Electrical Maxwell Demon and Szilard Engine Utilizing Johnson Noise, Measurement, Logic and Control

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1 Electrical Maxwell Demon and Szilard Engine Utilizing Johnson Noise, Measurement, Logic and Control Laszlo Bela Kish 1 *, Claes-Göran Granqvist 2 1 Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas, United States of America, 2 Department of Engineering Sciences, The Ångström Laboratory, Uppsala University, Uppsala, Sweden Abstract We introduce a purely electrical version of Maxwell s demon which does not involve mechanically moving parts such as trapdoors, etc. It consists of a capacitor, resistors, amplifiers, logic circuitry and electronically controlled switches and uses ermal noise in resistors (Johnson noise) to pump heat. The only types of energy of importance in is demon are electrical energy and heat. We also demonstrate an entirely electrical version of Szilard s engine, i.e., an information-controlled device at can produce work by employing ermal fluctuations. The only moving part is a piston at executes work, and e engine has purely electronic controls and it is free of e major weakness of e original Szilard engine in not requiring removal and repositioning e piston at e end of e cycle. For bo devices, e energy dissipation in e memory and oer binary informatics components are insignificant compared to e exponentially large energy dissipation in e analog part responsible for creating new information by measurement and decision. This result contradicts e view at e energy dissipation in e memory during erasure is e most essential dissipation process in a demon. Nevereless e dissipation in e memory and information processing parts is sufficient to secure e Second Law of Thermodynamics. Citation: Kish LB, Granqvist C-G (2012) Electrical Maxwell Demon and Szilard Engine Utilizing Johnson Noise, Measurement, Logic and Control. PLoS ONE 7(10): e doi: /journal.pone Editor: Dante R. Chialvo, National Research & Technology Council, Argentina Received July 24, 2012; Accepted September 6, 2012; Published October 15, 2012 Copyright: ß 2012 Kish, Granqvist. This is an open-access article distributed under e terms of e Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided e original auor and source are credited. Funding: The auors have no support or funding to report. Competing Interests: The auors have declared at no competing interests exist. * Laszlokish@tamu.edu Introduction Heat engines [1] utilize temperature differences to produce work while heat demons [2 8] employ information about instantaneous amplitudes of ermal fluctuations and execute control to produce a temperature difference and/or work. There has been an upsurge of interest in heat demons, as evidenced from an extensive recent literature [9 18]. Furermore ere is an old debate [8 14,19 21] on e question wheer e energy dissipation due to erasure of information in e memory is e fundamental process to save e Second Law of Thermodynamics or if e generation of new information via measurement and decision [2,13] and control requirements [9] is more important. This paper introduces new and improved types of demons: ey are purely electrical systems which employ e ermal voltage noise (Johnson noise) of resistors in ermal equilibrium. One of e demons pumps heat (similarly to Maxwell s demon) and e oer produces work (similarly to Szilard s engine). Their common characteristics is at noing but electrical energies appear in e fluctuations, measurement, decision, control and memory parts of e demons, and at all of e energy loss is due to dissipative electrical transport. This situation is very different from at of earlier versions of Maxwell s demon and Szilard s engine wherein various forms of energies are (implicitly) employed in e various components. In e case of Maxwell s demon, for example, e detection of an approaching molecule and its velocity require photoelectronics-based sensing at includes laser light (photon energy), photodiodes, electrical signals and energy, and an electromechanical system to drive e trapdoor, as well as related kinetic and potential mechanical energies. Our demon, which is a linear system, is also different in nature from ratchet-based demons [5] and Brillouin s diode-based ones [6,7] because ose utilize nonlinearity instead of measurement, decision and control. Similarly, in Szilard s engine, detecting e location of e molecule, controlling e lever and repositioning e piston require photonic, electrical and mechanical energies and imply related forms of energy dissipation. Complete evaluations of e various energy dissipation channels have not been performed in earlier work on Maxwell s demon and Szilard s engine, which may be associated wi difficulties to assess e transfer of energies between eir different forms. However our purely electrical demons, which are introduced in is paper, offer fundamental advantages and provide a convenient meodology for such analyses. Discussion and Results 1. Electrical Maxwell demon utilizing Johnson noise and control 1.1 General description. The electrical Maxwell demon described below employs e Johnson noise of resistors as well as measurement, decision, logic operations and control for pumping heat. One should note at control systems in general contain units for making measurements, decision and executing control. Figure 1 shows an electrical Maxwell demon in its starting stage 1 extracting energy from e resistor to e left. The switch is in position 1, and e Johnson noise current of e resistor yields voltage fluctuations U(t) in e capacitor wi variance PLOS ONE 1 October 2012 Volume 7 Issue 10 e46800

2 Figure 1. Electrical Maxwell demon wi Johnson noise in stage 1. The temperature T is e same in bo resistors. The ick line indicates e moving part of e switch and e double-ended arrow e switching positions. doi: /journal.pone g001 Ueff 2 ~SU 2 (t)t~ ð1þ C in accordance wi Boltzmann s energy equipartition eorem. It leads to a mean energy of =2 in e capacitor [1], where U eff is e effective noise voltage (root mean square, RMS, voltage) on e capacitor C. This voltage fluctuation is a Gaussian process wi exponential relaxation and correlation time (relaxation time) t expressed as t~rc, where R is e resistance, and wi an amplitude distribution function g(u) given by ð2þ " # g(u)~ 1 pffiffiffiffiffi exp { U 2 : ð3þ 2p U eff 2U 2 eff The voltage U(t) is monitored as illustrated in Figure 2, and e demon stays in stage 1 until e voltage reaches a chosen arbitrary reshold value U, which corresponds to e energy E ~0:5CU 2 on e capacitor. The switch is en flipped to position 2, see Figure 3, and e demon gets into its stage 2 wherein e capacitor ð4þ Figure 3. Electrical Maxwell demon wi Johnson noise in stage 2. doi: /journal.pone g003 is discharged and e energy E is pumped into e resistor on e right. In stage 2, U(t) is monitored on e capacitor, as shown in Figure 4, and when it reaches e zero reshold level wi U(t)~0 e switch is flipped into its position 1 shown in Figure 1. Thus e demon gets into stage 1 again, and e full cycle of e demon is completed rough is step. By e end of stage 1, e capacitor has extracted e energy E ~0:5CU 2 from e resistor on e left, and is energy is fully dissipated in e resistor on e right by e end of stage 2. When e demon gets back to stage 1 to start a new cycle, e energy in e capacitor is zero. Thus by e end of e new stage 1 process, e capacitor again extracts E ~0:5CU 2 from e resistor on e left, and is energy will once more be fully dissipated in e resistor on e right by e end of e new stage 2. Figure 5 illustrates voltage fluctuations during e whole cycle. It is important to note at, in an ideal system, e heat pumped during a single cycle does not have an upper limit because, according to Eq. 2, e reshold amplitude U (and us e energy E ) can be chosen to be arbitrarily large Cycle duration versus extracted heat/work during a cycle. The cycle frequency of Maxwell s demon is usually not part of its energy balance, but we note at an increase of e reshold U will cause an exponential slowdown of e demon as a result of e increased cycle time needed to reach U, as furer discussed later. The duration of e exponential discharge process in stage 2 scales logarimically wi e pumped energy (reshold energy) E, whereas e duration of stage 1 scales exponentially in e limit of large E, as discussed below. Thus e main question is e average duration of stage 1, i.e., e mean first-passage time T 1 of e voltage fluctuation of e capacitor between zero and U when is energy is much greater an e average ermal energy Figure 2. Voltage fluctuations in e capacitor during stage 1 for an electrical Maxwell demon wi Johnson noise. (Note at e real voltage time function typically exhibits more random fluctuation events, as discussed later.) doi: /journal.pone g002 Figure 4. Voltage fluctuations on e capacitor during stage 2 for an electrical Maxwell demon wi Johnson noise. (Note at real voltage fluctuations typically exhibit more random events, as discussed later.) doi: /journal.pone g004 PLOS ONE 2 October 2012 Volume 7 Issue 10 e46800

3 Figure 5. Voltage fluctuations in e capacitor, indicating e role of e two reshold levels U and 0 during e whole cycle, and e beginning of a new cycle. doi: /journal.pone g005 in e capacitor; e reshold is given by E ~0:5CU 2 ww0:5: This expectation agrees wi an earlier, analytic treatment of Gaussian noise wi exponential relaxation, wherein it was derived in an asymptotic limit [22]. Note at first passage times belong mostly to e field of unsolved problems of noise, and analytic results (e.g. [22 24]) exist only for e asymptotic limit indicated by Eq. 5. In e present analysis we use results from e analytic treatment in [22]. A dimensionless first passage time t 1 from zero amplitude to e level U is obtained [22] according to ð5þ pffiffiffi p t 1 (g)~ exp g2, ð6þ 2g where t 1 (g)~t 1 (g)=t. Here g is e dimensionless reshold T 1 and t is e relaxation time of e Lorentzian. In our system e dimensionless reshold is g~ p 1 ffiffi U p 2 U, where U eff ~ ffiffiffiffiffiffiffiffiffiffiffiffi =C is e eff effective value of e noise. Thus e real first passage time T 1 from zero amplitude to level U is pffiffiffi! p U 2 T 1 (f)~t 1 (g)t~rc 2 p 1 ffiffi U exp 2U 2 2 U eff eff sffiffiffiffiffiffiffiffiffiffiffiffi! p Ueff 2 U 2 ~RC 2 U 2 exp 2Ueff 2 ~ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ~RC 2 CU 2 exp CU 2 =2 sffiffiffiffiffiffiffiffiffiffiffiffi ~RC p exp E sffiffiffiffiffiffi!rc exp E : 4E In conclusion, even ough an arbitrarily large energy E can be transferred by e Maxwell demon type heat pump wiin a cycle, e cycle duration scales exponentially wi is energy, which means at increasing E results in an exponential scaling down of e mean power On energy requirement and dissipation. In is section we discuss e energy dissipation in e limit of high E ð7þ reshold, E ww (where Eq. 7 holds), and small error probability. A full assessment of e energy dissipation needed to run e demon requires consideration of all of its building elements. Such analyses have not been common in prior work because e demon s functional characteristics have usually not been described in sufficient detail. To avoid mistakes caused by such deficiencies, Figure 6 shows e complete functional block scheme required to realize Maxwell s demon. A possible physical realization of is scheme is given in Figure 7. It is important to identify e dominant causes of energy dissipation. The first two stages of e measurement, decision, logic and control unit in Figure 6 are e amplifier and e reshold device in Fig. 7, which represent e measurement and decision. For example, e output of e reshold device can be a positive or negative pulse representing e moment when e switch state must be changed from 1 to 2 or from 2 to 1 and triggering e memory to alter its state accordingly. The memory stores binary, single-bit information a high or low value and ese two-bit states signify e actual state (1 or 2) of e switch. During a single cycle of e demon, e single-bit information cycles rough its two states low-high-low or high-low-high depending on e choice of e designer of e system: wheer e low-bit represents e switch-state 1 and e high-bit e switch-state-2, or vice versa. The reader and switch driver device is basically an amplifier system at reads out e memory status and checks e voltage controlled switches, which eier make a connection between e capacitor and point 1 while breaking e former connection to point 2, or vice versa. Thus ese are two electronically controlled switches between e capacitor and 1 and between e capacitor and 2, respectively driven in alternate fashion. We now consider e elements for handling e binary information and note at according to Brillouin s negentropy principle [6,7] e processing of each bit of information will dissipate energy of at least ln (2), and processing wi acceptable error probability (vv0:5) will require even greater energy dissipation E d [9,25]. Wiin e correlation time (reciprocal bandwid) of ermal fluctuations and in e smallerror probability limit [9,25] e lower boundary of energy dissipation can be described by e Rice formula [25]. The required energy is logarimically divergent when e error probability goes to zero according to E d,bin & ln 1 e ½Joule=bitŠ, ð8þ where evv0:5 is e bit error probability during e correlation time t. Demanding e same numerical value of e error probability [9] for a longer cycle time (or clock period) T c wwt makes e energy dissipation increase logarimically as a result of e additivity of independent errors generated in different nonoverlapping correlation time intervals. It follows at E d,bin & ln 1 T c e c t ½Joule=bitŠ, ð9þ where e c v0:5 is e bit error probability during T c. Using e result of Eq. 7 we obtain e following relation for e lowest possible energy dissipation of each binary switching element (transistor, etc) in e demon: PLOS ONE 3 October 2012 Volume 7 Issue 10 e46800

4 Figure 6. Electrical Maxwell demon showing a block scheme of e building elements for measurement, decision and control. The voltmeter and e reshold device create new information, and e rest of e system only processes or utilizes is binary information. doi: /journal.pone g006 " sffiffiffiffiffiffi E d,bin ln 1 exp E # e c E " s ffiffiffiffiffiffi! # ~E z ln 1 : e c E ð10þ We conclude at e Second Law of Thermodynamics is satisfied by even a single binary unit (such as any logic gate or e memory) for e case of a fixed E = ratio and error probability approaching zero. The energy dissipation scales roughly wi e energy reshold E, i.e., in e small error limit it scales wi e pumped energy. Turning now to e major sources of energy dissipation, we consider e amplifier and e reshold device, which are e analog circuitry elements where new information is created. Their energy dissipation scale exponentially wi E and is much larger Figure 7. Possible realization of an electrical Maxwell demon. Some of e elements can be joined in oer realizations, but e energy dissipation sources described here remain. doi: /journal.pone g007 PLOS ONE 4 October 2012 Volume 7 Issue 10 e46800

5 an e amount required by e Second Law of Thermodynamics. Theoretically, if e chosen reshold level U is large enough, e amplifier is not needed and it is sufficient wi a reshold device. However, any such device can be modeled as an amplifier wi eier saturation and/or positive feedback, and us we keep e amplifier stage because it helps illustrate e point where e energy dissipation scales exponentially wi E. In order to elucidate e origin of e amplifier s energy dissipation it should be realized at any amplifier has a component of inertia. To choose a simple illustrative example, a hydraulic amplifier has inertia due to e mass of its components. In e present case of an electrical voltage amplifier, inertia ensues from parasitic capacitances and inductances. We demonstrate is issue of electrical inertia by considering e final capacitor, positioned at e point where e reshold device makes e decision about e reshold crossing, denoted C p in Figure 7. For e sake of simplicity we assume at e amplification is unity in e amplifier. Under ermal equilibrium, ermal voltage fluctuations (Johnson pffiffiffiffiffiffiffiffiffiffiffiffiffiffinoise) appear on e capacitor wi an RMS value =C p, which corresponds to a mean ermal energy of /2 in e capacitor. If is Johnson noise is not actively damped it must be kept at a much lower level an e main Johnson noise of e demon we are monitoring, i.e., pffiffiffiffiffiffiffiffiffiffiffiffi p =Cww ffiffiffiffiffiffiffiffiffiffiffiffiffiffi =C p. This means at C p wwc in order to reduce erroneous reshold crossings caused by e parasitic noise. We note in passing at it can be shown at active damping of e parasitic Johnson noise would cause greater energy dissipation an e value discussed below [26]. Figure 8 illustrates ree possible trajectories of e voltage wiin stage 1 as its amplitude goes from zero to U. Curves A and B represent monotonically increasing functions which reach U wi relatively modest energy investment, i.e., 0:5C p U 2 ww=2 plus e ermalization losses due to e resistances, etc (not shown in e circuitry). However curve C is qualitatively different and non-monotonic. This feature causes serious additional energy dissipation and all of e energy invested in e trajectory from points a to b is completely lost. For example, in e case of a symmetric power supply (i.e., wi electrodes: +, 2, and zero (ground)) e parts of e trajectory wi positive velocity (increasing amplitude) are supplied by e positive electrode of e power source, while e decreasing parts of e trajectory are supplied by e negative electrode, when in bo cases e current flows into e ground. Bo of ese currents represent positive power extracted from e positive and negative parts of e supply, respectively. Even ough a net energy was dissipated in e voltage-pa from a to b, e whole charging process is back to zero at b, and all of e earlier invested energy was lost. How many such dissipative random fluctuation events will e voltage on e capacitors exhibit, and how large are ese fluctuations? During e correlation time t~rc of e noise in e main capacitor e noise typically goes rough at least one large wave ranging ough e +/2 RMS amplitude interval. This means at e related energy dissipation E d1 during e correlation time of e main noise is of e order E d1,an & C p C : ð11þ According to Eq. 7 e expected number of correlation time (t) periods during e dominant stage 1 part of e clock cycle is rffiffiffiffiffiffi exp E so at e total energy dissipation is of e order E E d,an & C p C sffiffiffiffiffiffi E exp E! exp E : ð12þ Thus e dominant energy dissipation during a single cycle in e large-reshold energy limit is E d,tot we d,an ze d,bin sffiffiffiffiffiffi & C p C exp E " s ffiffiffiffiffiffi! # ze z ln 1, e c E and hence e approximate scaling in is limit is E ð13þ E d,tot! exp E : ð14þ In conclusion, e dominant energy dissipation in e electrical Maxwell demon takes place in e part which generates e new information and is due its analog processing yielding an exponential dependence of e number of random voltage oscillations during a single cycle. Figure 8. Illustration of e main source of energy dissipation in e electrical Maxwell demon. An exponential number of random fluctuation events as a function of e reshold energy take place before reaching e reshold. The energy dissipation between e points a and b represent complete energy loss and do not contribute to e process of attaining e reshold. doi: /journal.pone g008 Figure 9. Electrical Szilard engine extracting heat from e resistor and executing work on e piston, i.e., e movable capacitor plate. doi: /journal.pone g009 PLOS ONE 5 October 2012 Volume 7 Issue 10 e46800

6 Figure 10. Arbitrarily chosen starting point of e cycle for e electrical Szilard engine. The piston is at e right end and e capacitor at e right has e work charge Q w. The corresponding electrical energy is E w &/2. doi: /journal.pone g Electrical Szilard engine utilizing Johnson noise and control The principles of e Johnson noise driven electrical Szilard engine are e same as ose of e electrical Maxwell demon ough heat is not pumped but work is executed by a moving capacitor plate used as a piston [1,27]. The control and all oer considerations are inherently e same as for Maxwell s demon except e fact at e Szilard engine requires a twice as large system as e Maxwell demon. Thus two single-bit memories are needed and e related voltage monitoring and switch control must be performed over two capacitors. Figure 9 shows e main elements and circuitry of e electrical Szilard engine except e decision, logic and control components which are identical to e corresponding parts in Maxwell s demon. The electrical Szilard engine extracts heat from e resistor and executes work on e piston, which is e moving joint capacitor plate of e two capacitors at e middle. The left capacitor has its maximum capacitance C left ~C max when e piston is at e left end while e right capacitor en attains its minimum capacitance C right ~C min. In e opposite case, when e piston is at e right end, e left capacitor reaches its minimum value C left ~C min while e right capacitor has its maximum capacitance C right ~C max. The capacitor wi e Figure 11. Operation of e first half cycle for e electrical Szilard engine. (a): The left switch is closed to charge e left capacitor. (b): Johnson noise current yields charge fluctuations in e left capacitor, i.e., a Gaussian noise in e charge. (c): As soon as e left capacitor reaches Q w, e left switch is abruptly opened to keep e charge in e left capacitor and e right switch is closed to discharge e capacitor on e right. (d): The charge in e right capacitor decays exponentially and, wiin a time interval of a few RC, ermalizes and crosses e zero charge (zero voltage) level. (e): The right switch is abruptly opened and e zero charge (zero voltage) state is preserved in e right capacitor. (f): The force due to e working charge in e left capacitor moves e piston to e left end, which is e end of e half-cycle; e piston is at e left end and e capacitor at e left has e work charge Q w. doi: /journal.pone g011 PLOS ONE 6 October 2012 Volume 7 Issue 10 e46800

7 Figure 12. Operation of e second half cycle for e electrical Szilard engine. (a): The right switch is closed to charge e right capacitor. (b): Johnson noise current yields charge fluctuations in e right capacitor, i.e., a Gaussian noise in e charge. (c): As soon as e right capacitor reaches Q w e right switch is abruptly opened to keep e charge in e right capacitor and e left switch is closed to discharge e capacitor on e right. (d): The charge in e left capacitor decays exponentially and, wiin a time interval of a few RC, ermalizes and crosses e zero charge (zero voltage) level. (e): The left switch is abruptly opened and e zero charge (zero voltage) state is preserved in e left capacitor. (f): The force due to e working charge in e right capacitor moves e piston to e right end, which is e end of e full cycle and e same state as where e analysis started; see Figure 10. The piston is at e right end and e capacitor in e right has e work charge Q w. doi: /journal.pone g012 minimum capacitance will be charged up to e work charge Q w at different points of e cycle; ey correspond to a reshold voltage U ~Q w =C min as shown in Figure 2. The corresponding electrical charging energy analogously to e case of Maxwell s demon is much greater an e ermal equilibrium value, i.e., E w ~0:5Q 2 min =Cww=2: ð15þ Figure 10A shows an arbitrarily chosen starting point of e cycle. The piston is at e right end and e capacitor to e right has e work charge Q w. The voltage on e right capacitor is Q w =C max. The energy difference, compared to e situation wi e opposite position of e piston at e same charge, is 1 {DE~0:5Q w { 1 v0 ð16þ C max C min implying at e piston has a negative potential energy and sits in a potential well. We suppose at 1 DE~0:5Q w { 1 ww=2, ð17þ C min C max i.e., e piston remains at e bottom of is potential barrier, which is in e vicinity of e right end, while it executes its mechanical Brownian motion. Figure 11 illustrates e operation of e first half-cycle for e electrical Szilard engine. In Figure 11A e left switch is closed in order to charge e left capacitor and e Johnson noise current of e left resistor yields charge fluctuations in e left capacitor, which is a Gaussian noise of e charge; see Figure 10B. After an exponentially long waiting time characterized by Eq. 7 e left capacitor, which originally had zero charge, is charged up to e working charge Q w and energy E w given by Eq. 15. At at moment e left switch is abruptly opened to keep e charge in e left capacitor and e right switch is closed to discharge e capacitor on e right as shown in Figure 10C. The charge in e right capacitor decays exponentially and, wiin a time interval of a few RC, ermalizes and crosses e zero charge (zero voltage) level; see Figure 10D. At at moment, as illustrated in Figure 10E, e right switch is abruptly opened and e zero charge (zero voltage) state is preserved in e right capacitor. Then e PLOS ONE 7 October 2012 Volume 7 Issue 10 e46800

8 electrostatic force due to e working charge in e left capacitor moves e piston to e left end, as shown in Figure 10F, while, in accordance wi e energy conservation law, e piston executes positive work amounting to DE. This is e end of e half-cycle. The piston is at e left end and e capacitor at e left has e work charge Q w. The remainder of e full cycle comprises e same processes as during e first half wi e proper change of e indices signifying left and right of capacitors and switches, as elucidated in Figure 12. Thus e full cycle has been carried out wi purely electronic control based on e information from e measurement of charge or voltage in e capacitors. It should be noted at e system is free from e major deficiency of e original Szilard engine, which is at its piston must be artificially relocated into e initial position at e end of e cycle. The two switches represent four possible states and two bits of information, and consequently e unit for measurement, decision, logic and control is basically twice at of e system shown for e electrical Maxwell demon, while e energy dissipation essentially doubles. Thus e energy dissipation in e binary part, including in e memory, is negligible compared to at in e analog part at executes e measurement and creates e new information. Meods and Conclusions Two electrical demons utilizing Johnson noise, measurement, decision, logic and control were introduced; ey are new versions References 1. Kish LB (2011) Thermal noise engines. Chaos Solitons Fractals 44: Szilard L (1929) On e reduction of entropy in a ermodynamic system by e interference of an intelligent being. Z Phys 53: Maxwell JC (1871) Theory of Heat. London: Longmans, Green & Co. 4. Leff HS, Rex AS, editors. (2003) Maxwell s Demon 2: Entropy, Classical and Quantum Information, Computing. Bristol: Institute of Physics. 5. Chialvo DR, Millonas MM (1995) Asymmetric unbiased fluctuations are sufficient for e operation of a correlation ratchet. Phys Lett A 209: Brillouin L (1964) Scientific Uncertainty and Information. New York: Academic. 7. Brillouin L (1962) Science and Information Theory. New York: Academic. 8. Zurek WH (1986) Maxwell s demon, Szilard s engine and quantum measurements. In: Moore GT, Scully MO, editors. Proceedings of Frontiers of Non- Equilibrium Quantum Statistical Mechanics. New York: Plenum. pp Kish LB, Granqvist CG (2012) Energy requirement of control: Comments on Szilard s engine and Maxwell s demon. EPL 98: Norton JD (2011) Waiting for Landauer. Stud Hist Philos Mod Phys 42: Norton JD (2005) Eaters of e lotus: Landauer s principle and e return of Maxwell s demon. Stud Hist Philos Mod Phys 36: Norton JD (2012) The end of e ermodynamics of computation: A no go result. (Manuscript to appear in Philos Sci). Available at John Norton s website at Univ. of Pittsburg. Available: pdf. Accessed 2012 Sep Granger L, Kantz H (2011) Thermodynamic cost of measurements. Phys Rev E 84: Gea-Banacloche J, Leff HS (2005) Quantum version of e Szilard one-atom engine and e cost of raising energy barriers. Fluct Noise Lett 5:C39 C47. of Maxwell s demon and Szilard s engine. We showed all of e necessary building elements and analyzed e cycle time versus e energy output. Bo demons have an arbitrarily large energy output during a single cycle, and e cycle duration and energy dissipation scale exponentially wi is energy output. The exponentially scaling energy dissipation occurs in e analog parts of e demons, where e creation of new information (measurement and decision) takes place. In e binary part, including e memory, e energy dissipation scales only linearly wi e energy output and even is dissipation is enough to satisfy e Second Law of Thermodynamics due to e enhanced energy reshold requirements caused by e exponentially long cycle duration and e requirement of keeping e error probability small. Acknowledgments LBK is grateful to Katja Lindenberg for useful discussions about firstpassage problems and Eleonora Alfinito and Cecilia Pennetta for valuable remarks during a seminar about is topic. Useful discussions wi John Norton, Dave Ferry and Wolfgang Porod about e irreversibility of computation and e Landauer eorem are appreciated. Auor Contributions Analyzed e data: LK CGG. Wrote e paper: LK CGG. Concept: LK CGG. Theory: LK CGG. Analysis: LK CGG. Conclusions: LK CGG. 15. Horowitz J, Parrondo JMR (2011) Designing optimal discrete-feedback ermodynamic engines. New J Phys 13: Sagawa T, Ueda M (2012) Nonequilibrium ermodynamics of feedback control. Phys Rev E 85: Vaikuntanaan S, Jarzynski C (2011) Modeling Maxwell s demon wi a microcanonical Szilard engine. Phys Rev E 83: Van den Broeck C (2010) Thermodynamics of information: Bits for less or more for bits? Nature Phys 6: Porod W (1988) Energy requirements in communication Comment. Appl Phys Lett 52: Porod W, Grondin RO, Ferry DK (1984) Dissipation in computation. Phys Rev Lett 52: Porod W, Grondin RO, Ferry DK, Porod G (1984) Dissipation in computation Reply. Phys Rev Lett 52: Lindenberg K, Seshadri V (1979) Analytic eory of extrema. I. Asymptotic eory for Fokker-Planck processes. J Chem Phys 71: Palatella L, Pennetta C (2011) Distribution of first-return times in correlated stationary signals. Phys Rev E 83: Pennetta C (2006) Distribution of return intervals of extreme events. Eur Phys J B 50: Kish LB (2004) Moore s law and e energy requirement of computing versus performance. IEE Proc Circuits Devices Systems 151: Kish LB, Granqvist CG (2011) Energy requirement of control: Comments on Maxwell s demon and Szilard s engine. In: Alfinito E, Leuzzi M, Millialer J-F (editors). All e Colors of Noise: Essays in Honor of Lino Reggiani. Cormigliano del Brenta: Munari Edizione. pp Davis BR, Abbott D, Parrondo JMR (2001) Thermodynamic energy exchange in a moving plate capacitor. Chaos 11: PLOS ONE 8 October 2012 Volume 7 Issue 10 e46800