On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings

Home , Thermodynamic system

124 Maxwell's Demon t iOIl men \dl('~ ing Ii hy d A kllO\' "pel' chill "au ON THE DECREASE OF ENTROPY IN A THERMODYNAMIC SYSTEM wol'l BY THE INTERVENTION OF INTELLIGENT BEINGS trill fiuct! LEO SZILARD wou long' t 11(' WOl' nonl pl'n. play Wlll the Translated by Anatol Rapoport and Mechthilde Knoller from the original article "Uber die Entropievel'. 01' II Zeit~chrift mindenmg in einem thermodynamischen System bei Eingriffw intelligenter Wesen." /111' whe Physik, 1929, 53, 840-856. fol'll entropy in connection with the lIleasure· call The objective of the investigation is to ment, therefore, need not be greater than esta find the conditions which apparently allow Equation (1) requires. onl the construction of a perpetual-motion ma whf:' chine of the second kind, if one permits an tem HERE is an objection, already historiCILI, tive intelligent being to intervene in a thermo against the universal validity of tim veil dynamic system. When such beings make TSecond Law of Thermodynamics, which ill 81 measurements, they make the system behave deed looks rather ominous. The ohjectioll 1M fal' in a manner distinctly different from the way embodied in the notion of :\Iaxwell's dcnlllll, pcrl a mechanical system behaves when left to who in a different form appear... en'n nowlL chi! itself. We show that it is a sort of a memory days again and again; perhaps not umCRHOIl hut faculty, manifested by a system where reg measurements occur, that might cause a ably, inasmuch as behind tilt' prcciHllIy intl' permanent decrease of entropy and thus a formulated question quantitative <,Olnll'C' A violation of the Second Law of Thermody tions seem to be hidden which to date IIIL\'" pos namics, were it not for the fact that the not been clarified. The objection in itH odIC' of p measurements themselves are necessarily nal formulation concerns a tIl'IIIOII who as a accompanied by a production of entropy. At catches the fast molecules and lets till' ,dolY t.im first we calculate this production of entropy ones pass. To be sure, the objt:-ction mil hI! ing quite generally from the postulate that full met with the reply that man cnnlLot in pdll inte compensation is made in the sense of the ciple foresee the value of a thel'lllully 'hit' Itt R Second Law (Equation [1]). Second, by tuating parameter. However, onc' ('ILIlIIO! Thl\ using an inanimate device able to make deny that we can very well ml'Ul'lllrCl lit" measurements-however under continual value of such a fluctuating pal'UIIH'lm' I\llll entropy production-we shall calculate the therefore could certainly gain ellf'rl/;~' ILl. thI! resulting quantity of entropy. We find that expense of heat by arranging (Jill' illt (,I'Vt'l' it is exactly as great as is neees;;ary for full compensation. The actual production of l1P I

L Szilard 125

tion a('('())'{lil1g to til<' re"nlts of the measure as well as the motor nel'Vous systems a l11<'nt::;. I're"ently, of ("ourse, we do !lot know degradation of energy is always involved, whether \\'e ('omJllit an elTor by 1101. includ quite apart from the fact that the very ing tlw ilJterYening man into the system and existence of a nervous system is depC'ndent by di"regarding his biological phenomena. on continual dissipation of ellergy. Apart from this ulll'esolved matter, it is Whether--considering these circum- kno\\'11 toda~' that in a system left to itself no stances-'real living bC'ings could continually "perpetuUlIl mobile" (perpetual motion ma or at least regularly produce energy at the chiJl(') of tIlt' second kind (more exactly, no expense of heat of the lowest temperature ap rEM "automatic machine of continual finite pears very donbtful, even though our ignor \\'ork-yi<'ld which uses heat at the lowest ance of the biological phenomena does not temperatuJ'(''') can op<,rate in spite of the allow a definite answer. HowC'ver, the latte!' fluctuatioll phenomena. A perpetuum mobile questions lead beyond the scope of physics \\'ould haw to be a machine which in the in the strict sense. long nlll could lift a weight at the expense of II. appears that the ignorance of the bio tIl(' lH'at content of a reservoir. In other logical phenomena need not prevent us from words, if \\"l' want to use the fluctuation phe understanding that whieh seems to us to be nomella in order to gain energy at the ex the essential thing. We may be sure that pem;p of Iwat, we are in the samc position as intelligent living beings-insofar as we are playing a game of chance, in which we may dealing with their intervention in a ther win celtain amounts no\\' and then, although modynamic system-can be replaced by non the expcetation value of the winnings is zero living devices whose "biological phenomena" ler or !legative. The same applies to a system one could follow and determine whethm' in fIll' when> th£> intervention from outside is per fact a compensation of the entropy decrease formed st riel ly periodically, say by periodi takes plaee as a result of the intervention by eally moving machines. We consider I. his as such a device in a system. established (Szilard, 1923) and intend here In the first place, we wish to learn what only to ('onsider the difficulties that occur circumstance conditions the decrease of \\'hen illtelligent being'S intervene in a sys entropy which takes place when intelligent tem. We shall try to discover the quantita living beings intervene in a thermodynamic :al, tive relations having to do with this inter system. We shall see that this depends on a ;he ventiOl). certain type of coupling between different ill Smoluchowski (1914, p. 89) writes: "As parameters of the system. We shall consider lIS far as w(' know today, there is no automatic, an unusually simple type of these ominous In, permalJently effective perpetual motion ma couplings.! For brevity we shall talk about a va chine, in spite of the moleculal' fluctuations, "measurement," if we succeed in coupling )n but 8\1('h a device might, perhaps, function the value of a parameter y. (for instance' the ely regularly if it were appropriately operated by position co-ordinate of a pointer of a meas ec intelligent beings...." uring instrument) at one moment with the Lye A perpetual motion machine therefore is simultaneous value of a fluctuating parame igi possible if-according to the general method ter x of the system, in such a way that, from rho of physics-we view the experimenting man the value y, we can draw conclusions about ow as a sort of deus ex machina, one who is con the value that :/.; had at the moment of the be tinuously and exactly informed of the exist "measurement." Then let :I.; and y be un 'in ing state of nature and who is able to start or coupled after the measurement, so that x can ue intel'l'upt the macroscopic course of nature change, while y retains its value for some [lot at any moment without expenditure of work. time. Such measurements are not harmless the Therefore he would definitely not have to interventions. A system in which such md possess the ability to eatch single molecules measurements occur shows a sort of memory like }laxwell's demon, although he would the definitely be different from real living beings I The author e"identlY uses the word "omi ell- nous" in the sense that the possibility of realizing in possessing the abo\-e abilities. In eliciting the proposed arrangement threatens the validity any physical effect by action of the sensory of the Second Law.-T,;anslalo/, 126 Maxwell's Demon faculty, in the sense that one can recognize However, it is no longer constrailH'd to the by the state parameter y what value another upper part of the cylinder but bOUIH'PS many tion 01 state parameter :1' had at an earlier moment, times against the piston whieh is already tion c ' and we shall see that simply because of such moving in the I0'ver part of the f·ylinder. III In t a memory the Second Law would be vio this way the molecule does a certain amount "'ish t. lated, if the measurement could take place of work on the piston. This is tllt' work that 1. without compensation. We shall realize that corresponds to the isothermal expansion of piston the Second Law is not threatened as much an ideal gas--consisting of ailE' sing;le mole the c by this entropy decrease as one would think, cule-from volume VI to tlw volume either as soon as we see that the entropy decrease VI + V 2 • After some time, when the piston choose resulting from the intervention would be has reached the bottom of the container, the ately, compensated completely in any event if the molecule has again the full volume r I + 1-2 restric execution of such a measurement were, for to move about in, and the piston is then re ;f < 0 instance, always accompanied by produc moved. The procedure can be repeated as 2. r tion of k log 2 units of entropy. In that case many times as desired. The man moves the ment, it will be possible to find a more general piston up or down depending on wlwther the during' entropy law, which applies universally to all molecule is trapped in the upper or lower half down., measurements. Finally we shall consider a of the piston. In more detail, this motion of the' very simple (of course, not living) device, may be caused by a weight, that is to be the ori that is able to make measurements con raised, through a mechanism that tmnsmits if the tinually and whose "biological phenomena" the force from the piston to the wf'ight, in cylind we can easily follow. By direct calculation, such a way that the latter is always dis i.e., W one finds in fact a continual entropy produc placed upwards. In this way thl' potential on tht" tion of the magnitude required by the above· energy of the weight certainly increases part mentioned more general entropy law de constantly. (The transmission of force to the enel'gy rived from the validity of the Second Law. weight is best arranged so that the force x has t The first example, which we are going to exerted by the weight on the piston at any on the consider more closely as a typical one, is the position of the latter equals thf' average riod i following. A standing hollow cylinder, closed pressure of the gas.) It is clear that in this spondi at both ends, can be separated into two manner energy is constantly gained at the tion 0 possibly unequal sections of volumes VI and expense of heat, insofar as the biological nated

V 2 respectively by inserting a partition from phenomena of the intervening man are ig positio the side at an arbitrarily fixed height. This nored in the calculation. that d partition forms a piston that can be moved In order to understand the eSSCllte of the positio up and down in the cylinder. An infinitely man's effect on the system, one best imagines afterw large heat reservoir of a given temperature T that the movement of the piston i,; performed passes insures that any gas present in the cylinder mechanically and that the man's aetivity that il undergoes isothermal expansion as the consists only in determining tIll' ~ltitude of the co piston moves. This gas shall consist of a the molecule and in pushing a len:,r (which disapp single molecule which, as long as the piston steers the piston) to the right or left. depend Wt' is not inserted into the cylinder, tumbles ing on whether the molecule's height requires ramet about in the whole cylinder by virtue of its a down- or upward movement. This means varies thermal motion. that the intervention of the lllllllan being value Imagine, specifically, a man who at a given consists only in the coupling of two position etcr ~. time inserts the piston into the cylinder and co-ordinates, namely a co-ordinate :r, which proced somehow notes whether the molecule is determines the altitude of the moleeule, with effech' caught in the upper or lower part of the cyl another co-ordinate y, which determines the beings. inder, that is, in volume 1-1 or 1'2' If he OIlE' position of the lever and therefore also should find that the former is the case, then Ul'en\(,1 he would move the piston slowly downward whether an upward or dowll\yard motion is riated lIntil it reaches the bottom of the cylinder. imparted to the piston. It is best to imagine pl'oduc During this slow movement of the piston the the mass of the piston as large and its speed ance \\ molecule stays, of course, above the piston. sufficiently great, so that the thermal agilll' cntrop: of ('011I L Szilard 127

tion of the pi~ton at the temperature in ques mental amount, but not smaller. To put it tion can lot' neglected. precisely: we have to distinguish here be In the typical example presented here, we tween two entropy values. One of them, k~l, "'ish to dir4inguish two periods, namely: is produced when during the measurement y 1. Tht-' pt-'riod of measurement when the assumes the value 1, and the other, 82 , when piston has ju;,:t been inserted in the middle of y assumes the value -1. We eannot expect the cyli.lld('r and the molecule is trapped to get general information about Sl 01' 82 either in th" upper or lower part; so that if we separately, but we shall see that if the choose the origin of co-ordinates appropri amount of entropy produced by the "meas ately, tlw .r-co-ordinate of the molecule is urement" is to compensate the entropy de restricted to either the interval x > 0 or crease affected by utilization, the relation ;1" < 0; . must always hold good. 2. The period of utilization of the measure ment, ··tJl(' pt-'riod of decrease of entropy," (1) during whieh the piston is moving up or One sees from this formula that one can down. During this period the x-co-ordinate of the mol('('ule is certainly not restricted to make one of the values, for instance Sl' as small as one wishes, but then the othel' value the original interval x > 0 or x < O. Rather, becomes correspondingly greater. Fur if the mol"eule was in the upper half of the 82 thermore, one can notice that the magnitude cylinder during the period of measurement, of the interval under consideration is of no i.e., whell .1' > 0, the molecule must bounce consequence. One can also easily understand on the dowllward-moving piston in the lower that it cannot be otherwise. part of tJH:' cylinder, if it is to transmit Conversely, as long as the entropies 1 and energy to tlw piston; that is, the co-ordinate 8 82 , produced by the measurements, satisfy x has to t'nter the interval:r O. The lever, < the inequality (1), we can be sure that the on the eontrary, retains during the whole pe expected decrease of entropy caused by the riod its po:,ition toward the right, corre later utilization of the measurement will be sponding to downward motion. If the posi fully compensated. tion of th(· lewr toward the right is desig Before we proceed with the proof of in nated by H = 1 (and correspondingly the equality (1), let us see in the light of the position toward the left by y = -1) we see above mechanical example, how all this fits that durillg the period of measurement, the together. For the entropies Sl and 82 pro position .r > 0 corresponds to y = 1; but duced by the measurements, we make the afterwards !I = 1 stays on, even though x following Ansatz: passes into the other interval x < O. We see that in the utilization of the measurement (2) the coupling of the two parameters :r and y disappears. This ansatz satisfies inequality (1) and We shall :,ay, quite generally, that a pa the mean value of the quantity of entropy rameter !I "rneasures" a parameter x (which produced by a measurement is (of course in varies a('('ording to a probability law), if the this special case independent of the fre value of !I is directed by the value of param quencies WI , W2 of the two events): eter :1' at a gin-'n moment. A measurement = k log 2 procedure' underlies the entropy decrease S effected hy the intervention of intelligent In this example one achieves a decrease of beings. entropy by the isothermal expansion:2 One Illay ]"('asonably assume that a meas urenwnt proeedure is fundamentally asso VI - 81 = -klog TTl + TT ; ciated \yith a ecrtain definite average entropy 2 production, and that this restores concord (4) l'0 ance with the Second Law. The amount of - 82 = -k log VI V ' entropy gent-'rated by the measurement may, + 2 of courst-', ahyays be greater than this funda- 2 The entropy gene;ated is denoted h~' 81, 82· 128 Maxwell's Demon

depcndinl!: on whether the molecule was also shall look upon them as chemically dif it is tra! found in volume VIOl' V2 when the piston ferent, if they diffCl' only in that the y co the sell was inserkd. (The decrease of entropy ordinate is +1 for one and -1 for the other. which equals the ratio of the quantity of heat taken We should like to give the box in which the paralll from thl:' heat resl:'rvoir during the isothermal "molecules" are stored the form of a hollo\\' being, .' expansion, to the temperature of the heat cylinder containing four pistons. Pistons A. value reservoir in question). Sinee in the above and A' are fixed while the other two are mov during case the frl:'qul:'ncies WI , W2 are in the ratio of able, so that the distance BB' always equals origina the voluml's 1"1, 112, the mean value of the the distance A.1', as is indicated in Figure 1 We sh entropy gl:'llerated is (a negative number): by the two brackets. A', the bottom, and B, meIllOl the cover of the container, are impermeable just di' .~ = WI .( +~1) + W2' ( + 82) = for all "molecules," while A and B' are semi we sep 1"1 1"1 permeable; namely, A is permeable only for out dd 1';--+ l k log VV + those "molecules" for which the parameter ;1: taincl's tl 2 1+ 2 (;j) ' is in the preassigned interval, i.e., (XI, :1'2), B' one III V 2 1"1 is only permeable for the rest. other.

1'1 + 1'2 k log 'VI + V2 saIlle In one As one can see, we have, indeed B itself, ' 2 A 1 gard t VI k I 1'1 + F ---_.- COUl'se, 1"1 + 112 og VI + 1'2 VI + 1'2 (6) remain V2 to be ~ ·klng VI + 1'2 + k log 2 0 cntly (I) 1 ture, t and tlwrl'fore: tainly ~ S + 8 O. (7) _____ ---< B' release 2 cation, In the spceial case considered, we would A' actually have a full compensation for the de aCCOIll FIG. 1 crease of entropy achieved by the utilization both (. of the ml:'asurcment. the enl We shall not examine more special cases, In the beginning the piston B is at A and crease hut instead try to darify the matter by a therefore B' at A', and all "molecules" are entrop general argument, and to derive formula (1). in the space between. A certain fraction of the va We shall therefore imagine the whole sys the molecules have their co-ordinate X in the cull' is tem-in whieh the co-ordinate X, exposed to preassigned interval. We shall designate by some kind of thermal fluctuations, can be WI the probability that this is the case for a measured h:,' the parameter y in the way just randomly selected molecule and by W2 the (Th explained--as a multitude of particles, all probability that x is outside the interval. sign to enclosed in one box. Everyone of these par Then WI + W2 = 1. o('eur J ticles can moye freely, so that they may be Let the distribution of the parameter y be tolal n considered as the molecules of an ideal gas, over the values +1 and -1 in any propOl' one or which, hecausc of thermal agitation, wander tion but in any event independent of the XO\\ about in the common box independently of x-values. We imagine an intervention by an gases each other and exert a certain pressure on the intelligent being, who imparts to y the value pxppnd walls of the box-the pressure being deter 1 for all "molecules" whose X at that moment piHt.on mined by the temperature. We shall now is in the selected interval. Otherwise the tilil\('l' eonsider two of these molecules as chemi value -1 is assigned. If then, because of HN' eally different and, in principle, separable by thermal fluctuation, for any "molecule," thll olltHidt semipermeable walls, if the co-ordinate X for parameter x should come out of the preal! II'hidl one molecule is in a preassigned interval signed interval or, as we also may put it, if 111111/;<'1', while the corresponding co-ordinate of the the "molecule" suffers a monollloiccultu ,If'('I'('II other molccule falls outside that interval. We chemical reaction with regard to :r (by whidl 111I'1l11 I L Szilard 129 it is transformed from a specie;; that can pass long as we do /lot lise the fact that the molecules the semipel'lneable piston A into a species for in the container BB', by l'irtlle (~r their co which the piston is impermeable), then the ordinate y, "remember" that the .r-eo-ordinate param('tt'r y retains its value 1 for the time for the JIIolec1tles of this rontaillN originally being, ;;0 that the "molecule," becausp of the was in the preassigned interval, fllll cOIi/pensa valul' of the paranletel' y, "remembers" lion e:rists for the calculated de('reast ofentropy, dUl'ing thl' whole following process that ;!" by virtue of the fact that til<' partial pres originally was in the preassigned interval. SUl'es in the two eontaincrs al't' snlUllp!' than We ;;hall see immediately what part this in the original mixture. memory Illay play, Aft er the intervention But now we can use the fa('/ that all mole just diseussed, we move the piston, so that cules in the container BB' hat,(, the y-co-ordi we separate the two kinds of Illolecules with nate 1, and in the other arrurdillgly -1, to out doing work. This rpsults in two eon bring all molecules back agaill to the origt'nal tainers, of which the first eontains only the volulI/e. To accomplish this \H' only nced to one Illodification and the seeond only the rcplace the semipermeable wall .\ by a wall at her. Each modification now occupies the .1*, which is semipermeablp not \"ith regard sallie volume as the mixtUl'e did previously. to:r hut with regard to y, nanlely so that it i;; In one of th('se containers, if considered by pel'lneahle for the molecules \\'ith the y-co itself, there is now no equilibrium with re ordinate 1 and impermeahlP for til(' others. gard to thp two "modifications in :1'." Of Correspondingly we replacp R' by a piston course the ratio of the two modifications has B'*, which is impermeable' for t he molecules remained WI: W2 • If we allow this equilibriulll with y = -1 and permeable for the others. to be achipvpd in both containers independ Then both containers can 1)(' put into pach ently and at eonst ant volume and tempera other again without expenditul't, of energy. ture, then the entropy of the system cer The distribution of the y-co-ordinate with tainly has increased. FOl' the total heat regard to 1 and -1 no\\' has Iwcome sta release is 0, sinee the ratio of the two "modifi tistically indepcndent of the ;r-nllues and be cations in :1''' WI :W2 does not change. If we sides we are able to re-establish 1he original accomplish the equilibrium distribution in distribution over 1 and -1. Thus we would both containers in a reversiblc fashion then have gone through a eompl('le eycle. Thc the pntropy of the rpst of the world will de only change that we have to register is the crease by the samp amount. Thprefore the resulting decrease of entl'Op~' giwn by (n): entropy increases by a negative valup, and, the value of the entropy increase PP!' mole s = k(wl log WI + U'~ lo/.!: ll'2)' (10) cule i;; pxaf't Iy: If we do not wish to admit that the Second Law has been violated, \\'(' must conclude s = k(wl log WI W2 log wJ. (H) + that the 1'ntervention which cl'tabl ishes the (1'1](' entropy ('onstants that we must as coupling between y and :1', the /licaS/lI'ement of sign to the I\YO "modifieations in :1''' do not :1' by y, 'IIIust be accolI/panied !I,I/ a production occur here pxplicitly, as the proeess leaves the of entropy. If a definite way of achieving this total number of molecules bdonging tothe (~oupling is adopted and if tIll' quantity of one 01' the other species Ulwhanged.) entropy that is inevitably produeed is desig Xow of COUl'se we cannot bring the two nated by 8 1 and 8 2 , wh('1'(' "I stands for the gasps hack to the original volume without mean increasc in pntropy that ()('eurs when ?J exppnditure of work by simply moving the acquires the value 1, and accordingly 8 2 for piston hack, as there arc now in the con the increase that occurs when !J acquires the tainer-which is hounded hy the pistons value -1, we al'l'ive at the equation: BB'-also molecules whose x-co-ordinate lies (11 ) outside of the preassigned interval and for which thp piston A is not permeable any In order for the Second La\\' to remain in longpl'. Thus onp can spp that the calculated force, this quantity of pntl'Opy must be decreasp of entropy (Equation [9]) docs not greater than the decreas(~of entropy S, which mpan a eontradidion of the Second Lm\'. As according to W) is producpd by til(' utiliza- 130 Maxwell's Demon tion of thp nwasurement. Therefore the fol not possible. But we can try to describe sim the poil lowing unequality must be valid: ple nonliving devices that effeet such eou small a pIing, and see if indeed entropy is generated tUl'e 'J'N 8+8;:;;0 and in what quantity. Having already reeog pointer nized that the only important fattor is a interva WI'';I + W 2S2 (12) certain characteristie type of ('oupling, a while t. + !.'(lel log WI + W2 log wJ ;:;; 0 "measurement," we need not construct any no long' This ('quation must be valid for any values complicated models which imitate the inter the ene of WI and W2 ,3 and of course the constraint vention of living beings in detail. We can be gard t satisfied with the eonstruction of this par pointer 1))2 + W2 = 1 eannot be violated. We ask, in particular, for which WI and W2 and given ticular type of coupling which is accom sion wit S-valw's thp expression becomes a minimum. panied by memory. at the In our next example, the position co-or For the two minimizing values WI and W2 the llleaSUI' inequalit~, (l~) must still be valid. Under the dinate of an oscillating pointer is "mcasured" Aftel! above con::-t mint, the minimum occurs when by the energy content of a body K. The complis the follO\yin~ ('quation holds: pointer is supposed to connect, in a purely ally fu mechanical way, the body K -by whose cOllllec~ 8 1 52 energy content the position of the pointcr is and B ' -- + 10" WI = - + 10" W2 ( 13 ) Ie " 1.. " to be measured·-by heat conduction with the pUl' one of two intermediate pieces, A. 01' B. The is now' But then: body is connected with A as long as the co interlll 1k e-"~k ~ ('-"1 + 1. (14) ordinate-which determines the position of state. the pointer-falls into a certain preassigned, ate pie This is ea::-ily seen if one introduces thc no but otherwise arbitrarily large or small inter has be tation val u, and otherwise if the co-ordinate is in TB-to the interval b, with B. Up to a certain mo an iner ment, namely the moment of the "measure be avoi ment," both intermediate pieces will be to hea then: thermally connected with a heat reservoir at temper ( 16) temperature To. At this moment the inser the res tion A. will be cooled reversibly to the tem and to If one suhst,itutes these values into the in perature 1'.., e.g., by a periodically function measUl' equality (I2) one gets: ing mechanical device. That is, after the tw Ac'(C-$l/k + e-"~k) ;:;; O. (17) successive contacts with heat reservoirs of at that intermediate temperatures, A. will be brought had bee Therefore the following also holds: into contact with a heat reservoir of thc end. T temperature T.{ , At the same time the insel'· A ;:;; O. (18) weaho tion B will be heated in the same way to tween' If one puti' the values 'WI and W2 from (16) temperature T B • Then the intermediate Thn into the Njuation WI + 'W2 = 1, one gets pieces will again be isolated from the corrc tl'oPY sponding heat reservoirs.

the pointer at the time of "measurement"- HowevC'r, following this, tl1(' cntropy' with small at temperature 1'.. or great at tempera drawn from the J'(>servoir To by din'ct con ture 1'8 and will retain its value, even if the tact with it "'as pointer eventually leaves the preassigned ulT ) - u(1'.. ) interval or enters into it. After some time, o (:2:2) while the pointer is still oscillating, one can To no longer draw any definite conclusion from the energy content of the body K with re All in all the entropy was inereased by gard to thl' momentary position of the the amount pointer but one can draw a definite conclu sion with regard to the position of the pointer at the time of the measurement. Then the measurement is completed. Analogously, t he entropy will incrcase by After the measurement has been ac the following amount, if the body was in con complished, the above-mentioned periodic tact with the intermediate pi('cc B at the ally functioning mechanical device should time of the "measurement": connect the thermally isolated insertions A and B with the heat reservoir 1'0. This has s = u(TH ) - u(1'o) +jT"!dii 11' (:24) the purpose of bringing the body K -which B l'o TB l' (11'( . is now also connected with one of the two intermediate pieces--back into its original We shall now evaluate thesl' cxpressions state. The direct connection of the intermedi for the very simple ease, whl'l'l' the body ate pieces and hence of the body K -which which we use has only two ('lIl'rgy statcs, ~ has been either cooled to 1'.. 01' heated to lower and a higher state, If such u body is in 1'B-to the reservoir 1'0 consequently causes thermal contact with a heat l'l'SC1TOir at any an increase of entropy. This cannot possibly temperature 1', the probability I hat it is i~ be avoided, because it would make no sense the lower or upper state is giWll by re to heat the insertion A. reversibly to the spect ively : temperature 1'0 by successive contacts with 1 the reservoirs of intermediate temperatures peT) = ; and to cool B in the same manner. After the 1 + ge-u"T! measurement we do not know with which of ge- lI .'kT , the two insertions the body K is in contact 1j(T) = I at that I"flment; nor do we know whether it 1 + ge-u."kT: had been in connection with 1'.. or l'B in the Here u stands for the diffcl'('ncl' of cnergy end. Therefore neither do we know whether of the two states and g for thl' statistical we should use intermediate temperatures be weight. We can set the cncrgy of the tween 1'.. and To 01' between To and l'B. lower state equal to zero without 1m-is of The mean value of the quantity of en generality. Thereforc:4 tropy 8 1 and 8 2, per measurement, can be calculated, if the heat capacity as a function ( - 'I q( T .. ) p (TlI ) of the temperature u(1') is known for the 8.. = q l'A) I;~ OJ:!; - (1' ) (T) q 0 p -' , body K, since the entropy can be calculated from the heat capacity. We have, of course, klog~-'(T ) i neglected the heat capacities of the inter + p(To) mediate pieces. If the position co-ordinate r (:2tj) - (1'-)"1 q(To)p(TH ) • of the pointer was in the preassigned inter SB - P 8 '\ og , -- -- _ val at the time of the "measurement," and - q(1,,)p(Tn)! accordingly the body in connection with in i sertion A, then the entropy conveyed to the + k log Ij( TI!_ »)' heat reservoirs during successive cooling was I/( T,,) (TO ~ du Here Ij and pare the,func! ions of l' given JT_~ 1'd1" (21) 4 See the .\ppendix. 132 Maxwell's Demon by equat.ion (25), which are here to be taken and the mean energy of the body is p;iyen by: 'Ye obtain thl' for the arguments 1'0 , 1'.1 , or T B• if we replace tl If (as is neccssitated by the above concept uge-UlkT tain: iitT) = uq(T) = 1 + ge- ' (31) of a "measurement") we wish to draw a UlkT dependable conclusion from the energy con the following identity is valid: tent of the body K as to the position co-or dinate of the pointer, we have to see to it Formula (41) i 1 du d {U(T) ( ulkT)' - - = - -- + k log 1 + e- I (32) that the body surely gets into the lower TdT dT T ). SA, in the text. energy state when it gets into contact with We can brin what different T B • In otlH'r words: Therefore we can also write the, equation: q( p(T.t ) = 1, q(L) = 0; (27) p(TH ) = 0, q(TB ) = 1. as This of course cannot be achieved, but may _ u(TA) - utTo) be arbitrarily approximated by allowing 1'..1. 'AS - to approach absolute zero and the statis To (34) tical weight g to approach infinity. (In U(T) ukT }TO + --+ k log(l + ge- ), thL., limiting process, To is also changed, in { l' TA such a way that p(To) and '1(1'0) remain constant.) The equation (26) then becomes: and by substituting the limits we obtain:

S.t = -k log p(To); 1 1 ) 1 + ge-ulkTO S.t = ii.(TA ) -- -1' + k log 1 T' (35) (28) ( To A + ge-ulk A SB = -k log q(To) If we write the latter equation according to SAlk and if \H' form the expression e- + (25) : SB1 e- \ we find: 1 1 + ge-"lkT = (36) (29) peT)

for T A and To , then we obtain: Our foregoing considerations have thus just realized the smallest permissible limiting _, ( 1 1 ) peTA) = u(TA To - p(T ) (37) eare. The use of semipermeable walls ac SA ) 1'..1. +klog o cording to l~igUl'e 1 allows a complete utilization of the measurement: inequality and if we then write according to (31): (1) certainly cannot be sharpened. (38) As we haye scen in this example, a simple inanimate de'vice can achieve the same we obtain: essential r('sult as would be achieved by the p(TA ) u u) (39) intervention of intelligent beings. We have SA = '1(1' A) To - l'A k log p(T )• ( + o examined the' "biological phenomena" of a nonliving de'Yice and have seen that it gen If we finally write according to (25): erates exactly that quantity of entropy \\'hich is requircd by thermodynamics. ~ = -k 10 q(T) (40) T g tJp(T) APPENDIX for T A and To , then we obtain: In the case considered, when the frequency of the two statrs drpends on the temperature ac P (To) q(TA ) SA = '1(1' A) k log (To) peTA) cording to the equations: q (41) p(T. ) 1 ge-u1kT t + k log p(T ) . peT) = 1 + ge-ulkT ;qtT) = 1 + ge-ulkT (30) o L Szilard 133

We obtain thl:' corresponding equation for Sfj , expand and colll:'ct terms, thl:'n we get if we replace thl:' indpx A. with B. Then we ob tain: (44)

s = (IT) k 10 p(T.) 2!:.(TB ) + k I p(TB ) (42) B ~\ B g q(T.) p«TB) og peT. r This is the formula given in the text for SB .

Formula (41) is identical with (26), givm, for REFERENCES SA, in the text. We can bring the formula for Sfj into a some Smoluchowski, F. I-or/rage itber die kinetische what different form, if we write: Theorie der Jla/erie u. Elektl'izi/at. Leipzig: 1914. q(TB ) = 1 - p(TB ), (43) Szilard, L. Zeitschrift fur Physik, 1925,32,753.

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