with the history-free control system. If $\Phi \approx 0$, there is almost no influence from past history, consistent with Markovian behavior. A positive value means slower decay of information and a negative value means faster decay; in either case, a deviation from zero indicates that the environment's memory is affecting the motion.

In the experiment, $\Phi$ showed a clear maximum at a measurement interval of $\Delta t\approx 1$ second. This corresponds to the time window in which past information remains in the deformation of the micellar network and can still influence the particle's current motion. If the interval is too short, the difference between past and present cannot be captured well, and if it is too long, the environment's memory is lost. An interval of about 1 second was the condition under which the backflow of information from the hidden degrees of freedom could be captured most clearly.

What "three times the theoretical limit" means, and how results differ by measurement outcome

Having captured the environment's memory, the team next used that information to extract work. Immediately after the second position measurement, the trap center of the optical tweezers was moved at constant speed for 3 seconds. This operation translates only the position without changing the shape of the harmonic potential confining the particle. The work $W$ extracted from the system was calculated from the particle's position during this operation and the amount of potential displacement.

In the paper, positive work is defined as energy extracted from the system, and negative work as net work added to the system from outside. The team compared the work obtained for each measurement outcome with the theoretical maximum extractable work $W_m^{\text{*}}$ when the same kind of potential translation is performed on a memoryless Markovian reference system. For this harmonic potential, the limit is expressed as $W_m^{\text{*}}=\kappa ⟨X{⟩}_m^2/2$, using the particle's mean position after measurement $⟨X{⟩}_m$ and the spring constant $\kappa$ of the optical tweezers.

The figure reported in the paper, "up to about three times the theoretical limit," was obtained from this comparison. It does not mean that three times the total energy present in the system was extracted, nor that the limit set by the second law of thermodynamics was exceeded. It is purely a comparison with the theoretical limit of a specific control operation in a reference system that does not account for the environment's memory.

Furthermore, this roughly threefold result was obtained only for the $|10⟩$ case among the four measurement outcomes. For the other outcomes, the amount of extractable work and the contribution of the environment's memory differed greatly. The differences among the states reported in the paper are summarized below.

Measurement outcome	Relaxation and effect of memory	Comparison with Markovian limit	Characteristics of work extraction
$|00⟩$	No notable energy backflow; close to ordinary relaxation	Work is negative; none can be extracted	Work is added from outside as the potential is moved
$|01⟩$	Memory speeds up the decay of information	Positive work extracted, but below the theoretical limit	Extracted amount is small relative to the initial potential energy
$|10⟩$	Non-monotonic relaxation and pronounced energy backflow from environment to particle	Up to about three times the theoretical limit	Large contribution from free energy stored in the hidden degrees of freedom
$|11⟩$	Large initial potential energy of the particle; memory also affects relaxation	Positive work extracted, but below the theoretical limit	Largest absolute amount of extracted work among the four states

It is important to note that the ratio to the theoretical limit and the absolute amount of work actually extracted are different measures. The normalized work ratio was highest for $|10⟩$, but the absolute amount of work was largest for $|11⟩$. Therefore, it would be wrong to interpret the "threefold work" result as meaning that this state recovered the most energy.

In $|11⟩$, the particle was on the right side of the threshold in both measurements, and a relatively large average displacement remained immediately after the second measurement. The particle itself therefore has large potential energy, and moving the potential allows relatively more work to be extracted. The environment's memory also affects work extraction, but the extracted amount fell below the theoretical limit of the Markovian reference system.

In $|10⟩$, by contrast, a particle that was first on the right had moved to the left by the second measurement. Although the theoretical limit calculated from this state's mean position is small, the experiment extracted up to about three times that amount of work. Because work was obtained that cannot be explained by the potential energy calculable from the particle's position alone, the team concluded that free energy stored in the hidden micellar network contributed substantially.

For $|01⟩$, the particle's initial potential energy is close to that of $|11⟩$. However, the extractable work was far smaller than for $|11⟩$, and the time to reach its maximum was also shorter. This is consistent with the environment's memory speeding up the particle's relaxation in this state, so that energy recoverable as work is lost sooner. In $|00⟩$, the work was consistently negative, and no energy could be extracted.

Thus, the environment's memory does not always act to increase work extraction. Depending on the measurement outcome, it can speed up relaxation and reduce usable energy. What matters is not only knowing that memory exists but choosing appropriate control according to its state and time evolution. Numerical simulations performed by the team reproduced these outcome-dependent differences well, supporting the experimental results.

Why using a fluid's memory matters, and the challenges before practical use

The key achievement of this study is that it demonstrated, in an experiment, that environmental memory, which had mostly been discussed theoretically, can be used as a "thermodynamic resource." Without directly observing the state of the surrounding micellar network, the team read out how its memory affects the particle's motion simply by measuring the microscopic particle's position twice with a time lag. They also confirmed that by moving the optical tweezers according to the measurement results, part of the free energy stored in the environment can be extracted as work.

However, this result cannot be directly linked to practical microscopic generators or nanomachines. The experiment was simplified to verify the principle: the position measurement was limited to a binary determination of whether the particle was left or right of a threshold. Work extraction was also carried out with a constrained method, translating the harmonic potential at constant speed. The paper's authors point out that increasing the number of measurements, improving position measurement precision, or introducing more flexible feedback control could increase the extracted work further.

At the same time, such improvements would not let the extractable work grow without limit. The energy used in this experiment is constrained by the "excess free energy" present in the composite system of the observed particle and the hidden environmental degrees of freedom at the start of control. Using the environment's memory makes it possible to exceed the limit calculated from the particle's state alone, but not the thermodynamic constraints of the composite system as a whole.

There is also an important caveat regarding the energy balance of the whole experimental apparatus. The amount of work extracted here is extremely small, tied to the thermal fluctuations of a microscopic particle, and the electric power to drive the optical tweezers' laser, the power for the cameras and instruments that detect the particle's position, and the power consumed by the computer that processes measurement results and performs feedback control are not included in the evaluation of work extraction. Whether net energy could be extracted once these are included was not tested in this experiment.

In other words, this research did not develop a power generator that produces net electricity. It demonstrated that, in a complex fluid environment, free energy thought to be unusable in conventional simplified models can be recovered by using the environment's memory as a clue. It suggests that combining measurement and control that take the environment's behavior into account could improve the performance of microscopic heat engines.

These findings are also relevant to microscopic motion systems in living organisms. Environments such as the cytoplasm, blood, and mucus contain complex structures including polymers and cytoskeletal elements, and their mechanical responses can show temporal correlations. In such environments, unlike in simple water, past information retained by the surrounding structure affects the motion of particles and molecules, so non-Markovian effects can sometimes not be ignored.

The result offers new clues for research on microscopic machines that exploit such complex environments and on information-based control technologies. Going forward, the focus will be on how far work extraction efficiency can be raised through measurement methods that acquire more information and adaptive feedback control that changes its actions to match the environment's memory. At the same time, evaluating the overall balance, including the energy needed for measurement and control, will be essential for judging the prospects for practical use.

The environment is both a destination into which energy is lost and a holder of information about past interactions. This study showed experimentally that this memory can be read out by measurement and converted into work. It reaffirms the importance, in microscale thermodynamics, of treating the environment not merely as a place of dissipation but as an element actively involved in the exchange of information and energy.