Optimized finite-time information machine
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Optimized finite-time information machine
Michael Bauer, Andre C. Barato and Udo Seifert
II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, GermanyPACS numbers: 05.70.Ln, 05.10.Gg
Abstract.
We analyze a periodic optimal finite-time two-state information-drivenmachine that extracts work from a single heat bath exploring imperfect measurements.Two models are considered, a memory-less one that ignores past measurements andan optimized model for which the feedback scheme consists of a protocol depending onthe whole history of measurements. Depending on the precision of the measurementand on the period length the optimized model displays a phase transition to a phasewhere measurements are judged as non-reliable. We obtain the critical line exactlyand show that the optimized model leads to more work extraction in comparison tothe memory-less model, with the gain parameter being larger in the region where thefrequency of non-reliable measurements is higher. We also demonstrate that the modelhas two second law inequalities, with the extracted work being bounded by the changeof the entropy of the system and by the mutual information.
1. Introduction
The thermodynamics of information processing is a very active area of research. Whereascentral concepts in this field have been developed a while ago [1–3], more recentlythe fluctuation relation obtained by Sagawa and Ueda [4] has shown that stochasticthermodynamics [5] provides a convenient framework to study the relation betweeninformation and thermodynamics. Moreover, ingenious experiments with small systems[6,7] verifying second law inequalities that involve information have played an importantrole in triggering the recent avalanche of papers. These works deal with the derivationof fluctuation relations and second law inequalities [4, 8–19] and the study of specificmodels [20–29].In finite-time thermodynamics the issue of optimal protocols is of centralimportance. A recent result within stochastic thermodynamics has been the observationthat the optimal protocol has discontinuities at the beginning and end of the finite-timeprocess [30–36]. In information processing, optimal protocols have so far been analyzedfor the maximal work extraction in a feedback driven system described by an one-dimensional over-damped Langevin equation [37] and for the minimum dissipated heatin an erasure process [38, 39].In this paper we study a paradigmatic discrete two-state model [28,34,40,41], wherethe work extraction, performed by lifting and lowering one energy level, is driven by ptimized finite-time information machine τ = 0 − − p p τ = 0 + E . . . τ = t − E t τ = t + − p t p t Figure 1.
Representation of the finite-time process. Initially, at τ = 0 − , the entropyof the system is H ( p ) = − p ln p − (1 − p ) ln(1 − p ). At τ = 0 + the level with lowerprobability p ≤ / ≤ τ ≤ t this energy level is lowered with protocol E τ . At time τ = t this energy level is set from E t back to 0 extracting work E t if thislevel had been occupied at τ = t − . feedback. Besides applying the optimal protocol leading to the maximal work during oneperiod, this information machine will also be optimized in the sense that the protocoltakes the whole history of measurements into account.We show that this optimized feedback strategy leads to more work extraction incomparison to a simple memory-less machine. Moreover, we observe a phase transition,where in one phase the machine always lifts the state indicated by the last measurementas empty and in the other phase the state measured as occupied is lifted with a certainfrequency. The extracted work is observed to be bounded by two quantities: the familiarmutual information between system and controller and the change of the entropy of thesystem. While the second bound is valid for every measurement trajectory, the firstbecomes valid only after an average over measurement trajectories is taken. Finally,we show that the memory-less model allows for a different physical interpretation ofthe system interacting with a tape, i.e., a sequence of bits. This memory-less modelthen corresponds to a generalization of the model recently introduced in [42] (seealso [41, 43–45]).The paper is organized as follows. In Sec. 2 we obtain the optimal protocol fora single period. The full feedback driven models are defined in Sec. 3. The phasetransition and gain parameter for the optimized model are studied in Sec. 4. In Sec. 5the different second law inequalities valid for the model are analyzed. We conclude inSec. 6.
2. Two-state finite-time process
The model analyzed in this paper is a two-state system where the time dependent energyof the upper level E τ ≥ ≤ τ ≤ t . The lower level has always energy zero. This system is connected to a heatbath at temperature T and a work reservoir. We consider a finite-time process withduration t , where both energy levels are zero immediately before starting τ = 0 − andimmediately after finishing τ = t + , see Fig. 1. These initial and final jumps of E τ area generic feature of optimal protocols [30]. ptimized finite-time information machine p τ , the time derivativeof the average internal energy reads ddτ E τ p τ = ˙ E τ p τ + E τ ˙ p τ , (1)where the dot represents a time derivative throughout the paper. This is the first lawof thermodynamics, where ˙ w = − ˙ E τ p τ is identified as the rate of extracted work and˙ q τ = E τ ˙ p τ as the rate of absorbed heat. This identification means that if a jump occursheat is exchanged with the heat bath and if the energy level changes work is exchangedwith the work reservoir. The extracted work in the time interval t then becomes W t = − Z t ˙ E τ p τ d τ + E t p t − E p , (2)where the boundary terms comes from the discontinuities in E τ represented in Fig. 1.Since the variation of the internal energy is zero, the extracted work equals the heatabsorbed from the heat bath, i.e., W t = Q t = Z t E τ ˙ p τ d τ. (3)Even though the system is connected to a single heat bath and the variation of theinternal energy is zero, it is still possible to extract work due to the increase in theentropy of the system. More precisely, the second law for such isothermal processestablishes that the extracted work is bounded by the change of the entropy of thesystem, i.e., W t ≤ k B T [ H ( p t ) − H ( p )] , (4)where H ( p ) ≡ − p ln p − (1 − p ) ln(1 − p ). In this paper we set k B T = 1 and, in order tohave work extraction, we restrict to the case p ≤ p t ≤ / The optimal protocol E τ that leads to the maximal work extraction for given timeinterval t and initial occupation probability p is calculated in the remaining of thissection. The master equation reads˙ p τ = − ω − p τ + ω + [1 − p τ ] , (5)where ω + ( ω − ) is the time dependent transition rate to (from) the upper level. Theserates must fulfill the detailed balance relation ω − /ω + = e E τ . For convenience, we choose ω + = e − E τ and ω − = 1 . (6)Following the analysis for a symmetric choice of rates [34], the optimal protocol and thecorresponding maximal extracted work is found by considering the Lagrangian L ( p, ˙ p ) ≡ ˙ p ln (cid:18) − pp + ˙ p (cid:19) , (7) ptimized finite-time information machine W t = Z t L ( p, ˙ p )d τ. (8)Since L ( p, ˙ p ) does not explicitly depend on τ , we have the following constant of motion, K ≡ L − ˙ p ∂L∂ ˙ p = ˙ p p + ˙ p ≥ . (9)Introducing the variable r τ ≡ p τ /K ≥
0, equation (9) becomes˙ r τ = 12 + 12 √ r τ ≥ . (10)The solution of this equation is˙ r τ = −
12 plog − ( − r e − r e − τ ) ≡ f τ ( ˙ r ) . (11)where plog − ( x ) is the lower branch of product logarithm [46]. Using relations r τ = ˙ r τ − ˙ r τ , K = p /r = p / [ ˙ r − ˙ r ], and (11), the extracted work (8) becomesa function of the single variable ˙ r . The maximal work is then obtained for ˙ r = a ,where dd ˙ r W | ˙ r = a = 0. In this way, a ( p ) is given by the solution of the transcendentalequation f t ( a ) (exp[1 /f t ( a )] + 1) − f t ( a ) = a ( a − p . (12)For convenience the optimal protocol and corresponding maximal work are simplydenoted by E τ ( p ) and W t ( p ), respectively. From (8) the maximal work that can beextracted for fixed t and p is W t ( p ) = − ln − p a − a [ f t ( a ) − f t ( a )]1 − p ! + p (cid:20) ln (cid:18) p a (1 − p )( a − (cid:19) + a − (cid:21) , (13)and the corresponding optimal protocol reads E τ ( p ) = ln (cid:18) p f τ ( a ) + a − ap f τ ( a ) − (cid:19) , (14)where a ( p ) is given by the solution of equation (12). In Fig. 2, we plot the maximalwork, the power and the discontinuities of the optimal protocol as a function of p forgiven t . The optimal work is a decreasing function of p , with full knowledge of theinitial state leading to the maximal work extraction for fixed t . By increasing t , thework increases whereas the power W t ( p ) /t decreases, going to zero in the limit t → ∞ .The initial and final energy jumps decrease with p , being maximal for p = 0. Theinitial jump E ( p ) increases with t , while the final jump E t ( p ) decreases with t . Moreprecisely, for t → E ( p ) = E t ( p ) and the difference between the jumps growswith t , with E ( p ) reaching its maximal value and E t ( p ) → t → ∞ .Finally it is useful for the following discussion to give p t for the optimal protocolexplicitly as p t = p a − a [ f t ( a ) − f t ( a )] . (15) ptimized finite-time information machine . W t p t = 0 0 . . ∞
05 0 0.25 0.5 E p /e W t / t p E t p Figure 2.
Maximal work W t ( p ) for different values of t on the left, with the power W t ( p ) /t plotted in the inset. The initial energy jump E ( p ) and the final energyjump E t ( p ) in the inset are plotted on the right.
3. Feedback driven machine
An information driven machine periodically repeats the process explained in the previoussection using feedback control. Measurements and feedback drive the work extractionby resetting the entropy of the system at the end of the time interval. We denote thestate of the system just before starting period i by x i and the measurement by m i ,where x i = − x i = +1) means that the left (right) state is occupied while m i = − m i = +1) corresponds to measuring the left (right) state as being occupied. Theconditional probability of the measurement is defined as P ( m i | x i ) ≡ ( − ǫ if m i = x i ,ǫ if m i = x i , (16)where ǫ is the measurement error. The machine never knows the real state of the system x i and has access only to the history of measurements m i = { m , m , . . . , m i } . Hence,in all calculations that follow the state of the system is always averaged out. First we consider a feedback procedure with a protocol taking the whole measurementtrajectory m i = { m , m , . . . , m i } into account. We are interested in the probability ofbeing at state x i given the history of measurements m i , which is denoted P ( x i | m i ). Forthis feedback scheme the initial occupation probability of the level that will be raisedat the beginning of period i is p ( i )0 = min { P ( x i = m i | m i ) , P ( x i = − m i | m i ) } ≤ . (17) ptimized finite-time information machine − m i independentof the measurement history it is also possible to make the unusual choice of lifting thelevel m i . In this second case, the level indicated by the last measurement as occupied islifted: the measurement is judged to be not reliable. Moreover, the machine applies theprotocol E τ ( p ( i )0 ), which takes into account the whole history of measurements by usingthe history dependent initial probability p ( i )0 .In Appendix A we show that the initial probability p ( i )0 fulfills a nonlinear recursionrelation. Denoting by p ( i − t the probability at the end of the period i − p ( i − , we define the functions F + ( p ( i − ) ≡ ǫp ( i − t − q ( i − t (18)and F − ( p ( i − ) ≡ ǫ (1 − p ( i − t ) q ( i − t , (19)where q ( i − t ≡ p ( i − t + ǫ (1 − p ( i − t ) . (20)The recursion relation for p ( i )0 then reads p ( i )0 = ( F + ( p ( i − ) if ˜ z i = 1 , min { F − ( p ( i − ) , − F − ( p ( i − ) } if ˜ z i = − . (21)As explained in Appendix A, the variable ˜ z i has the purpose of identifying whether themeasurement outcome m i corresponds to the upper or the lower level of the interval i −
1, with ˜ z i = − m i and ˜ z i = 1 if the lower level is m i . Wecall this machine taking the history of measurements m i into account the optimizedmachine because, as we will see in Sec. 4, it leads to more work extraction then a simplememory-less machine which we define next. A memory-less feedback scheme that only takes the last measurement into accountwould be to simply apply a protocol for which the level raised for the next period isjust the state measured as empty. Hence, for a measurement outcome m i , the level − m i is lifted at the beginning of period i . As we show in Appendix B, where the memory-less machine is more explicitly defined, the average initial occupation probability of theupper level is ǫ , independent of the protocol. Therefore, the appropriate choice for aprotocol that must be independent from the whole measurement history and correspondsto the memory-less version of the optimized machine is E τ ( ǫ ), which is obtained from(14) with p = ǫ . ptimized finite-time information machine
4. Gain and phase transition
The work extracted during period i with the optimized machine is denoted by W ( i ) t = W t ( p ( i )0 ). For a given measurement realization m N we define W t ≡ N N X i =1 W ( i ) t . (22)The average work h W t i is obtained by considering the limit N → ∞ and averaging overall measurement trajectories, where the brackets denote this average over measurementtrajectories. Numerical simulations for large enough N indicate that W t (andother observables we calculate below) is independent of the numerically generatedmeasurement history, i.e., self-averaging. Therefore, we calculate the average work bygenerating a single long measurement history.For the memory-less machine the average work is just W t ( ǫ ), as demonstrated inAppendix B. The improvement of the optimized in relation to the memory-less machineis quantified by the gain parameter α ≡ − W t ( ǫ ) h W t i . (23)Naively one expects the optimized machine that takes the history of measurements intoaccount to extract more work than the simple machine. This expectation is confirmedby numerical simulations, from which we observe that α ≥
0. For α = 0 the workextraction in the memory-less model would be the same as in the optimized model andfor α → α in the ( t, ǫ )-plane. The gain approaches 1 for small t and ǫ close to 1 /
2, wherenon-reliable measurements are more likely to occur.It turns out that the optimized model displays a phase transition. The orderparameter for this transition φ is the frequency at which the state m i is lifted, i.e., φ ≡ * N N X i =1 (1 − σ i ) + , (24)where σ i = − σ i = 1 if the measurementis reliable (see Appendix A for a precise definition). The numerical calculation of thisorder parameter is also shown in Fig. 3. We can clearly see a phase transition with φ = 0below a critical threshold ǫ c ( t ). Numerics indicates a second order phase transition.The optimized machine has two advantages in relation to the memory-less machine:it lifts the level m i if the last measurement is not reliable and it uses a history dependentprotocol E τ ( p ( i )0 ). By comparing α with φ in Fig. 3, we see that in the phase φ > φ = 0 the average initial occupation probability is ǫ . Hence, α > W t ( p ) plotted in Fig. 2 isconvex, implying h W ( i ) t i i ≥ W t ( ǫ ), where the average h . i i is defined in (A.11). ptimized finite-time information machine t ǫ t t ǫ t ǫ φ > α φ = 0 Figure 3.
The gain parameter α (left) and the order parameter φ (right) as functionsof the time interval t and the measurement error ǫ . The results are obtained bynumerically generating a measurement trajectory of length 10 . The full black criticalline is obtained analytically from (C.1) and the dotted line on the right panel from(C.2). As we show in Appendix C, the critical line ǫ c ( t ) can be obtained analytically fromthe transcendental equation (C.1). It is in perfect agreement with numerical results, asshown in Fig. 3.
5. Second law inequalities
The second law for feedback driven systems [10] states that the average extracted workis bounded by the average mutual information between system and controller due tomeasurements. The mutual information between the system and the controller due tothe measurement m i is defined as I ( i ) t ≡ X m i ,x i P ( m i , x i | m i − ) ln P ( m i , x i | m i − ) P ( m i | m i − ) P ( x i | m i − )= H ( q ( i − t ) − H ( ǫ ) . (25)We denote the average mutual information by h I t i , so that the efficiency of the optimalmachine reads η ≡ h W t ih I t i . (26)In Fig. 4 we show the numerically calculated efficiency η and power h W t i /t for theoptimized model in the ( t, ǫ )-plane. Increasing the time period t increases the efficiencybut decreases the power of the machine. For fixed t , the efficiency increases for increasingmeasurement error ǫ . Hence, maximum power is obtained for small ǫ and small t , whichis, however, a rather inefficient case with η . . ptimized finite-time information machine t ǫ t ǫ η h W t i t Figure 4.
The efficiency η and the average power h W t i /t as functions of the timeinterval t and the measurement error ǫ . The results are obtained by numericallygenerating a measurement trajectory of length 10 . Another bound on the extracted work is provided by the Shannon entropy change, asexpressed in (4), which for the optimized model can be written as W ( i ) t ≤ ∆ H ( i ) t , (27)where the Shannon entropy change in interval i is∆ H ( i ) t ≡ H ( p ( i ) t ) − H ( p ( i )0 ) . (28)The inequality (27) is then valid for a fixed measurement trajectory, whereas thestandard second law for feedback driven systems h W t i ≤ h I t i is valid only after anaverage over measurement trajectories is taken. By numerical inspection we observe that I ( i ) t (or I ( i +1) t ) can be smaller than W ( i ) t . Furthermore, by taking the average for large N we find h I t i = h ∆ H t i within numerical errors. This equality can be demonstrated withthe following heuristic argument. Rearranging the terms in the mutual information weget I ( i +1) t = H ( q ( i ) t ) − H ( ǫ )= H ( p ( i ) t ) − q ( i ) t H ǫ (1 − p ( i ) t ) q ( i ) t ! − (1 − q ( i ) t ) H ǫp ( i ) t − q ( i ) t ! = H ( p ( i ) t ) − h H ( p ( i +1)0 ) i i +1 . (29)where the average h . i i +1 is defined in (A.11). From this equation it is clear that theaverage mutual information and the average Shannon entropy change differ only byboundary terms, which for large N should be irrelevant, implying h I t i = h ∆ H t i . ptimized finite-time information machine As in Appendix B we now consider a memory-less machine using an arbitrary protocol˜ E τ . Equation (29) is also valid for the memory-less case and, therefore, the averageShannon entropy change should be equal to the average mutual information for large N . This result was confirmed numerically for the protocol ˜ E τ = E τ ( ǫ ) and for ˜ E τ = E ,which corresponds to the energy level held fixed during the whole time interval. Similarto (25), the mutual information depending on m i − reads˜ I ( i ) t = H (˜ q ( i − t ) − H ( ǫ ) , (30)where ˜ q ( i − t is defined in (B.5). Denoting by ˜ p t the solution of the master equation (5)with protocol ˜ E τ and initial probability ǫ we define˜ I t ≡ H (˜ q t ) − H ( ǫ ) , (31)where ˜ q t = ˜ p t + ǫ (1 − p t ). Since ˜ q ( i ) t (and ˜ p ( i ) t ) is linear in ˜ p ( i )0 , it follows that h ˜ q ( i ) t i i = ˜ q t ,where the average h . i i is defined in (B.6). From the fact that the Shannon entropy isconcave we obtain that ˜ I t provides an upper bound on the average mutual information h ˜ I t i ≡ * N N X i =1 ˜ I ( i ) t + . (32)As the average extracted work is equal to the work extracted in the first period(see Appendix B), if before the first measurement both states are equally probable, theaverage extracted work is also bounded by the Shannon entropy change in the firstperiod ∆ ˜ H t ≡ H (˜ p t ) − H ( ǫ ) . (33)Comparing with the other bounds we have ∆ ˜ H t ≤ ˜ I t and, numerically, for the protocols˜ E τ = E τ ( ǫ ) and ˜ E τ = E , we observe ∆ ˜ H t ≤ h ˜ I t i . We conjecture that this entropychange provides the strongest bound on the extracted work.The inequality˜ W t ( ǫ ) ≤ ∆ ˜ H t (34)for the protocol ˜ E τ = E has been recently studied in [41]. In this reference it hasbeen shown that the two-state model can also be interpreted as a tape, i.e., a sequenceof bits, interacting with a thermodynamic system. In this interpretation the entropychange is dumped to a tape or information reservoir. The inequality (34) means that theextracted work is bounded by the Shannon entropy change of the tape, which is initially H ( ǫ ) and becomes H ( p t ) after the system has written information to it. This model fora tape interacting with a thermodynamic system has been introduced by Mandal andJarzynski [42], for a model with six instead of two states and a protocol that is also heldfixed during the whole time interval. By showing that inequality (34) is valid also forarbitrary ˜ E τ protocols, we thus obtain that their model can be generalized to arbitrarytime-dependent protocols. ptimized finite-time information machine
6. Conclusion
We have studied a two-state finite-time optimized information-driven machine. Besidesutilizing the optimal protocol, this machine is also optimized in the sense that thefeedback scheme takes into account the whole history of measurements. We have shownthat the optimized machine leads to more work extraction in comparison to a simplememory-less machine that does not take the full measurement trajectory into account.This optimized model displays a phase transition with the frequency at whichnon-reliable measurements occur being the order parameter. In the region of thephase diagram where non-reliable measurements occur with a higher frequency thegain parameter, characterizing the improvement of the optimized in relation to thememory-less machine, was found to be high. Hence the possibility of lifting the statelast measured as occupied if the measurement is non-reliable is the key feature thatmakes the optimized model perform better. Moreover, analyzing the recursion relationsfor the initial occupation probability of the upper level we have obtained the criticalline exactly.We have shown that the work extraction is bounded both by the Shannon entropychange and the mutual information. While the first bound is valid for every measurementtrajectory the second is valid only after averaging over the measurements. In thiscase, both bounds become the same. Moreover, for the memory-less model we havedemonstrated that the average extracted work is bounded by the Shannon entropychange of the first period. This inequality allows for an interpretation of the model as athermodynamic system interacting with a tape, thus generalizing the model introducedin [42].
Appendix A. Iterative relation for the optimized model
In this appendix, we obtain the nonlinear recursive relation for the initial occupationprobability of the upper level of the optimized machine (21). From relations P ( m i | m i − ) = X x i P ( x i , m i | m i − ) (A.1)and P ( x i , m i | m i − ) = P ( m i | x i , m i − ) P ( x i | m i − ) = P ( m i | x i ) P ( x i | m i − ) , (A.2)we obtain P ( m i | m i − ) = (1 − ǫ ) P ( x i = m i | m i − ) + ǫP ( x i = − m i | m i − )= P ( x i = m i | m i − ) + ǫ [1 − P ( x i = m i | m i − )] , (A.3)where we used the definition of measurement error (16). Using the relation P ( x i | m i ) = P ( x i , m i | m i − ) /P ( m i | m i − ) (A.4) ptimized finite-time information machine P ( x i | m i ) can then be written as P ( x i | m i ) = (1 − ǫ ) P ( x i = m i | m i − ) P ( x i = m i | m i − )+ ǫ [1 − P ( x i = m i | m i − )] if x i = m i , ǫ [1 − P ( x i = m i | m i − )] P ( x i = m i | m i − )+ ǫ [1 − P ( x i = m i | m i − )] if x i = − m i . (A.5)Depending on the past measurements the probability P ( x i = m i | m i − ) on the right sideof this equation is either p ( i − t or 1 − p ( i − t , where p ( i − t is obtained from p ( i − andequation (15). From equation (17), the state indicated by the measurement as occupied m i is lifted provided P ( x i = m i | m i ) < P ( x i = − m i | m i ), which from (A.5) is equivalentto P ( x i = m i | m i − ) < ǫ . Since 1 − p ( i − t ≥ /
2, a necessary condition for lifting thestate m i at the beginning of period i is that p ( i − t < ǫ .It is convenient to define the variables z i ≡ m i m i − , for i >
1, and σ i , which takesthe value 1 ( −
1) if the level − m i ( m i ) is lifted at the beginning of period i , i.e., σ i ≡ ( P ( x i = m i | m i ) > P ( x i = − m i | m i ) , − P ( x i = − m i | m i ) > P ( x i = m i | m i ) . (A.6)Furthermore, we define ˜ z i ≡ z i σ i − . This last variable identifies whether for given m i the probability P ( x i = m i | m i − ) is p ( i − t or 1 − p ( i − t : P ( x i = m i | m i − ) = ( − p ( i − t if ˜ z i = 1 ,p ( i − t if ˜ z i = − . (A.7)In words, this equation means that if ˜ z i = 1 (˜ z i = −
1) then m i corresponds to the lower(upper) level of period i − p (1)0 = ǫ and σ = 1. Numerical simulations of the measurementtrajectory are then performed with the following algorithm:1) For period i randomly choose a measurement according to the probability P ( m i | m i − ) given by equations (A.3) and (A.7);2) with p ( i − t , the variable ˜ z i , and equations (17), (A.5) and (A.7) calculate σ i and p ( i )0 ;3) from relation (15) and p ( i )0 calculate p ( i ) t . Go back to the first step with thesubstitution i → i + 1.This algorithm can be translated into a recursion relation for the initial probability.Using (A.5) and (A.7), relation (17) becomes p ( i )0 = ( F + ( p ( i − ) if ˜ z i = 1 , min { F − ( p ( i − ) , − F − ( p ( i − ) } if ˜ z i = − , (A.8)where F + ( p ( i − ) ≡ ǫp ( i − t − q ( i − t (A.9) ptimized finite-time information machine F − ( p ( i − ) ≡ ǫ (1 − p ( i − t ) q ( i − t , (A.10)with q ( i − t ≡ p ( i − t + ǫ (1 − p ( i − t ). Note that the function 1 − F − ( p ( i − ) is minimalwhen F − ( p ( i − ) > /
2, which implies ǫ > p ( i − t . Only in this case, the state measuredas occupied m i can be lifted at the beginning of period i .Moreover, from (A.3), (A.7), and (A.8), the average initial occupation probabilityconditioned on m i − is h p ( i )0 i i ≡ X m i p ( i )0 P ( m i | m i − ) = min { p ( i − t , ǫ } . (A.11)If the machine never lifts the level m i , i.e., F − ( p ( i − ) > / h p ( i )0 i i = ǫ . Appendix B. Extracted work for the memory-less machine
For the memory-less machine we denote the initial occupation probability at period i by ˜ p ( i )0 . The final occupation probability at period i is ˜ p ( i ) t : as the memory-less machinedoes not use the optimal protocol, ˜ p ( i ) t is not obtained from (15) but rather it is thesolution of the master equation (5) for a given protocol ˜ E τ and initial condition ˜ p ( i )0 .Another difference in relation to the optimized model considered in Appendix A isthat the variable σ i is not necessary for the memory-less machine, since here σ i = 1 forall i . Hence, for the memory-less machine, equation (A.7) becomes P ( x i = m i | m i − ) = ( − ˜ p ( i − t if z i = 1 , ˜ p ( i − t if z i = − . (B.1)The iterative relation (A.8) then simplifies to˜ p ( i )0 = ( ˜ F + (˜ p ( i − ) if z i = 1 , ˜ F − (˜ p ( i − ) if z i = − , (B.2)where ˜ F + (˜ p ( i − ) ≡ ǫ ˜ p ( i − t − ˜ q ( i − t (B.3)and ˜ F − (˜ p ( i − ) ≡ ǫ (1 − ˜ p ( i − t )˜ q ( i − t , (B.4)with ˜ q ( i − t ≡ ˜ p ( i − t + ǫ (1 − p ( i − t ) . (B.5)Similar to (A.11), the average initial probability for fixed measurement history is h ˜ p ( i )0 i i ≡ X m i ˜ p ( i )0 P ( m i | m i − ) = ǫ. (B.6) ptimized finite-time information machine p ( i − p ( i )0 p † p ∗ . . . . F + F − p ( i − p ( i )0 − F − p † p ∗ . . . . F + F − p ( i − p ( i )0 − F − p † p ∗ . . . . F + F − Figure C1.
Cobweb diagram of all possible first five iterations, starting with p (1)0 = ǫ ,of the relation (21). In the left panel t = 2 and ǫ = 0 .
2, in the central panel t = 1 . ǫ = 0 .
4, and in the right panel t = 0 . ǫ = 0 .
4. For the first two cases φ = 0,whereas in the third case φ > We denote by ˜ W t (˜ p ( i )0 ) the work that is obtained from (3) with protocol ˜ E τ and initialprobability ˜ p ( i )0 . The average work is then given by h ˜ W t (˜ p ( i )0 ) i i = ˜ W t ( h ˜ p ( i )0 i i ) = ˜ W t ( ǫ ) , (B.7)where in the first equality we used the fact that ˜ W t (˜ p ( i )0 ) is linear in ˜ p ( i )0 . Since, ˜ W t ( ǫ ) isindependent of the history m i − , it follows that the average work is simply ˜ W t ( ǫ ). Forthe memory-less machine we compare with the optimized one ˜ E τ = E τ ( ǫ ), which is givenby (14), and the average work is W t ( ǫ ), which is given by (13). Moreover, assuming thatbefore the first measurement both states are equally probable, which leads to ˜ p (1)0 = ǫ ,the average work equals the work extracted in the first period. Appendix C. Critical line
We now obtain the critical line exactly by analyzing the iterative relation for p ( i )0 (21).As discussed in Appendix A, the level m i will be lifted at the beginning of interval i onlyif F − ( p ( i − ) > /
2. Hence, in the phase φ = 0 the condition F − ( p ( i − ) < / F + ( p † ) = p † and F − ( p ∗ ) = p ∗ . The possible trajectories in the cobweb diagram for the first five iterationsof relation (21) are shown in Fig. C1. It is clear that p ( i )0 does not go below the fixedpoint p † . Therefore, the critical line ǫ c ( t ) can be obtained analytically by setting F − ( p † ) = 1 / , (C.1)which leads to a cumbersome transcendental equation. Solving this equation we obtainthe full black line in Fig 3, in perfect agreement with the numerical simulations.Moreover, the phase ǫ < ǫ c can be further separated by two distinct regions, wherethe line (dotted line in Fig. 3) separating these two regions is obtained from F − (0) = 1 / . (C.2)In the region closer to the critical line with F − (0) > / p † is not small enough, i.e., F − ( p † ) < / ptimized finite-time information machine References [1] Szilard L, 1929
Z. Phys.
840 – 856[2] Landauer R, 1961
IBM J. Res. Dev. Int. J. Theor. Phys. Phys. Rev. Lett.
Rep. Prog. Phys. Nature Phys. Nature
Phys. Rev. Lett. Physica A
Phys. Rev. E Phys. Rev. E Phys. Rev. E Phys. Rev. Lett.
Phys. Rev. E Phys. Rev. E Phys. Rev. Lett.
Phys. Rev. Lett.
J. Stat. Mech.
P09011[19] Hartich D, Barato A C and Seifert U, 2014
J. Stat. Mech.
P02016[20] Cao F J, Dinis L and Parrondo J M R, 2004
Phys. Rev. Lett. EPL New J. Phys. EPL Phys. Rev. E EPL EPL Phys. Rev. Lett.
Phys. Rev. Lett.
EPL
Phys. Rev. Lett. Phys. Rev. E EPL J. Chem. Phys.
EPL Phys. Rev. Lett.
Phys. Rev. E J. Phys. A: Math. Theor. Phys. Rev. E Phys. Rev. E Phys. Rev. E Phys. Rev. Lett.
Proc. Natl. Acad. Sci. U.S.A.
Phys. Rev. Lett.
EPL
Phys. Rev. X ptimized finite-time information machine [46] Corless R, Gonnet G, Hare D, Jeffrey D and Knuth D, 1996 Adv. Comput. Math.5