Noise Breaks the Trajectory Bounds
Context
Information thermodynamics grew out of the Maxwell's demon paradox and the Szilard engine, and modern experiments with colloidal particles, electronic boxes, and micromechanical oscillators have made it possible to test generalized second laws. Around this work lie studies comparing work with different information measures: from transfer entropy to unavailable information and Markov mutual information. Neighboring papers in the archive deal with the entropy of climate swings, a Maxwell's demon with indistinguishable photons, and a gyroscope for climate, meaning the general vector is aimed at the connection between information, noise, and irreversibility. This paper stands somewhat apart because for the first time it directly compares three competing bounds on the same experimental setup with inertia, measurement noise, and temporal correlations all present at once.
Methods
The experiment is built on a conducting microcantilever that oscillates as a harmonic oscillator with a resonance frequency of 1100 Hz and a relaxation time of 2 ms. The cantilever position is measured by an interferometer with precision better than 1 pm, and an electrostatic force creates a potential whose center switches depending on the sign of the measured coordinate. Noise is introduced intentionally: uniform noise with amplitude 2Δx is added to the coordinate, and only then is the sign taken. Measurements are taken at intervals Δtm that vary from 0.22 to 5 relaxation times, allowing a transition from a strongly nonequilibrium regime with memory to a nearly equilibrium one. To estimate the entropy of the control sequence and the unavailable information, finite-memory estimates are used, and for mutual information the k-nearest-neighbors method is applied.
Results
Under nearly ideal measurements, the unavailable information bound is the tightest, but it does not coincide with work because control depends only on the sign of the coordinate, not on the full state. As noise increases, trajectory-based bounds degrade faster: the entropy of the control sequence grows, and unavailable information cannot keep up. As a result, Markov mutual information, which does not account for temporal correlations at all, becomes the tightest over a wide range of noise. Its sign change precisely tracks the disappearance of work extraction, that is, the moment when the system ceases to be an engine. The transfer entropy bound always remains weaker than the Markov one, but at sufficiently strong noise it can overtake unavailable information. The crossover is observed at all sampling intervals studied, although as the interval decreases, the difference between the bounds intensifies.
Implications
The work shows that a more detailed description of trajectories does not always yield a tighter thermodynamic bound. Unlike neighboring studies, which usually consider either ideal measurements or Markov control, here for the first time both noise and memory are systematically varied. This reveals a general mechanism: noise selectively destroys access to temporal structure, leaving instantaneous correlations relatively robust. Such a conclusion limits the applicability of trajectory measures under real conditions and shifts the focus to operationally useful quantities.
Future development
The topic could develop toward designing feedback strategies that exploit temporal correlations but remain robust to noise. Another path is the search for information measures that automatically balance between the detail of trajectory description and robustness to measurement uncertainty. Hybrid bounds may emerge that include only those correlations that can still be reliably estimated at a given noise level.
Impact
The results are important for all experimental information engines where noise and inertia are inevitable: from colloidal systems to nanoelectromechanical devices. They also provide a practical guide for choosing an information measure when analyzing data.
Next steps
Test the crossover on other feedback architectures, for example with continuous control or multivalued decisions. Investigate whether the coarseness of measurements can be adaptively changed to keep the system in a regime where trajectory bounds remain useful.
Key open problems
In statistical physics, it remains an open question which information measure is thermodynamically relevant beyond idealized limits. This work shows that the answer depends on the noise level and temporal correlations, that is, on context rather than on intrinsic properties of the measure. This is directly related to the problem of defining the cost of information in nonequilibrium systems with memory.
The microcantilever oscillates with an amplitude of about 1 nanometer, comparable to the size of a small molecule, and the interferometer's precision allows detecting a displacement of one-thousandth of that amplitude.
The idea of a demon that extracts work from thermal motion but makes measurement errors echoes Stanisław Lem's story 'The Second Kind of Demon,' where the demon faces the limitations of its own perception.
Key numbers
- resonance frequency: 1100 Hz
- relaxation time: 2 ms
- measurement precision: 1 pm
- thermal oscillation amplitude: 1 nm
- noise range: up to several σ