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Feedback!control and! fluctuation! theorems! in! classical systems! Juan MR Parrondo! GISC and Universidad Complutense de Madrid Observation ➔ Operation Jordan Horowitz Feedback!control and! fluctuation! theorems! in! classical systems! Juan MR Parrondo! GISC and Universidad Complutense de Madrid The Szilard engine.! Fluctuation theorems.! Feedback control.! Optimal control protocols.! Multiparticle Szilard engine. Jordan Horowitz The Szilard engine A measurement is necessary to implement a reversible expansion T Wextract = Z Wextract Vfin P dV = Vinit Z Vfin Vinit kT Vfin dV = kT log V Vinit 1 = kT log = kT log 2 > 0 1/2 Thermal bath Q W “Soft” Szilard 1 T Wextract 1 = kT log >0 1/2 Thermal bath ↵ ↵ Q W Feedback vs. blind protocols Feedback protocol: action depends on measurement. Wextract 1 = kT log 1/2 Wextract = kT log 1 1/2 Blind protocol: no measurement. L Wextract = kT log ? ? R Wextract = kT log Average work: Wextract h p R L Wextract + Wextract = = kT log 2 (1 2 i ) 0 1 1/2 1/2 Fluctuation theorems L Wextract = kT log ? ? R Wextract = kT log 1 1/2 1/2 Kawai, JMRP, van den Broeck, PRL (2007) Q+ E+ F = m hWextract i hWextract i = X m Probability of m T S tot + F m pF (m)hWextract i pF (m) kT log pB (m) kT X m pF (m) log Probability of m in the reverse process pF (m) 2nd law for cyclic 0 pB (m) isothermal processes Relative entropy 0 Fluctuation theorems and control Wextract = kT log Wextract 1 1/2 1 = kT log 1/2 Probability of m m hWextract i BUT now: + X m hWextract i = X m F pF (m) kT log m pB (m) pm B (m) 6= 1 m pF (m)hWextract i ( kT X ) The process depends on m pm B (m) = 1 m X m 0 Probability of m in the reverse process pF (m) log pF (m) pm B (m) Extracted work can be positive Not a relative entropy! Optimal control protocol hWextract i kT X m pF (m) pF (m) log m pB (m) m p The maximum work is obtained for: B (m) = 1 hWextract imax = kT X for all m pF (m) log pF (m) = kT Hshannon (m) m The reverse of this step prepares the system in state m Information gathered in the measurement T Wextract = kT H = kT log 2 Optimal control protocol The recipe: pm B (m) = 1 for all m The protocol after measuring m must be such that, when it is run backwards in time, prepares the system in state m The reverse of this step prepares the system in state m T An example: multiparticle Szilard engines PRL 106, 070401 (2011) Selected for a Viewpoint in Physics PHYSICAL REVIEW LETTERS week ending 18 FEBRUARY 2011 Quantum Szilard Engine Sang Wook Kim,1,2 Takahiro Sagawa,2 Simone De Liberato,2 and Masahito Ueda2 1 Department of Physics Education, Pusan National University, Busan 609-735, Korea 2 Department of Physics, University of Tokyo, Tokyo 113-0033, Japan (Received 23 June 2010; revised manuscript received 28 November 2010; published 14 February 2011) The Szilard engine (SZE) is the quintessence of Maxwell’s demon, which can extract the work from a heat bath by utilizing information. We present the first complete quantum analysis of the SZE, and derive an analytic expression of the quantum-mechanical work performed by a quantum SZE containing an arbitrary number of molecules, where it is crucial to regard the process of insertion or removal of a wall as a legitimate thermodynamic process. We find that more (less) work can be extracted from the bosonic (fermionic) SZE due to the indistinguishability of identical particles. DOI: 10.1103/PhysRevLett.106.070401 Maxwell’s demon is a hypothetical being of intelligence that was conceived to illuminate possible limitations of the second law of thermodynamics [1,2]. Szilard conducted a classical analysis of the demon, considering an idealized heat engine with a one-molecule gas, and directly associated the information acquired by measurement with a physical entropy to save the second law [3]. The basic working principle of the Szilard engine (SZE) is schematically illustrated in Fig. 1. If one acquires the information concerning which side the molecule is in after dividing the box, the information can be utilized to extract work, e.g., via an isothermal expansion. The crucial question here is how this cyclic thermodynamic process is compatible with the second law. Now it is widely accepted that the measurement process including erasure or reset of demon’s memory requires the minimum energy cost of at least kB ln2 (kB is the Boltzmann constant), associated with the entropy decrease of the engine, and that this saves the second law [4–7]. Although the SZE deals with a microscopic object, namely, an engine with a single molecule, its fully quantum analysis has not yet been conducted except for the measurement process [8,9]. In this Letter we present the first complete quantum analysis of the SZE. The previous literature takes for granted that insertion or removal of the wall costs no energy. This assumption is justified in classical mechanics but not so in quantum mechanics [10] because the insertion or removal of the wall alters the boundary condition that affects the eigenspectrum of the system. As shown below, a careful analysis of this process leads to a concise analytic expression of the total net work H= PACS numbers: 05.30.!d, 03.67.!a, 05.70.!a, 89.70.Cf performed perfectly. The case of imperfect measurement is discussed in terms of mutual information in Ref. [7]. To define the thermodynamic work in quantum mechanics, let us consider a closed system described as H c n ¼ En c n , where H, c n , and En are the Hamiltonian of the system, its nth eigenstate, and eigenenergy, respectively. The P internal energy U of the system is given as U ¼ n En Pn , where Pn is the mean occupation number of the nth eigenstate. In equilibrium Pn obeys the canonical distribution. From the derivative of U, one obtains dU ¼ P n ðEn dPn þ Pn dEn Þ. Analogous to the classical thermodynamic first law, TdS ¼ dU þ dW, where S and W are the entropy and work done by the system, respectively, the quantum P thermodynamic work (QTW)Pcan be identified as dW ¼ ! n Pn dEn [11,12]. Note that n En dPn should be 1 1 1 3 log 4 + log 4 + log 2 = log 2 4 4 2 2 Wextract = kT ✓ ◆ 2 2 log 2 + log 2 = kT log 2 4 4 We waste half a bit.Can we extract work from ? L=2. One finds f0( ¼ f1( ¼ 1 since in these cases the wall m reaches the end of the box so that Zðlm eq Þ ¼ Zm ðleq Þ ( ¼ 1 is always true (m ¼ 0; 1) is satisfied. Note that fm for m ¼ 0 and N. Together with f0 ¼ f1 ¼ 1=2, we obtain Wtot ¼ kB T ln2, implying the work performed by the quanSelected for a Viewpoint in Physics week ending tum SZE is equivalent SZE. P H Y S I C A LtoR Ethat V I E W of L E T Tthe E R S classical 18 FEBRUARY 2011 PRL 106, 070401 (2011) However, consideration of individual processes reveals an Quantum Szilard Engine important distinction between the classical and quantum Sang Wook Kim, Takahiro Sagawa, Simone De Liberato, and Masahito Ueda SZEs. For Department the quantum SZE one obtains WinsKorea ¼ %! þ of Physics Education, Pusan National University, Busan 609-735, Department of Physics, University of Tokyo, Tokyo 113-0033, Japan kB T ln2, Wexp ¼2010;!,revised and Wrem ¼28 0November for 2010; each process, where (Received 23 June manuscript received published 14 February 2011) P1 demon,%!E The Szilard engine (SZE) is the quintessence of Maxwell’s whichncan ðlÞextract the work from a ! ¼ ln½zðLÞ=zðL=2Þ&, zðlÞ e analysis,of theand Ederive heat bath by utilizing information. We present¼ the first complete SZE, and n ðlÞ ¼ n¼1 quantum an analytic expression of the quantum-mechanical work performed by a quantum SZE containing an 2 2 2 h n =ð8Ml Þ with hwherebeing the Planck constant. Inas the arbitrary number of molecules, it is crucial to regard the process of insertion or removal of a wall a legitimate thermodynamic process. We find that more (less) work can be extracted from the bosonic low-temperature ! isof simply given as E1 ðL=2Þ % (fermionic) SZE due to limit, the indistinguishability identical particles. PACS numbers: 05.30.!d, 03.67.!a, 05.70.!a, 89.70.Cf E1 ðLÞ. DOI: If 10.1103/PhysRevLett.106.070401 the insertion process were ignored in the classical SZE, the second law would be violated because ! ) kB T performed perfectly. The case of imperfect measurement Maxwell’s demon is a hypothetical being of intelligence in Ref. [7]. that was to illuminate possible limitations of the Inis discussed in theconceived low-temperature limit. fact, in!terms forof mutual the information expansion To define the thermodynamic work in quantum mechansecond law of thermodynamics [1,2]. Szilard conducted a ics, let us consider a closed for systeminserting described as H c ¼ classical analysis the demon, considering anby idealized process isof compensated the work required heat engine with a one-molecule gas, and directly associE c , where H, c , and E are the Hamiltonian of the ated the information acquired by measurement with a system, its eigenstate,between and eigenenergy, respectively. the wall. In the end, a tiny difference ofnthwork these physical entropy to save the second law [3]. The basic The P internal energy U of the system is given as U ¼ working principle of the Szilard engine (SZE) is schematiP , where P classical is the mean occupation number of two processes results in the Eprecise value, cally illustrated in Fig. 1. If one acquires the information the nth eigenstate. In equilibrium P obeys the canonical whichAs side the molecule is in after dividing the distribution. From derivative of U, one obtains dU ¼ kconcerning the temperature increases, thetheclassical results B T ln2. P box, the information can be utilized to extract work, e.g., ðE dP þ P dE Þ. Analogous to the classical thermovia an individual isothermal expansion. The crucial question here of processes areis recovered, Wins 0,W are dynamic first law, TdSi.e., ¼ dU þ dW, where! S and how this cyclic thermodynamic process is compatible with the entropy and work done by the system, respectively, the the exp second! law. k Now it isln2, widely accepted the rem mea- ¼quantum thermodynamic (QTW)Pcan be identified as W and thatW 0 since !work approaches BT P surement process including erasure or reset of demon’s dW ¼ ! P dE [11,12]. Note that E dP should be requires thethis minimum energy cost of at least T ln2 in limit. kmemory B k ln2 (k is the Boltzmann constant), associated with the entropy the engine, and that this saveswith the For decrease the ofquantum SZE more than one particle, second law [4–7]. Although the SZE deals with a effects microscopic object, dramatic quantum come into play. Let us consider namely, an engine with a single molecule, its fully quantum has not yet been conducted for the mea-in a symmetric potential well aanalysis two-particle SZE except confined surement process [8,9]. In this Letter we present the first ( ¼ f( ¼ 1 for the completeaquantum analysis of¼ the L=2. SZE. The One previous also litand wall at l finds f 0 2 erature takes for granted that insertion or removal of the distinguishability for bosons (fermions) as the temperature increases. The inset of Fig. 3 clearly shows that one can extract more work from the bosonic SZE but less work from the An example: multiparticle Szilard engines 1,2 2 2 2 1 2 n n n n n n n n n n n n n n n B n n n n n n B wall costs no energy. This assumption is justified in classical mechanics but not so in quantum mechanics [10] because the insertion or removal of the wall alters the boundary condition that affects the eigenspectrum of the system. As shown below, a careful analysis of this process leads to a concise analytic expression of the total net work performed by the quantum SZE. If more than one particle is present in the SZE, we encounter the issue of indistinguishability of quantum identical particles. Indeed, how much work is extracted from the quantum SZE strongly depends crucially on whether it consists of either bosons or fermions. We also show that the crossover from indistinguishability to distinguishability occurs as the tem- FIG. 3 (color online). f0 as a function of T for bosons (solid curve), fermions (dashed curve), and classical particles (dashdotted line) in the case of the infinite potential well. The temperature is given in units of E1 ðLÞ=kB . (a) Three possible ways in which two identical bosons are assigned over two states. (b) Four possible ways in which two distinguishable particles are allocated over two places. The inset shows Wtot =Wc as a function of T. 070401-3 FIG. 1 (color online). Schematic diagram of the thermodynamic processes of the classical SZE. Initially a single molecule is prepared in an isolated box. (A) A wall depicted as a vertical gray bar is inserted to split the box into to two parts. The molecule is represented by the dotted circles to indicate that at this stage we do not know in which box the molecule is. (B) By the measurement, we find where the molecule is. (C) A load is attached to the wall to extract a work via an isothermal expan- We waste half a bit. Can we extract work from ? An example: multiparticle Szilard engines Designing Optimal Feedback Engines 8 WB IB on the compressed box size. To simplify our analysis, we only consider boxes such that lx = 2ly . In figure 3, we plot WB IB as a function of the box size parameter = lx /d = 2ly /d. The smaller the smaller the box. Notice that WB IB < 0. We also 0.0 WB IB 0.1 0.2 0.3 3.0 3.5 4.0 4.5 5.0 ⇥ Figure 3. Plot of the deviation from reversibility WB IB for the two-particle Szilard engine protocol implemented in response to measuring each particle in a separate half of the box (outcome B) as a function of the box size parameter = lx /d = 2ly /d. observe that the process becomes reversible (WB IB = 0) when < 4 (lx < 4d and ly < 2d); the box is so small when < 4 that both particles cannot fit into the same No information is wasted (W=kTH) if the box is small and the particle have a finite size.! Many particles (Hal Tasaki) Traps This protocol is optimal if the number of traps exactly matches the number of particles in each side of the box. A simple preparation protocol measurement Slow evolution Conclusions hWextract i kT X m pF (m) pF (m) log m pB (m) Not normalized The protocol after measuring m must be such that, when it is run backward in time, prepare the system in state m Thanks!