Carnot's theorem and Szilárd engine. School of energy and power engineering, Huazhong University of Science & Technology. Wuhan, China.

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1 Carnot's teorem and Szilárd engine Liangsuo Su 1,2, Xiaokang Liu 1, Suyi Huang 1, Siping Jin 1,2,3 1 Sool of energy and power engineering, Huazong University of Siene & enology. Wuan, Cina. 2 Innovation Institute, Huazong University of Siene & enology. Wuan, Cina. 3 Cina-Europe Institute for Clean and Renewable Energy, Huazong University of Siene & enology. Wuan, Cina. Abstrat In tis work, te relationsip between Carnot engine and Szilárd engine was disussed. By defining te available information about te temperature differene between two eat reservoirs, te Carnot engine was found to ave a same pysial essene wit Szilárd engine: lossless onversion of available information. us, a generalized Carnot's teorem for wider sope of appliation an be desribed as all te available information is 100% oded into work. Introdution As one of te most fundamental pysis laws, te seond law of termodynamis an trae its origin bak to Carnot's teorem obtained by Sadi Carnot in 1824, wi limits te maximum effiieny for any possible engine. However, James Clerk Maxwell reated a tougt experiment (Maxwell's demon) in wi te seond law migt be violated in In 1929, Leó Szilárd invented an engine wi an transform eat from an isotermal environment into work at te ost of te information onsumption of a Maxwell's demon[1]. After te efforts of many researers[2 5], Maxwell's demon was proved to be onsistent wit te seond law. In reent year, benefit from advanes in tenology, various kinds of Maxwell's demons ave been realized in laboratories wit metods of bot pysis[6,7] and emistry[8 10]. In biology, te speial annel struture of aquaporin-1 was also found to enable it to work as a suspet Maxwell's demon by reognizing and onsuming te information about te size differene between different solute moleules[11]. More teoretial semes of eat engine based on Maxwell s demon ave been put forward[5,12 17]. However, tere is still mu ontroversy on weter tese kinds of eat engines (espeially te quantum ones) respet Carnot's teorem[15,16,18] or not[19 22]. In a previous work, we ave generalized te priniple of entropy inrease for isolated system to te liangsuo_su@ust.edu.n

2 priniple of available information derease for any system. e available information of an observer about a system is te quantifiational measurement of te auses of any observable effets. In tis work, we analyze te available information in te Carnot engine and Szilárd engine and find tat tey two ave a same pysial essene: lossless onversion of available information. Carnot's teorem was generalized as te available information is 100% oded into work. Available information in Carnot's yle and Szilárd engine Fig.1 Forward yle of Carnot engine(a) and Szilárd engine(b). For Szilárd engine, a Maxwell s demon extrats tis I* and odes it in te eat pumped from te environment (Q ) to output work(w Szilárd); For Carnot engine, te isotermal endoterm is a proess of writing available information of I*, wi is oded in output work (W Carnot) subsequently and rejets te waste eat of Q witout available information. Supposing tere are two eat reservoirs, a ot one and a old one, wit temperature of and respetively, for a Carnot engine (Fig.1.a) driven by an amount of eat (Q ) from ot reservoir, te maximum work W Carnot will be, WCarnot Q Carnot Q (1 ) (1) were η Carnot is te Carnot effiieny, te maximum effiieny for any possible engine aording to te seond law of termodynamis. e eat got by old reservoir will be, Q Q W Q Carnot (2) e available information about te temperature differene arried by te amount of eat Q at liangsuo_su@ust.edu.n

3 an be desribed as below 1, * Q 1 1 IQ ( ) (3) k In a eat transfer witout any work output, I * Q will be onsumed and results in an entropy inrease of k I * Q. If I * Q serves as te information soure of te Maxwell s demon in a Szilárd engine (Fig.1b) in old reservoir, te eat pumped from te old reservoir and transformed into work will be, * WSzilard k IQ Q (1 ) (4) Comparing Eq. (1) and (4), we an find tat te Szilárd engine obeys Carnot's teorem. In a yle, te available information of eat reservoir dereases, * * Q 1 1 I IQ ( ) 0 (5) k te available information of old reservoir remains unanged, * Q 1 1 I ( ) 0 (6) k All te available information about te temperature differene is taken away by te work, leaving no available information in Q. Comparing Eq. (4), (5) and (6), Carnot's teorem an be desribed in te language of available information as, te available information about te temperature differene between two eat reservoirs arried by te eat is 100% oded into work. For Szilárd engine ere, in order to make te Maxwell s demon able to get all of I *, te eat transfer of Q from to must be a reversible proess experiening an infinite number of intermediate eat soures to avoid additional onsumption of available information during eat transfer. e net eat obtained by old reservoirs will be, 1 康神整理工作 liangsuo_su@ust.edu.n

4 (7) ' Q Q Q Q WSzilard Q erefore, after a omplete yle of a Carnot engine or a Szilárd engine, te states of te two eat reservoirs are te same. In fat, if bot of tem are plaed in blak boxes, an outside observer will find no metod to distinguis tem. Heat engine onneting eat reservoirs in non-equilibrium states ad been disussed by Hasegawa et al. [23]and akara et al.[24], and were found to be able to output more work tan tose onneting equilibrium states. In fat, beside te available information about temperature differene desribing te imbalane between system and its environment, te internal imbalane of a system is also one kind of available information. For Szilárd engine, tis internal available information of ot eat reservoirs an also at as part of information soure wile some residual available information must be retained as te origin of te internal available information of old reservoirs. For a system omposed of nearly-ideal gas of moleules, its internal available information an be desribed by Boltzmann's H funtion 2. erefore, te possible maximum available information oded in work in te yle will be, * * Inon equilibrium I H H (8) were H and H are te H of te ot and te old reservoirs respetively. If H =0 (final state is an equilibrium state), te work will take its maximum value(w max(non-eq)), wi an be greater tan te value of a lassi Carnot engine (Eq.(1) and Eq.(4)). Wmax( non eq) k ( I * H) (9) Beside temperature differene, oter kinds of internal and external imbalanes are also available information. erefore, Carnot's teorem an be generalized as, All te available information is 100% oded into work. Classi Carnot engine annot utilize H of a non-equilibrium state. In te ase wi loal equilibrium assumption an be met, one possible approa to utilize it is to split te ot eat soure into a number of internal balaned subsystems between wi many small Carnot engines or moleular information ratets[8 10] an run. ey annot violate te generalized Carnot's teorem 2 liangsuo_su@ust.edu.n

5 above. Reversed yle If Q is moved from a old reservoir to a ot reservoir, te ot reservoir will get bot Q and te available information about te temperature differene between te old reservoir and itself arried by Q, wi must be replenised from outside beause Q from te old reservoir as no available information (Eq. (6)). * * Q 1 1 I IQ ( ) (10) k For te reversed yle of bot te Carnot engine (Fig.2.a) and Szilárd engine (Fig.2.b), tis replenisment an be aieved troug external work (W ). For Carnot engine, te work is, W ' Carnot Q ( 1) (11) For Szilárd engine, te work is W k I (12) ' * Szilard were is te operating temperature of information writing proess. wen is equal to, W Szilárd will take its minimum value (if is lower tan, te information writing proess annot be ompleted). min ( 1) (13) ' * WSzilard k I Q Fig.2 Reversed yle of Carnot engine(a), Szilárd engine(b) and general Maxwell s demon. In te above two ases, te total eat got by te ot reservoir will be Q = Q /. erefore, in a reversed yle, an outside observer still annot distinguis Szilárd engine from Carnot engine. For liangsuo_su@ust.edu.n

6 general Maxwell s demon (Fig.2.) beside Szilárd engine, te replenisment an also be aieved by onsuming oter available information su as te information stored in DNA or a omputer s ard drive and te information about te differene between AP and ADP. Conlusion and disussion From te disussion above, it an be found tat te pysial essene of Carnot engine and Szilárd engine is te same: 100% transmission of available information. Compared wit equal amount of eat at te same temperature of, work arries additional available information, * * W( Q, I ) Q I (14) I (15) * W k were is te information operating temperature. For a forward yle of bot Carnot engine and Szilárd engine, I *, arried by Q about te temperature differene between ot and old reservoir, is enoded in work; In a reversed yle, I * from work at as te available information soure to over te sortfall of available information. From Eq. (4), (10) and (15), it an be found tat te amount of information arried by work anges wit te operating temperature. is means tat te stored information an also be used as a working medium of a yle. Fig.1a as sown te available information proess of Carnot engine from te view of eat reservoir regarding te engine as a blak box and taking no aount of kineti details of te yle. In te supplementary materials, tis problem was analyzed from te view of Carnot engine and te generalized Carnot's teorem is found to still be followed. In a omplete yle, a eat engine based on Maxwell s demon, weter a lassi or quantum one, an only use existing available information (already stored in demon s memory or extrated from outside information soure) but unable to reate any new available information. erefore, no one an violate te generalized Carnot's teorem (at least statistially). Referene [1] L. Szilard, On te derease of entropy in a termodynami system by te intervention of intelligent beings, Beav. Si. 9 (1964) doi: /bs [2] R. Landauer, Irreversibility and Heat Generation in te Computing Proess, IBM J. Res. Dev. 5 (1961) doi: /rd liangsuo_su@ust.edu.n

7 [3] C.H. Bennett, e termodynamis of omputation-a review, Int. J. eor. Pys. 21 (1982) doi: /bf [4]. Sagawa, M. Ueda, Minimal energy ost for termodynami information proessing: Measurement and information erasure, Pys. Rev. Lett. 102 (2009) 1 4. doi: /pysrevlett [5] K. Maruyama, F. Nori, V. Vedral, Colloquium: e pysis of Maxwell s demon and information, Rev. Mod. Pys. 81 (2009) doi: /revmodpys [6] S. oyabe,. Sagawa, M. Ueda, E. Muneyuki, M. Sano, Experimental demonstration of information-to-energy onversion and validation of te generalized Jarzynski equality, Nat. Pys. 6 (2010) doi: /npys1821. [7] A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillensneider, E. Lutz, Experimental verifiation of Landauer s priniple linking information and termodynamis, Nature. 483 (2012) doi: /nature [8] A. Carlone, S.M. Goldup, N. Lebrasseur, D.A. Leig, A. Wilson, A ree-ompartment Cemially-driven Moleular Information Ratet, J. Am. Cem. So. 134 (2012) [9] V. Serreli, C.-F. Lee, E.R. Kay, D. a Leig, A moleular information ratet., Nature. 445 (2007) doi: /nature [10] J.P.S. Peterson, R.S. Sartour, A.M. Souza, I.S. Oliveira, J. Goold, K. Modi, et al., Experimental demonstration of information to energy onversion in a quantum system at te Landauer limit, Pro. R. So. A Mat. Pys. Eng. Si. 472 (2016) doi: /rspa [11] L. Su, Y. Li, Xiaokang, L.X. Qian, S. Huang, S. Jin, et al., Aquaporin-1 is a Maxwell s Demon in te Body, (2015) ttp://arxiv.org/abs/ [12] P. Strasberg, G. Saller,. Brandes, M. Esposito, ermodynamis of a Pysial Model Implementing a Maxwell Demon, Pys. Rev. Lett. 110 (2013) doi: /pysrevlett [13] D. Mandal, C. Jarzynski, Work and information proessing in a solvable model of Maxwell s demon, Pro. Natl. Aad. Si. 109 (2012) doi: /pnas [14] A.M. Jayannavar, Simple model for Maxwell s-demon-type information engine, Pys. Rev. E. 53 (1996) doi: /pysreve [15] M.O. Sully, M.S. Zubairy, G.S. Agarwal, H. Walter, Extrating Work from a Single Quantum Coerene, Siene (80-. ). 299 (2003) doi: /siene [16] M.O. Sully, Extrating work from a single termal bat via quantum negentropy., Pys. Rev. Lett. 87 (2001) doi: /pysrevlett [17] H.J. Jeon, S.W. Kim, Optimal work of te quantum Szilard engine under isotermal proesses wit inevitable irreversibility, New J. Pys. 18 (2016) 0. doi: / /18/4/ [18].D. Kieu, e seond law, Maxwell s demon, and work derivable from quantum eat engines, Pys. Rev. Lett. 93 (2004) 1 4. doi: /pysrevlett [19]. Sagawa, M. Ueda, Seond Law of ermodynamis wit Disrete Quantum Feedbak Control, Pys. Rev. Lett. 100 (2008) doi: /pysrevlett [20] S. Hormoz, Quantum ollapse and te seond law of termodynamis, Pys. Rev. E - Stat. Nonlinear, Soft Matter Pys. 87 (2013) 1 9. doi: /pysreve [21] M. Esposito, G. Saller, Stoasti termodynamis for Maxwell demon feedbaks, EPL (Europysis Lett. 99 (2012) doi: / /99/ liangsuo_su@ust.edu.n

8 [22] E. Boukobza, H. Rits, Breaking te Carnot limit witout violating te seond law: A termodynami analysis of off-resonant quantum ligt generation, Pys. Rev. A - At. Mol. Opt. Pys. 87 (2013) 1 6. doi: /pysreva [23] H.H. Hasegawa, J. Isikawa, K. akara, D.J. Driebe, Generalization of te seond law for a nonequilibrium initial state, Pys. Lett. Set. A Gen. At. Solid State Pys. 374 (2010) doi: /j.pysleta [24] K. akara, H.H. Hasegawa, D.J. Driebe, Generalization of te seond law for a transition between nonequilibrium states, Pys. Lett. Set. A Gen. At. Solid State Pys. 375 (2010) doi: /j.pysleta Supplementary materials Fig.s1 Carnot yle A omplete Carnot yle onsists of following four steps: isotermal eat rejetion (1 2), isentropi ompression (2 3), isotermal eat absorption (3 4), and isentropi expansion (4 1). From te view of Carnot engine, te work from te environment will bring available information into it, wile te output work to environment will take away available information. di * W dw pdv k k (s.1) were I * W is te available information arried by work. Assuming te working medium is ideal single-atom gas, putting ideal gas equation of state (pv=nr) into te above equation and integrating it along te proess of isotermal eat rejetion (1 2) and isentropi ompression (2 3), we an get, * v2 v3 v3 IW in nr(ln ln ) nr ln (s.2) v v v During te isotermal eat absorption, I * Win is eated, wi will be oded in te output work. integrating Eq. (s.1) along te proess of isotermal eat absorption (3 4) and isentropi expansion (4 1), we will get liangsuo_su@ust.edu.n

9 * v4 v1 v1 IW out nr(ln ln ) nr ln (s.3) v v v Comparing Eq. (s.3) and Eq. (s.2), it an be found tat, I * * W W out I (s.4) in Eq. (s.4) sows tat from te view of Carnot engine, te input and output available information are equal during a omplete yle (te negative sign indiates te diretion is different). erefore, te generalized Carnot's teorem still olds. For an Otto yle, te eat absorption and rejetion of wi are aieved at a onstant-volume, te input and output proess and eat treatment proess of available information are ompleted separately. From te view of an engine, a termodynami yle onsists of four links of available information: inputting, eating, outputting and erasing residue to reset. e first tree links an be mixed, wile te last link is not neessary to ideal yles su as Carnot yle. liangsuo_su@ust.edu.n