Mechanical work extraction from an error-prone active dynamic Szilard engine

Holger Stark; Luca Cocconi; Paolo Malgaretti

REVIEW 3 major objections 4 minor 39 references

Mechanical work extraction from an error-prone active dynamic Szilard engine

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims a realistic, mechanically coupled Szilard engine can extract positive work from one actively propelled particle despite noisy measurements, and can convert measurement information into work at efficiency above Landauer's…

desk verdict The one-shot engine analysis is careful and worth engaging, but the headline cyclic efficiency gain rests on a free-dynamics approximation the authors admit is inexact at the operating point, so that part is not yet established. read the letter →

arxiv 2505.09523 v1 pith:YCOCFGL7 submitted 2025-05-14 cond-mat.stat-mech

Luca Cocconi , Paolo Malgaretti , Holger Stark This is my paper · ORCID

classification cond-mat.stat-mech

keywords activeSzilardenginerun-and-tumbleparticleinformation-to-workconversionLandauerboundmeasurementerrormechanicalpistoninformationefficiencystochasticthermodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a minimal model of a dynamic Szilard engine in which a demon measures the position and direction of a single run-and-tumble active particle, then places a damped mechanical piston ahead of it to lift a weight. It claims that even with finite measurement errors, positive work and power survive, and that the extracted work is maximized when the protocol lasts about a third of the persistence time and the piston is placed roughly two measurement-error widths ahead of the estimated position. The central thermodynamic assertion is that the one-shot information efficiency—extracted work divided by the information cost of the measurement—can exceed the equilibrium Landauer limit and grows with the Péclet number. Cyclic operation lowers the effective measurement cost by reusing residual mutual information between consecutive measurements, raising efficiency by a factor of four to five. If correct, this establishes a physically concrete route to active information engines that outperform their equilibrium counterparts.

What carries the argument

The load-bearing object is the one-shot average work expression $W_{\rm os} = (1-\varepsilon)p_{\rm fnt}W^+ + [\varepsilon + (1-\varepsilon)(1-p_{\rm fnt})]W^-$, assembled from the probability of placing the piston on the correct side, the distance-dependent contact delay $\tau_\delta$, and the exponential tumbling statistics of the run-and-tumble particle. The piston is a local steric element subject to an external force set at $F_w=1/2$ in units of the stall force, so misses and dissipation during free piston motion are intrinsic to the design. On the information side, the one-shot cost is $\Delta S_{\rm os} = -\ln(\sigma_x\sqrt{2\pi e}/L) + \ln 2 - s_\varepsilon$, and the cyclic cost is carried by the residual-information formula $\Delta S_\tau \simeq D_p\tau/(2\sigma_x^2) + \tau(1-2\varepsilon)\ln\big((1-\varepsilon)/\varepsilon\big)$, derived under free run-and-tumble dynamics between measurements. These pieces combine into the efficiency $\eta = W/(\mathrm{Pe}^{-1}\Delta S)$, and the claim that $\eta$ can exceed 1 follows from the scaling $\eta^*_{\rm os} \sim \mathrm{Pe}$ at fixed measurement precision.

What would settle it

Measure the information gain per cycle in a feedback experiment on a single tracked active particle: Eq. (28) predicts $\Delta S_\tau$ grows linearly in $\tau$ with slope $D_p/(2\sigma_x^2)+(1-2\varepsilon)\ln((1-\varepsilon)/\varepsilon)$, so repeating the measurement at two different positional-noise levels $\sigma_x$ and checking the predicted ratio of slopes would settle whether the cyclic advantage is real or an artifact of the free-dynamics approximation.

Watch

Extended reading notes

Core claim

The paper aims to establish that an active Szilard engine whose work extraction is mediated by a finite mechanical piston, rather than a time-dependent virtual potential, can convert measurement information into mechanical work with an information efficiency $\eta = W_{\rm os}/(\mathrm{Pe}^{-1}\Delta S_{\rm os})$ exceeding the Landauer value of 1 for sufficiently large Péclet number. It identifies nontrivial optima in the protocol duration $\tau$ and piston offset $\delta_m$: the best average work occurs near $\tau \simeq 0.3\,\tau_M$ and $\delta_m \simeq 2\sigma_x$, reaching about 15% of the ideal error-free power, and positive one-shot work is confined to a neighborhood of this optimum. Measurement imperfections act asymmetrically: the power optimum is independent of the direction-measurement error $\varepsilon$, while both work and power optima respond to the positional error $\sigma_x$. In cyclic operation, the mutual information from the previous measurement is not fully erased, so refreshing the estimate costs less than a fresh one-shot measurement, and the efficiency optimum is higher by a factor of 4–5 while also acquiring a negative-efficiency regime at large errors.

Load-bearing premise

The cyclic efficiency result depends on treating the particle's motion between measurements as free run-and-tumble dynamics, ignoring collisions with the piston and walls; if those collisions scramble the particle's state enough, the residual information that lowers the next measurement's cost would be smaller than estimated.

Editorial extensions

If this is right

At fixed measurement precision, optimal protocols are robust: the locations of the work and power maxima shift only weakly with $\varepsilon$ and $\sigma_x$, so no fine-tuning is needed.
The one-shot information efficiency exhibits a continuous transition: below a critical information gain (about 5 nats in the example), the nontrivial optimum has zero efficiency, then efficiency rises, peaks, and eventually falls as measurement precision is pushed further.
Cyclic operation dominates one-shot operation: residual mutual information between successive measurements reduces the effective information cost, giving a factor 4–5 higher efficiency, and any refractory pause between cycles strictly lowers efficiency.
The Landauer bound $\eta=1$ is crossed at $\mathrm{Pe}\simeq 10^3$ for the example parameters, and $\eta^*_{\rm os}\sim\mathrm{Pe}$ overall.
The independence of the power optimum from the directional error $\varepsilon$ is exact, because the factor $1-\varepsilon$ factors out of the extremization condition for the power.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The predicted work peak near $\delta_m \simeq 2\sigma_x$ and $\tau \simeq 0.3\,\tau_M$ suggests a concrete experimental search: scan piston offset and protocol duration under video-microscopy feedback on a single active colloid, and check whether the work contour and the zero-work boundary match the model.
The cyclic efficiency gain is computed with the piston and boundaries omitted from the inter-measurement dynamics; modifying the Green's function to include reflection off the piston is the natural next calculation, and it could shrink the factor 4–5 advantage.
The efficiency metric used here deliberately omits the steady dissipation needed to sustain the active self-propulsion; counting that cost as an additional operational expense would define a smaller efficiency, so the Landauer-violation result is specific to information-to-work accounting rather than total-budget accounting.
In two or three dimensions, a tumble does not immediately detach the particle from the piston, so the same mechanical design is likely to tolerate directional measurement errors better than the one-dimensional version treated here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: the one-shot analysis in this paper is careful and worth your time, but the headline cyclic result — the factor 4-5 efficiency gain — rests on an approximation the authors themselves flag as inexact at the operating point. That claim needs a simulation or a better calculation before it carries weight.

What is actually new: the mechanical piston coupling, the finite measurement error analysis for position and direction, the nontrivial optima in tau and delta_m, and the non-monotonic one-shot efficiency. Previous work on active Szilard engines used fully controlled external potentials. The steric piston introduces a 'miss' probability and a finite buffer distance, which is a real and underexplored element. The derivation of W_os in Sec. III is internally consistent given the one-tumble and large-Peclet approximations. I also give them credit for being explicit about when the approximations are loose, including the free-dynamics assumption in Sec. IV.B and the alternative efficiency that includes active dissipation.

Where the soft spots are, in order. First, the cyclic information gain in Eq. (28) uses free run-and-tumble dynamics with no piston and no boundaries, and the authors note it "won't be exact" at F_w = 1/2. The stress-test note is right: during the work phase the piston acts as a hard constraint, so the particle's position at the next refresh is conditioned by the piston placement in a way the free Green's function misses. The difference in Delta S_tau can be O(1), not a small correction. So the advertised factor 4-5 improvement is not established by this calculation. Second, the Landauer violation claim is partly conventional: Eq. (22) defines efficiency with a Pe^-1 factor, so eta ~ Pe is essentially put in by hand. The authors acknowledge this, but the abstract states it as a clean result. Third, the abstract calls the one-shot efficiency transition "discontinuous" while Sec. IV.A describes it as continuous. That is a contradiction a careful reader will trip on.

The one-shot results — the optima, the cost of precision, the non-monotonic efficiency — do not depend on the shaky cyclic step. If the paper is revised with a simulation of the cyclic protocol or a piston-constrained estimate of Delta S_tau, the cyclic claim becomes testable.

Who this is for: people working on active matter information engines and stochastic thermodynamics. It deserves a serious referee. My recommendation: send it out, but make sure the referee asks for the cyclic calculation to be backed up or softened, and the abstract/body inconsistency fixed.

Referee Report

3 major / 4 minor

Summary. The paper studies a one-dimensional run-and-tumble active Szilard engine in which a demon measures the particle position and direction with finite errors and places a mechanical piston ahead of the particle. The piston is pushed by the particle against an external force, extracting work over a protocol of duration τ. The authors derive an explicit one-shot average-work expression, locate nontrivial optima in (τ, δ_m), and analyze how these optima depend on the position error σ_x and direction error ε. They then define an information efficiency η = W_os/(Pe^{-1}ΔS_os), report that it can exceed 1 at large Péclet number, and interpret this as a violation of Landauer's bound. Finally, for cyclic operation they approximate the refreshed mutual information ΔS_τ using free run-and-tumble dynamics between measurements, obtaining a factor-of-4–5 improvement in efficiency at optimal protocol parameters.

Significance. If the results are fully established, the paper would be a useful contribution to the active-information-engine literature: it models a genuinely steric, local mechanical coupling rather than a globally imposed virtual potential, includes finite measurement errors in both position and direction, and provides closed-form one-shot work statistics in Appendix A. The one-shot derivation in Sec. III is internally consistent given the stated one-tumble and high-Péclet assumptions, and the authors are transparent about several limitations, including the inexactness of the cyclic information calculation at the operating force F_w = 1/2 and the choice not to count active dissipation in the efficiency. However, the headline cyclic-efficiency gain and the Landauer-violation claim rest on assumptions that are either explicitly invalid at the operating point or sensitive to the efficiency normalization, so the paper needs revision before the central claims can be accepted.

major comments (3)

[Sec. IV.B, Eqs. (24)–(29) and Fig. 5] The cyclic information gain ΔS_τ, which drives the advertised factor-of-4–5 efficiency improvement, is computed under approximation (i): free run-and-tumble dynamics with no piston interaction and no domain boundaries. The authors explicitly state that this is accurate for L/d_M ≫ 1 and F_w ≪ 1, but the relevant operating force is F_w = 1/2, where they say the results “won’t be exact.” This is load-bearing, not a minor caveat: at F_w = 1/2 the piston is a hard steric constraint that blocks the particle during contact and shifts its trajectory after collisions, so the unconstrained Green’s function in Eq. (27) can mis-estimate the posterior width and the residual mutual information by O(1). The factor-4–5 improvement in Fig. 5(b) is therefore not established for the work-optimal regime. I ask the authors to compute or bound ΔS_τ under the piston-constrained dynamics for F_w = 1/2 (for instance by numerical simulation of the constrained Langevin dynamics) or to restrict the cyclic-efficiency claim to the parameter regime where approximation (i) is controlled.
[Sec. IV.A, Eq. (22), and abstract] The claim of Landauer-bound violation is based on the efficiency η_os = W_os/(Pe^{-1} ΔS_os), where Pe = γ_p v_0 d_M/k_B T. Because W_os is expressed in units of the active energy scale γ_p v_0 d_M, this definition makes η_os ∼ Pe whenever the reduced work and information gain are O(1) and independent of Pe. The statement that η_os can be made arbitrarily large by increasing Pe is therefore a consequence of excluding the active dissipation from the cost, rather than a violation of the equilibrium Landauer bound. The authors do mention an alternative efficiency η̃_os that includes the active dissipation, but they do not use it in the main claim. The abstract should be reworded to say that the engine can exceed the equilibrium information-to-work bound for an information efficiency that excludes the active maintenance cost, not that it violates Landauer’s bound in the usual thermodynamic sense.
[Sec. IV.B, Eq. (28)] The small-τ expansion for ΔS_τ retains only the diffusive term D_p τ/(2σ_x^2) and the binary-state term, while neglecting tumbling contributions at O(τ^2). For the parameter values used in Fig. 5, τ* is indeed small, so the truncation is plausible. However, the derivation also uses the free-dynamics posterior from Eq. (27), which permits the particle to occupy both sides of the origin after a tumble and ignores the piston-induced shift of no-tumble trajectories. Since ΔS_τ is an entropy difference between this free-evolved posterior and the post-measurement Gaussian, the structural difference with the constrained process can change the result by more than the stated O(τ^2) error. The numerical verification in Fig. 5(a) checks only the approximation against the full free-dynamics expression, not against the constrained dynamics, so it does not resolve this concern.

minor comments (4)

[Eq. (27)] The delta function δ(x_τ + t) contains an undefined variable t; it should presumably be δ(x_τ + τ), consistent with the other terms in that equation.
[Eq. (A2)] The symbols γ2 and 2γ2 in C1 and C2 are undefined; they should presumably be γ_w and 2γ_w, respectively.
[Sec. II.A and Eq. (22)] The phrase “the natural unit of efficiency is the active Péclet number Pe” is confusing, since Pe is a dimensionless parameter, not a unit; the sentence could be rephrased to say that the reduced thermal energy scale is Pe^{-1}, which makes η naturally of order Pe in reduced units.
[Sec. IV.B, Eq. (26)] The notation G_τ^B(x′ − τ′) mixes a physical time τ′ with a Green’s-function duration τ′ in a way that is hard to follow; using separate symbols for the tumble time and the propagation time would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the work and efficiency calculations are self-contained, and the flagged items are an acknowledged approximation and a transparent normalization convention, not reductions of the derivation to its inputs.

full rationale

The derivation chain is self-contained. The one-shot work W_os is computed from the model probabilities (Eqs. (7)-(17)) with no parameter fitted to the claimed predictions; the optima in Figs. 2-3 follow from the analytic expression in Appendix A. The information cost Delta S_os is computed directly from the Gaussian measurement model (Eqs. (19)-(20)), and the cyclic cost Delta S_tau is derived from a stated free run-and-tumble Green's function approximation (Eqs. (23)-(28)), with the approximation and its F_w = 1/2 caveat explicitly acknowledged. The linear-in-Pe behavior of eta_os is a transparent consequence of the normalization in Eq. (22), stated as such by the authors ('as suggested by Eq. (22)'), and the Landauer comparison uses the independently computed W_os > 0; this is a modeling convention, not a reduction of the result to its inputs. Self-citations (Refs. [13,14,16]) support context and prior results, but the central work and efficiency calculations are re-derived in this paper. The free-dynamics approximation for Delta S_tau is a robustness limitation, not circularity.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The central calculation rests on standard stochastic thermodynamics and on several explicitly stated approximations: one-tumble truncation, high-Peclet neglect of diffusion in work, and free dynamics for the cyclic information gain. No parameters are fitted to external data, so the ledger is dominated by model assumptions rather than fitted constants.

free parameters (6)

Position measurement error sigma_x = varied; e.g. sigma_x/d_M = 0.01 in Fig. 2, 10^-2 in Fig. 5
Model input setting the Gaussian spread of position measurements, Eq. (4); scanned to map optima, not fitted to data.
Direction measurement error epsilon = e.g. 0.1 in Fig. 2 and Fig. 5
Probability of reading the self-propulsion direction incorrectly; scanned in Figs. 3-5, not fitted.
Piston force F_w = 1/2 in units of gamma_p v_0
Set to the value that maximizes contact power in Eq. (3); a control choice, not a fitted constant.
Piston friction gamma_w = 1 in numerical figures
Dimensionless piston-to-particle friction ratio; example value used in plots, not optimized.
Peclet number Pe = large; eta proportional to Pe
Activity scale; the claim that efficiency can exceed Landauer's bound uses Pe as an adjustable dial.
Domain length L = unspecified, with sigma_x much less than L
Enters the one-shot measurement cost Eq. (20) through ln(L/d_M); must be large for the Gaussian-posterior entropy approximation.

assumptions (8)

domain assumption Run-and-tumble dynamics with dichotomous noise and Gaussian translational noise, Eqs. (1)-(2)
Standard active particle model that underlies all trajectory and Green's function calculations.
domain assumption High Peclet number, Pe >> 1, so translational diffusion is neglected in the work calculation
Stated at the end of Sec. II; needed for deterministic piston-contact analysis in Sec. III.
domain assumption At most one tumbling event per protocol duration tau
Assumed in Sec. III.B and used in Eqs. (13)-(17); accurate for tau of order tau_M but not for longer cycles.
domain assumption Free run-and-tumble dynamics between successive measurements, ignoring the piston interaction and boundaries
Explicitly stated in Sec. IV.B(i) for the cyclic information gain; authors note it is inexact for F_w = 1/2.
domain assumption Uniform pre-measurement distribution P(x,w) = 1/(2L)
Used in Sec. IV.A to compute one-shot mutual information in Eq. (20).
domain assumption Gaussian measurement posterior, Eq. (4)
Model of finite position precision; real feedback systems could have non-Gaussian noise.
domain assumption Landauer cost of measurement k_B T_m Delta S
Standard information-thermodynamic result invoked from Refs. [17,20-23] to define efficiency.
domain assumption For short protocols, theta-function terms in Eq. (27) are negligible
Needed for the linearized Delta S_tau in Eq. (28); relies on tau/tau_M < 1.

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Cite this review

Pith. "Pith review of Mechanical work extraction from an error-prone active dynamic Szilard engine." pith.science (2026). https://pith.science/paper/YCOCFGL7

@misc{pith2026250509523,
  author       = {Pith},
  title        = {Pith review of: Mechanical work extraction from an error-prone active dynamic Szilard engine},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCOCFGL7}},
  note         = {Machine review of arXiv:2505.09523}
}

read the original abstract

Isothermal information engines operate by extracting net work from a single heat bath through measurement and feedback control. In this work, we analyze a realistic active Szilard engine operating on a single active particle by means of steric interaction with an externally controlled mechanical element. In particular, we provide a comprehensive study of how finite measurement accuracy affects the engine's work and power output, as well as the cost of operation. Having established the existence of non-trivial optima for work and power output, we study the dependence of their loci on the measurement error parameters and identify conditions for their positivity under one-shot and cyclic engine operation. By computing a suitably defined information efficiency, we also demonstrate that this engine design allows for the violation of Landauer's bound on the efficiency of information-to-work conversion. Notably, the information efficiency for one-shot operation exhibits a discontinuous transition and a non-monotonic dependence on the measurement precision. Finally, we show that cyclic operation improves information efficiency by harvesting residual mutual information between successive measurements.

Figures

Figures reproduced from arXiv: 2505.09523 by the authors.

**Figure 2.** FIG. 2. Average total work output (top) and work rate (bot [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Dependence of the nontrivial maxima of work and power output on the error parameters, shown for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Cost of measurement and optimum efficiency under [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Efficiency at optimum for cyclic operation. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Contours for work and power as a function of the er [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

C. G. Knott, Quote from undated letter from Maxwell to Tait, Life and Scientific Work of Peter Guthrie Tait. Cambridge University Press , 215 (1911)

work page 1911

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