Efficiency fluctuations and noise induced refrigerator-to-heater transition in information engines

Understanding noisy information engines is a fundamental problem of non-equilibrium physics, particularly in biomolecular systems agitated by thermal and active fluctuations in the cell. By the generalized second law of thermodynamics, the efficiency of these engines is bounded by the mutual information passing through their noisy feedback loop. Yet, direct measurement of the interplay between mutual information and energy has so far been elusive. To allow such examination, we explore here the entire phase-space of a noisy colloidal information engine, and study efficiency fluctuations due to the stochasticity of the mutual information and extracted work. We find that the average efficiency is maximal for non-zero noise level, at which the distribution of efficiency switches from bimodal to unimodal, and the stochastic efficiency often exceeds unity. We identify a line of anomalous, noise-driven equilibrium states that defines a refrigerator-to-heater transition, and test the generalized integral fluctuation theorem for continuous engines.

The work is defined as the difference between the energy of the particle in the trap potential before and after the pulse. Among the results is the demonstration of the possibility of extracting work defined this way "beyond the bound set by the generalized second law" and that the "generalized fluctuation theorem" is valid only in special circumstances.
The theory is linear and is based on the familiar solution of the Fokker-Planck equation. The obtained simple explicit expressions are discussed at length. However, a problem with the theory and the experiment is the concept of averaging. It seems that the averaging of the particle energy in the harmonic trap is not done over time, rather the energy change is calculated only during the pulse, Eq. (B20). However, the energy is fluctuating and, as seen from Eq. (B5), the variance of the fluctuations varies in time. I do not immediately see how the employed definition of the average energy and thus the average work can be justified.
The meaning and the importance of the "stochastic efficiency" needs to be explained, particularly that it seems to exist only in a limited parameter range. It is also necessary, if at all possible, to justify the relation to the DNA recognition, and more generally, to biological motors that indeed operate in the noisy environment, but nevertheless produce well-defined work.
Noisy feedback loops are common in technology and in nature. The effect of the noise is very well understood, in particular in various generators, atomic clocks, etc. Calling a feedback loop a "demon" adds hype, but not a deeper insight.
Except for the excessive hype, the scientific results are well described. On the positive side, the advantageous feature of the system is its simplicity, but then the motion of optically trapped Brownian particles has been studied in much detail. I find it important that the paper demonstrates a limited value of the extensions of thermodynamics to systems far away from thermal equilibrium. However, given the above criticism, I am not sure this is sufficient for justifying publishing the paper in Nature Communications, unless the authors can rebut the criticism. The paper may be more appropriate for a more specialized journal Reviewer #3: Remarks to the Author: In their manuscript, Paneru and colleagues reported on their investigation of noisy information engines with a colloidal particle. Their approach directly built upon a recent paper from the same team [17], where they built a colloidal engine with a noise-free feedback protocol. In the current work, they added random noise to the measured position after performing a nearly error-free position measurement of a colloidal particle. So the "demon-measured" position of the particle is y=x + error. The authors found that the system was in a "cooling" regime if the noise was less than the variance of the equilibrium distribution, and in the "heating" regime if the noise was more than the variance of the equilibrium distribution. The authors then performed detailed study of efficiency fluctuations due to the stochasticity of the extracted work and mutual information. The experiment and data analysis seem well-conducted. However, I personally do not find this work exciting. The experimental system is basically the same as the one in their recent paper [17]. The results are also not surprising.

Reviewer #1
The authors consider a Brownian particle subjected to a feedback control that allows to extract work from a single thermal reservoir. In other words they study an experimental model of an nonautonomous Maxwell demon. Since the demon measurements contain some error, the mutual information between the measurement outputs and the system quantifies the amount of work one may extract. An efficiency is introduced to quantify how close the actual conversion is from the ideal one (efficiency one). Efficiency fluctuations are also studied. Surprisingly, in finite time a finite error rate is found to improve work extraction. Overall the paper is well written and interesting. However, I have various concerns that I would like to see addressed.
We thank the Referee for the encouragement, and for constructive and thoughtful comments on our manuscript. The Referee comments that our work "is well written and interesting" but also raised some important points, which helped us to significantly improve the manuscript. We address the points one-byone in the following.
-My main concern is that the theoretical results that are used should be explained better in order to better appreciate the experimental results. I understand that most of it is in other papers, but this paper should be readable by non-experts and thus more self-contained.
Following the advice by the Referee, we have revised the manuscript accordingly as detailed below. Theoretical details are in the supplementary. The main text is made more self-contained by defining all the quantities like work, information and efficiency in support of the experimental work.
-The mutual information is never clearly defined.
The mutual information is now explicitly defined in the manuscript as "the mutual information between the particle position x and the measured outcome y". Its explicit calculation is also included in the text.
-What is work and which steps extract work it not discussed.
Following the comment by the Referee, this is now described clearly in the revised manuscript which reads: "In the overdamped regime the kinetic energy of the particle can be ignored, so the change in the potential energy when the trap shifts, ( ) V x Δ , is fully converted into heat and work. However, the potential is shifted much faster (within 20 μs ) than the typical relaxation time such that the particle has no time to move and dissipate energy. Therefore, all the potential energy gained by the shift is converted into work. During the relaxation step, since the trap center remains fixed, no work is performed on the particle, and only heat is dissipated. Thus, the work done on the particle during each shifting of the potential center is -In (4) it is said that the validity of the FT is tested. But it seems that the FT is only valid when S^*=S. I did not find a discussion about why this is the case. What is the meaning of the rhs of (4)?
Following this comment, we have now added discussion about the test of IFT and explained the meaning of rhs of Eq. (4) in the manuscript. Briefly, we tested experimentally and theoretically, the generalized integral fluctuation theorem, ( ) 1, which is valid for system under measurement and feedback control whose initial and final states are in equilibrium, and checked how far the average deviates from unity for our cyclic information engine with non-equilibrium initial and final states. The RHS of Eq. (4) shows amount of deviation of from unity as a measure of how far away the system is from equilibrium at the end of each cycle.
-The discussion on efficiency fluctuations is really brief. Is the transition from simple to double peak caused by essentially the same mechanism as the one discussed in Phys. Rev. Lett. 114, 050601 (2015)? If this is indeed the case, this point should be discussed.
We thank the Reviewer for this very helpful comment. We have added a new Section and a new plot (  (2016)) could also be mentioned.
We thank the Referee for these very relevant references; they have been added in the current version of the manuscript.
-Heat production in feedback controlled information erasure was always increased by errors in Diana et al. Phys. Rev. E 87, 012111 (2013). I wonder why work extraction is different and to what extend this is a system specific feature.
The work extraction and heat dissipation steps are now described in detail in the manuscript. Briefly, The average heat supplied to the system Q β during the relaxation step is minus the average work performed on the system during the feedback, . This shows that for , N S < the system is cooled immediately after the feedback control, and net heat flows from the reservoir to the system during the relaxation. The effective cooling decreases with increasing the error level until N S = at which 0. Q β = For , N S > the work performed on the system during the feedback is positive (heating), and net heat flows from the system to the reservoir during the relaxation. We provide example showing that heating and cooling are protocol dependent.

Reviewer #2
We thank the Referee for the encouragement, and for constructive, thoughtful comments on our manuscript. The Referee appreciates our work "the scientific results are well described. On the positive side, the advantageous feature of the system is its simplicity" but raises concern that trapped Brownian motion has been studied before. Additionally, the Reviewer comments, "I find it important that the paper demonstrates a limited value of the extensions of thermodynamics to systems far away from thermal equilibrium." At the same time, the Referee raises several important concerns, which we address in detail in the following, in particular the concern regarding the novelty compared to existing literature.
-The theory is linear and is based on the familiar solution of the Fokker-Planck equation. The obtained simple explicit expressions are discussed at length. However, a problem with the theory and the experiment is the concept of averaging. It seems that the averaging of the particle energy in the harmonic trap is not done over time, rather the energy change is calculated only during the pulse, Eq. (B20). However, the energy is fluctuating and, as seen from Eq. (B5), the variance of the fluctuations varies in time. I do not immediately see how the employed definition of the average energy and thus the average work can be justified.
Following the comment by the Reviewer, we have elaborated in the revised manuscript about the work and its averaging: Each engine cycle of duration τ consists of measurement of the particle position, shifting of the trap center, and relaxation. In the overdamped regime the kinetic energy of the particle can be ignored, so the change in the potential energy when the trap shifts, , is fully converted into heat and work, following the first law of thermodynamics. However, in our engine, the potential is shifted much faster (within 20 μs ) than the typical relaxation time such that the particle has no time to move and dissipate energy. Therefore, all the potential energy gained by the shift is converted into work. During the relaxation step, since the trap center remains fixed, no work is performed on the particle, and only heat is dissipated. Thus, the work done on the particle per cycle is equal to the work done during each shift of the potential center, , and the averging is done per cycle. Since resetting makes each cycle independent, our time averaging is equivalent to averaging over cycles. Our concept of averaging is consistent with prior theoretical works (ref. 6 and 10).
-The meaning and the importance of the "stochastic efficiency" needs to be explained, particularly that it seems to exist only in a limited parameter range. It is also necessary, if at all possible, to justify the relation to the DNA recognition, and more generally, to biological motors that indeed operate in the noisy environment, but nevertheless produce well-defined work.
Following the Reviewer's comment, the meaning and the significance of "stochastic efficiency" are explained in the revised manuscript as "Our measurement shows that the average efficiency / W I η β ≡ − is maximal for finite error level and long cycle period. However, this maximal efficiency is practically useless due to vanishing average power / 0 P W β τ ≡ − → in this limit. On the other hand, thermal fluctuations and fluctuations in the detector strongly affect the operation of these microscopic engines. For example, we can calculate from Eq. (2) that for / 0 N S = and large τ , the average work is maximal, the average values alone are not sufficient for understanding and designing information engines, and one must take into account fluctuations in the thermodynamic quantities such as work, heat, and information. Typical to fluctuating systems, the most probable efficiency, at the peak of the distribution, is more informative than the average. For small systems like ours, we find that the average and the most probable values have quite distinct physical behavior. Recent studies demonstrated that, due to the fluctuations in work and heat, the efficiency of a stochastic heat engine driven by nonequilibrium protocol is not bounded and often exceeds the limit of Carnot efficiency. Here, we study the stochastic efficiency / W I η β = − of an information engine owing to the fluctuations in work and mutual information (in the N S ≤ regime)." We studied efficiency fluctuations in the N S ≤ regime for which the engine extract positive work in average. Its study in N S > may not provide important insight because the engine cannot extract positive work in average there.
Related to DNA-recognition, we added following discussion in the revised manuscript: "… the efficiency measures how much information about the sequence can be extracted from one unit of binding energy. In other words, DNA recognition transforms energy to information, but still, the most efficient regime is similar to that of the colloidal engine that transforms information to energy." We also cited recent theoretical work by Ito and Sagawa (ref 14) that combines Maxwell's demon and signal transduction in living cells.
-Noisy feedback loops are common in technology and in nature. The effect of the noise is very well understood, in particular in various generators, atomic clocks, etc. Calling a feedback loop a "demon" adds hype, but not a deeper insight.
We utterly agree with the Reviewer that the notion of "demons" is equivalent to these of information engines or feedback loop. Therefore, following the Reviewer's advice we removed about 90% of the references to demons in the text and replaced them by information engines or feedback loops. We only kept the historical context in introduction, since the concepts of Maxwell demon and information engines are intertwined, and some readers are used to this terminology.
-Except for the excessive hype, the scientific results are well described. On the positive side, the advantageous feature of the system is its simplicity, but then the motion of optically trapped Brownian particles has been studied in much detail. I find it important that the paper demonstrates a limited value of the extensions of thermodynamics to systems far away from thermal equilibrium. However, given the above criticism, I am not sure this is sufficient for justifying publishing the paper in Nature Communications, unless the authors can rebut the criticism.
We thank the Reviewer for the positive assessment of the work. The important points of criticism raised by the Reviewer are answered in detail above. Indeed, optically trapped Brownian particles were studied before, so it is important to highlight the significance of the present work: We have proposed an exactly solvable continuous model of information engine. We show the first direct measurement of mutual information in nonequilibrium steady state and experimental verification of generalized integral fluctuation theorem. We demonstrate that measurement related nonequilibrium introduces unusual thermodynamic features, including the possibility of both heating and cooling regimes. Our studies of fluctuations in work, mutual information and efficiency are new and provide deeper and general insight into information engines. Particularly, the observation of bimodality of efficiency, and understanding its origin would be of broader interest to information theorists, nonequilibrium physicists and quantitative biologists. Additionally, most existing models are discrete, so our work would draw out the differences in physics with discrete systems.