Generalization of the Landauer Principle for Computing Devices Based on Many-Valued Logic

entropy

Article Generalization of the Landauer Principle for Computing Devices Based on Many-Valued Logic

Edward Bormashenko

Department of Chemical Engineering, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel; [email protected]; Tel.: +972-074-729-68-63 Received: 7 October 2019; Accepted: 22 November 2019; Published: 25 November 2019

Abstract: The Landauer principle asserts that “the information is physical”. In its strict meaning, Landauer’s principle states that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer bound W = kBTln2, where T is the temperature of a thermal reservoir used in the process and kB is Boltzmann’s constant. Modern computers use the binary system in which a number is expressed in the base-2 numeral system. We demonstrate that the Landauer principle remains valid for the physical computing device based on the ternary, and more generally, N-based logic. The energy necessary for erasure of one bit of information (the Landauer bound) W = kBTln2 remains untouched for the computing devices exploiting a many-valued logic.

Keywords: Landauer principle; binary logic; ternary logic; Landauer bound; trit

1. Introduction Modern computers use the binary system, whereby a number is expressed in the base-2 numeral system. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. The base-2 is ubiquitous in computing devices because of its straightforward implementation in digital electronic circuitry using binary logic gates. However, one of the first computing machines was based on the ternary logic. In 1840, Thomas Fowler, a self-taught English mathematician and inventor, created a unique ternary mechanical calculating machine, completely manufactured of wood [1]. Ternary logic based computers based in the “trit” unit of information were successfully developed in Soviet Union by Nicolay Brousentsov [2]. The Setun computer, based on the ideas of ternary logic, ternary symmetrical number system and ternary memory element (“flip-flap-flop”) was designed in 1958 in Moscow University [2,3]. In principle, computer may be based on a many-valued logics, exposed in recent years to a growing interest due to the fundamental aspects and numerous applications [4,5]. Ternary computer TERNAC was reported in 1973 by Frieder et al. in Reference [6]. The present paper does not come into the mathematical details of the ternary (or another) many-valued logics, but extends the Landauer principle to the erasing of the information by the computing machine, based on the many-valued logics. Informational theory is usually supplied in a form that is independent of any physical realization. In contrast, Rolf Landauer, in his papers, argued that “information is physical” and has an energy equivalent [7–9]. It may be stored in physical systems, such as books and memory chips, and it is transmitted by physical devices exploiting electrical or optical signals [6–8]. Therefore, he concluded, it must obey the laws of physics, and first and foremost, the laws of thermodynamics. The Landauer principle [7–9] established the energy equivalent of information and remains a focus of investigations in the last decade [10–18]. In its strictest, tightest, and simplest meaning, the Landauer principle states that the erasure of one bit of information requires a minimum energy cost equal to kBTln2, where T is the temperature of a thermal reservoir used in the process and kB is Boltzmann’s constant [7–14]. The Landauer principle is usually demonstrated with

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DiscussionConsider the computing device exploiting a particle enclosed within a chamber (cylinder) dividedConsider by half theby acomputing partition, as device shown exploiting in Figure 1.a Findingparticle the enclosed particle within M in the a certainchamber (left (cylinder) or right) Consider the computing device exploiting a particle enclosed within a chamber (cylinder) divided halfdivided of the by halfchamber by a partition, corresponds as shown to the in recording Figure 1. Finding of 1 bit the of particleinformation. M in the When certain the (left partition or right) is by half by a partition, as shown in Figure1. Finding the particle M in the certain (left or right) half removed,half of the the chamber location corresponds of the particle to theis uncertain, recording andof 1 this bit correspondsof information. to theWhen erasure the partitionof 1 bit ofis of the chamber corresponds to the recording of 1 bit of information. When the partition is removed, information.removed, the The location location of theof a particleparticle ison uncertain, a certain halfand ofthis the corresponds chamber corresponds to the erasure to “1”, of and1 bit the of the location of the particle is uncertain, and this corresponds to the erasure of 1 bit of information. uncertaininformation. location The location of the particle of a particle corresponds on a certai to “0n”, half thus of our the particle-based chamber corresponds computer to works “1”, and on the The location of a particle on a certain half of the chamber corresponds to “1”, and the uncertain location binaryuncertain logical location system. of the The particle work ofcorresponds this computer to “0 may”, thus be exemplifiedour particle-based by the computersingle-particle works thermal on the of the particle corresponds to “0”, thus our particle-based computer works on the binary logical system. engine,binary logical suggested system. by TheLeo workSzilard of thisin 1929 computer [19] anmayd depicted be exemplified in Figure by the 2. single-particleThe smallest possiblethermal The work of this computer may be exemplified by the single-particle thermal engine, suggested by thermodynamicengine, suggested machine by Leo consists Szilard of ain single 1929 particle[19] an ofd massdepicted m in ina closed Figure cylinder, 2. The whichsmallest has possible contact Leo Szilard in 1929 [19] and depicted in Figure2. The smallest possible thermodynamic machine withthermodynamic a thermal reservoirs. machine consists Consider of the a single “evergreen” particle Carnotof mass cycle, m in a performed closed cylinder, by a minimal which has engine, contact is consists of a single particle of mass m in a closed cylinder, which has contact with a thermal reservoirs. depictedwith a thermal in Figure reservoirs. 1 from Consider an informational the “evergreen” point of Carnot view cycle,[9,20,21]. performed At the firstby a minimalstage, the engine, particle is Consider the “evergreen” Carnot cycle, performed by a minimal engine, is depicted in Figure1 from contactsdepicted thein Figurethermal 1 reservoirfrom an informational(bath) T1 and pointundergoes of view a reversible[9,20,21]. Atisothermal the first stage,expansion, the particle which an informational point of view [9,20,21]. At the first stage, the particle contacts the thermal reservoir doublescontacts itsthe available thermal volume reservoir [21]. (bath) Note, T that1 and the undergoes particle initially a reversible occupies isothermal the left side expansion, of the cylinder. which (bath) T1 and undergoes a reversible isothermal expansion, which doubles its available volume [21]. Heatdoubles kBT 1its ln2 available is drawn volume from the [21]. bath Note, and that work the 𝑘 particle𝑇𝑙𝑛2 initially is extracted. occupies This processthe left side is equivalent of the cylinder. to the Note, that the particle initially occupies the left side of the cylinder. Heat kBT1 ln2 is drawn from the removalHeat kBT 1of ln2 the is partitiondrawn from at the the midpoint bath and ofwork the 𝑘cylinder,𝑇𝑙𝑛2 isthus, extracted. one bit This of information process is equivalent is erased, ifto one the bath and work kBT ln2 is extracted. This process is equivalent to the removal of the partition at the bitremoval finds particleof the partition m 1at a certain at the sidemidpoint (left in of our the case) cylinder, of the thus, cylinder, one bit as of shown information in Figure is erased2 [10]., ifThus, one midpoint of the cylinder, thus, one bit of information is erased, if one bit finds particle m at a certain side heatbit finds kBT1 particleln2 spent m by at thea certain thermal side bath (left was in our exploite case)d of for the erasure cylinder, of 1 as bit shown of information. in Figure 2 [10]. Thus, (left in our case) of the cylinder, as shown in Figure2[ 10]. Thus, heat kBT1 ln2 spent by the thermal heat kBT1 ln2 spent by the thermal bath was exploited for erasure of 1 bit of information. bath was exploited for erasure of 1 bit of information.

Figure 1. Finding of the particle M in the certain (left or right) half of the chamber corresponds to the Figure 1. Finding of the particle M in the certain (left or right) half of the chamber corresponds to the recordingFigure 1. Finding of 1 bit ofof information.the particle M in the certain (left or right) half of the chamber corresponds to the recording of 1 bit of information. recording of 1 bit of information.

Figure 2. SketchSketch of of the minimal single-particle thermal machine is depicted. Particle Particle MM moves the piston. The machine works between the hot (T ) and cold (T ) thermal reservoirs which may be ﬁnite. piston.Figure The2. Sketch machine of theworks minimal between single-particle the hot (T11) andthermal cold machine(T22) thermal is depicted. reservoirs Particle which mayM moves be finite. the The conditions of “thermalization” (randomization) of the particle motion are discussed. Thepiston. conditions The machine of “thermalization” works between (randomization) the hot (T1) and ofcold the ( Tparticle2) thermal motion reservoirs are discussed. which may be finite. TheAt the conditions second stageof “thermalization” our engine is (randomization) exposed to the adiabatic of the particle expansion motion and are thediscussed. additional mechanical At the second stage our engine is exposed to the adiabatic expansion and the additional work is made. At this stage, the entropy of the working body and the thermal reservoir remain mechanicalAt the worksecond is made.stage Atour this engine stage, is the exposed entropy to of the the adiabaticworking bodyexpansion and the and thermal the additional reservoir unchanged, and there is no informational change in both of them (thermal reservoir T1 is disconnected remainmechanical unchanged, work is made.and there At this is no stage, informationa the entropyl change of the inworking both of body them and (thermal the thermal reservoir reservoir T1 is from the engine at this stage). At the next stage, the engine is connected to the thermal bath T2 and disconnectedremain unchanged, from the and engine there at is this no stage).informationa At the lnext change stage, in theboth engine of them is connected (thermal toreservoir the thermal T1 is exerted to the reversible isothermal compression. A piston reversibly and isothermally compresses the bathdisconnected T2 and exerted from the to enginethe reversible at this stage).isothermal At the compression. next stage, the A pistonengine reversibly is connected and to isothermally the thermal space occupied by the particle m from full to half volume. One bit of information is recorded by the compressesbath T2 and theexerted space to occupied the reversible by the isothermal particle m compression.from full to half A piston volume. reversibly One bit andof information isothermally is engine. Heat Q2 = kBT2 ln2, is delivered to the heat bath, and work kT2 ln2 is consumed. At the last recordedcompresses by the the space engine. occupied Heat by𝑄 the=kB particleT2 ln2, ism fromdelivered full toto half the volume. heat bath, One and bit ofwork information kT2 ln2 is stage of the cycle the engine is disconnected from the reservoir T2 and the system is adiabatically heated consumed.recorded by At the the engine. last stage Heat of the𝑄 =cyclekBT2 theln2, engine is delivered is disconnected to the heat from bath, the andreservoir work TkT2 and2 ln 2the is to the temperature T1. No entropy and informational changes take place at this stage. The work of the consumed. At the last stage of the cycle the engine is disconnected from the reservoir T2 and the

Entropy 2019, 21, 1150 3 of 6 minimal Carnot engine illustrates the Landauer principle: Recording/erasing one bit of information demands kBTln2 units of energy. The non-trivial problems of “thermalization” of the motion of the particle in the minimal Carnot engine are out of the scope of our paper [10,20,21]. The Carnot engine is fully reversible; actually, the erasure/recording of information is asymmetric and there may be no entropy cost to the acquisition of information, but the destruction of information does involve an irreducible entropy cost [21]. This erasure/recording asymmetry is essential [10,22]. However, it is not in the focus of the present paper. Note, that the efficiency of the engine equals η = 1 T2 , as demonstrated in Reference [21]. − T1 This result is quite expected, due to the fact that the efficiency of the Carnot machine is insensitive to working substances in the engine and depends only on the temperatures of the thermal reservoirs [21]. The additional exemplification of the Landauer principle is supplied by the Brownian particle in a double-well potential, as shown in Figure3 and discussed in detail in References [ 8,10]. When the barrier is much higher than the thermal energy, the particle will remain in either well for a long time [8,10,20]. Thus, the particle being in the left or right well can serve as the stable informational states, “0” and “1” of a bit. A Brownian particle trapped in either left or right well represents the informational states m = 0 and m = 1, as shown in Figure3, where m is the parameter, characterizing the statistical state of the system. The average work W to change the statistical state of a memory from the state Ψ with the distribution pm to Ψ0 with distribution pm0 is given by Equation (1a,b):

W F(Ψ0) F(Ψ) (1a) ≥ − X X F(Ψ) = pmFm + kBT pmlnpm (1b) m m where Fm = Em TSm is the free energy of the conditional state [10]. For a symmetrical well and − 1 a random bit p0 = p1 = 2 , we immediately recover the Landauer bound W = kBTln2, as shown in Reference [10] and checked experimentally in References [23–25]. The colloidal particle in a double-well potential was used as a generic model of a one-bit memory in Reference [23]. The experimental veriﬁcation of the Landauer principle was carried out with an optical tweezer, which trapped a silica bead (2 µm in diameter) at the focus of a laser beam [23]. It was demonstrated that that the mean dissipated heat saturated at the Landauer bound in the limit of long erasure cycles [23]. Reference [24] reports testing of the Landauer principle with a colloidal particle conﬁned in a time-dependent, virtual potential which is created by a feedback trap. The extension of the Landauer principle to the quantum realm is carried out by using a crystal of molecular nanomagnets as a quantum spin memory, as demonstrated in Reference [25]. Employing a trapped ultracold ions enabled experimental validation of a quantum version of the Landauer principle [26]. In contrast, the experiment, which combinational logic realized with a micro-electromechanical cantilever, is reported in Reference [27]. The authors stated that the logical device can be operated with energy well below kBT, at room temperature, if the operation is performed slowly enough and friction losses are minimized [27]. Thus, it was suggested that no fundamental energy limit need be associated with irreversible logic computation in general [27]. These results were criticized in Reference [28], in which it was shown that the approach, reported in Reference [27], neglects the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode, which totally dissipates during the full cycle of logic values [28]. The analysis of the asymmetrical potential well, performed in Reference [10], is out of the scope of our paper. Now consider the computing device based on the ternary logic, and using the “trit” computing element, as presented in Figure4 and discussed in References [ 1–3,6]. Finding particle m in the certain one third part of the chamber corresponds to the recording of 1 trit of information. When both the partitions are removed, the location of the particle is uncertain, which corresponds to the erasure of 1 trit of information. The analysis of the minimal Carnot engine which work is analogical to removing/introducing the partition immediately yields that the work necessary for erasing of the “trit” of information equals W = kBTln3. The same conclusion arises from the analysis of the “trit”, based on Entropy 2019, 21, 1150 4 of 6

Entropy 2019, 21, 1150 4 of 6 basedEntropy on the2019 Brownian, 21, 1150 particle in a triple-well symmetrical potential, analogical to that depicted4 of 6 in Figure 3 and shown in Figure 5. Indeed, in this case 𝑝 =𝑝 =𝑝 = , and again, we obtain the based on the Brownian particle in a triple-well symmetrical potential, analogical to that depicted in the Brownian particle in a triple-well symmetrical potential, analogical to that depicted in Figure3 LandauerFigure 3 andbound shown 𝑊=𝑘 in Figure𝑇𝑙𝑛3. It5. seems Indeed, from in athis first case glance 𝑝 =𝑝that the=𝑝 ternary= , and computer again, devicewe obtain is well- the = = = 1 expectedand shown to be inenergetically Figure5. Indeed, unfavorable, in this case whenp0 comparedp1 p2 to3 , andthe again,computing we obtain device, the Landauerbased on the Landauer bound 𝑊=𝑘 𝑇𝑙𝑛3. It seems from a first glance that the ternary computer device is well- bound W = kBTln3. It seems from a ﬁrst glance that the ternary computer device is well-expected binary logic. However, this conclusion is erroneous. Indeed, “trit” equals to log23 bits of information expectedto be to energetically be energetically unfavorable, unfavorable, when compared when compared to the computing to the computing device, based device, on the based binary on the [29]. Thus, an energy bound for erasing of one bit of information for the ternary computers equals: binarylogic. logic. However, However, this this conclusion conclusion is erroneous. is erroneous. Indeed, Indeed, “trit” equals “trit” toequals log23 to bits log of23 information bits of information [29]. [29].Thus, Thus, an an energy energy bound bound for for erasing erasing of one of bitone of𝑘 bi information𝑇𝑙𝑛3t of information for the for ternary the ternary computers computers equals: equals: 𝑊 = =𝑘𝑇𝑙𝑛2 (2) 𝑘𝑙𝑜𝑔𝑇𝑙𝑛33 kBTln3 𝑊Wbit = ==𝑘kBTln𝑇𝑙𝑛22 (2) (2) 𝑙𝑜𝑔log233

Figure 3. The qubit model of a memory exploiting a Brownian particle M in a symmetrical double- well Figurepotential 3. The with qubit position model ofy which a memory can exploiting be stably a trapped Brownian in particle eitherM leftin or a symmetrical right well, double-wellcorresponding Figure 3. The qubit model of a memory exploiting a Brownian particle M in a symmetrical double- to informationalpotential with states position 𝑚=0;y which 𝑚=1 can be(see stably Reference trapped [9]). in either left or right well, corresponding to wellinformational potential with states positionm = 0; ym which= 1 (see can Reference be stably [9 ]).trapped in either left or right well, corresponding to informational states 𝑚=0; 𝑚=1 (see Reference [9]). It is recognized from Equation (2) that the erasing of 1 bit of information for the ternary computer It is recognized from Equation (2) that the erasing of 1 bit of information for the ternary computer equals to that inherent for the binary-memory-based one. Generalization of Equation (2) for the N- equalsIt is recognized to that inherent from Equation for the binary-memory-based (2) that the erasing of one. 1 bit Generalization of information of for Equation the ternary (2) for computer the based memory is straightforward: equalsN-based to that memory inherent is straightforward: for the binary-memory-based one. Generalization of Equation (2) for the N- 𝑘 𝑇𝑙𝑛𝑁 based memory is straightforward: kTlnN 𝑊 = B ==𝑘𝑇𝑙𝑛2 (3) Wbit 𝑙𝑜𝑔𝑁 kBTln2 (3) 𝑘log𝑇𝑙𝑛𝑁2N 𝑊 = =𝑘 𝑇𝑙𝑛2 (3) We conclude that the Landauer bound, which𝑙𝑜𝑔 𝑁 is necessary for erasing one bit of information We conclude that the Landauer bound, which is necessary for erasing one bit of information 𝑊=𝑘𝑇𝑙𝑛2 remains the same for the computers that are based on a many-valued logic. WWe= concludekBTln2 remains that the sameLandauer for the bound, computers which that areis necessary based on a for many-valued erasing one logic. bit of information

𝑊=𝑘𝑇𝑙𝑛2 remains the same for the computers that are based on a many-valued logic.

FigureFigure 4. Finding 4. Finding of the of the particle particle mm inin the the certain certain one-third part part of of the the chamber chamber corresponds corresponds to the to the recording of 1 bit of information. Thus, the “trit”-based computation becomes possible. recording of 1 bit of information. Thus, the “trit”-based computation becomes possible. Figure 4. Finding of the particle m in the certain one-third part of the chamber corresponds to the recording of 1 bit of information. Thus, the “trit”-based computation becomes possible.

Entropy 2019, 21, 1150 5 of 6 Entropy 2019, 21, 1150 5 of 6

FigureFigure 5. TheThe trit-based trit-based model model of ofa memory a memory exploiting exploiting a Brownian a Brownian particle particle M in Ma symmetricalin a symmetrical triple- triple-wellwell potential potential with withposition position y whichy which can canbe bestably stably trapped trapped in ineither either central, central, left left or or right well, corresponding to the informational states, namely: m𝑚=−1;= 1; m =𝑚=0;0; m = 1.𝑚=1 corresponding to the informational states, namely: − . 3. Conclusions 3. Conclusions The physical roots, justification, and precise meaning of the Landauer principle remain debatable The physical roots, justification, and precise meaning of the Landauer principle remain and were exposed to the turbulent discussion recently [7–12,30–32]. The present paper is devoted to debatable and were exposed to the turbulent discussion recently [7–12,30–32]. The present paper is the very particular question: If we assume that the Landauer principle holds for the binary-logic based devoted to the very particular question: If we assume that the Landauer principle holds for the computing device, should it hold for the many-valued logic computer? In other words, if we adopt the binary-logic based computing device, should it hold for the many-valued logic computer? In other principle that a minimum possible amount of energy required to erase one bit of information within a words, if we adopt the principle that a minimum possible amount of energy required to erase one bit binary logic computer equals the Landauer bound W = kBTln2, what minimal energy should be spent of information within a binary logic computer equals the Landauer bound 𝑊=𝑘𝑇𝑙𝑛2, what for the same purpose within the many-valued-logic-based computer? Starting from the analysis of minimal energy should be spent for the same purpose within the many-valued-logic-based ternary-logic-based computing device [1–3,33], we demonstrated that the Landauer limit, necessary for computer? Starting from the analysis of ternary-logic-based computing device [1–3,33], we erasing of one bit of information W = kBTln2 remains the same for computers based on a many-valued demonstrated that the Landauer limit, necessary for erasing of one bit of information 𝑊=𝑘𝑇𝑙𝑛2 logic. Thus, the universality of the Landauer principle is shown. remains the same for computers based on a many-valued logic. Thus, the universality of the Landauer Funding:principleThis is shown. research received no external funding. Conflicts of Interest: The author declares no conflict of interest. Funding: This research received no external funding.

ReferencesConflicts of Interest: The author declares no conflict of interest.

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