In memory of Leo Szilard, who passed away on May 30,1964, we present an English translation of his classical paper ober die Enfropieuerminderung in einem thermodynamischen bei Eingrifen intelligenter Wesen, which appeared in the Zeitschrift fur Physik, 1929,53,840-856. The publica- tion in this journal of this translation was approved by Dr. Szilard before he died, but he never saw the copy. At Mrs. Szilard’s request, Dr. Carl Eckart revised the translation. This is one of the earliest, if not the earliest paper, in which the relations of physical to information (in the sense of modem mathematical theory of communication) were rigorously demonstrated and in which Max- well’s famous demon was successfully exorcised: a milestone in the integra- tion of physical and cognitive concepts. ON THE DECREASE OF ENTROPY IN A THERMODYNAMIC SYSTEM BY THE INTERVENTION OF INTELLIGENT BEINGS by Leo Szilard Translated by Anatol Rapoport and Mechthilde Knoller from the original article “Uber die Entropiever- minderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen.” Zeitschrift fur Physik, 1929, 65, 840-866.

0*3 resulting quantity of entropy. We find that The objective of the investigation is to it is exactly as great as is necessary for full find the conditions which apparently allow compensation. The actual production of the construction of a perpetual-motion ma- entropy in connection with the measure- chine of the second kind, if one permits an ment, therefore, need not be greater than intelligent being to intervene in a thermo- Equation (1) requires. dynamic system. When such beings make F(J measurements, they make the system behave in a manner distinctly different from the way HERE is an objection, already historical, a mechanical system behaves when left to Tagainst the universal validity of the itself. We show that it is a sort of a memory Second Law of , which in- faculty, manifested by a system where deed looks rather ominous. The objection is measurements occur, that might cause a embodied in the notion of Maxwell’s demon, permanent decrease of entropy and thus a who in a different form appears even nowa- violation of the Second Law of Thermody- days again and again; perhaps not unreason- namics, were it not for the fact that the ably, inasmuch as behind the precisely measurements themselves are necessarily formulated question quantitative connec- accompanied by a production of entropy. At tions seem to be hidden which to date have first we calculate this production of entropy not been clarified. The objection in its origi- quite generally from the postulate that full nal formulation concerns a demon who compensation is made in the sense of the catches the fast molecules and lets the slow Second Law (Equation [l]).Second, by ones pass. To be sure, the objection can be using an inanimate device able to make met with the reply that man cannot in prin- measurements-however under continual ciple foresee the value of a thermally fluc- -we shall calculate the tuating parameter. However, one cannot Requests for reprints of this paper should be deny that we can very well measure the sent to: value of such a fluctuating parameter and Mrs. Leo Szilard 2380 Torrey Pines Road therefore could certainly gain at the La Jolla, California 92038 expense of by arranging our interven- 301 302 LEO SZILAHI) tiori according to the results of the ineasure- as well as the motor nervous a nients. Presently, of course, we do not know degradation of energy is always involved, whether we commit an error by not includ- quite apart from the fact that the very ing the intervening man into the system and existence of a nervous system is dependent by disregarding his biological phenomena. on continual of energy. Apart froin this unresolved matter, it is Whether-considering these circuin- known today that in a system left to itself no stances---real living beings could continually “perpetuuni mobile’’ (perpetual niotion ma- or at least regularly produce energy at the chine) of the second kind (more exactly, no expense of heat of the lowest ap- “automatic of continual finite pears very doubtful, even though our ignor- -yield which uses heat at the lowest ance of the biological phenomena does not temperature”) can operate in spite of the allow a definite answer. However, the latter fluctuation phenomena. A perpetuuni mobile questions lead beyond the scope of physics would have to be a machine which in the in the strict sense. long run could lift a weight at the expense of It appears that the ignorance of the bio- thc heat content of a reservoir. In other logical phenomena need not prevent us frorn words, if we want to use the fluctuation phe- understanding that which seems to us to be noinena in order to gain energy at the ex- the essential thing. We may be sure that pense of heat, we are in the same position as intelligent living beings-insofar as we are playing a ganic of chance, in which we may dealing with their intervention in a ther- win certain amounts now and then, although niodynaniic systen-can be replaced by non- the expectation value of the winnings is zero living devices whose “biological phenomena” or negat>ive.The same applies to a system one could follow and determine whether in where the intervention from outside is per- fact a compensation of the entropy decrease formed strictly periodically, say by periodi- takes place as a result of the intervention by cally moving . We consider this as such a device in a system. established (Szilard, 1925) and intend here In the first place, we wish to learn what only to consider the difficulties that occur circumstance conditions the decrease of when intelligent beings intervene in a sys- entropy which takcs place when intelligent tem. We shall try to discover the quantita- living beings intervene in a therniodynamic tive relations having to do with this inter- system. We shall see that this depends on a vention. certain type of coupling between different Snioluchowski (1914, p. 89) writes: “As parameters of the system. We shall consider far as we know today, there is no automatic, an unusually simple type of these ominous permanently effective perpetual motion ma- coup1ings.l For brevity we shall talk about a chine, in spite of the molecular fluctuations, “measurement,” if we succeed in coupling but such a device might, perhaps, function the value of a parameter y (for instance the regularly if it were appropriately operated by position co-ordinate of a pointer of a meas- intelligent beings.. . .” uring instrument) at one nioment with the A perpetual motion machine therefore is simultaneous value of a fluctuating parame- possible if-according to the general method ter x of the system, in such a way that, from of physics-we view the experimenting man the value y, we can draw conclusions about as a sort of deus ex machina, one who is con- the value that x had at the moment of the tinuously and exactly informed of the exist- “measurement.” Then let x and y be un- ing state of nature and who is able to start or coupled after the measurement, so that 5 can interrupt the macroscopic course of nature change, while y retains its value for some at any rnornent without expenditure of work. time. Such measurements are not harmless Therefore he would definitely not have to interventions. A system in which such possess the ability to catch single molecules measurements occur shows a sort of memory like Maxwell’s demon, although he would ’The author evidently uses the word “omi- definitely be different from real living beings nous” in the sense that the possibility of realizing in possessing the above abilities. In eliciting the proposed arrangement threatens the validity any physical effect by action of the sensory of the Second Law.-Translator ON THE DECREASEOF ENTROPY 303 faculty, in the sense that one can recognize However, it is no longer constraincd to the by the state parameter y what value another upper part of the cylinder but bounces inany state parameter z had at an earlier moment, times against the piston which is already and we shall see that simply because of such moving in the lower part of the cylinder. In a memory the Second Law would be vio- this way the molecule does a certain amount lated, if the measurement could take place of work on the piston. This is the work that without compensation. We shall realize that corresponds to the isothermal expansion of the Second Law is not threatened as much an ideal --consisting of one single - by this entropy decrease as one would think, cule-from V, to the volume as soon as we see that the entropy decrease V1 + V2 . After some time, when the piston resulting from the intervention would be has reached the bottom of the container, the compensated completely in any event if the molecule has again the full volume V1 + Vz execution of such a measurement were, for to move about in, and the piston is then re- instance, always accompanied by produc- moved. The procedure can be repeated as tion of k log 2 units of entropy. In that case many times as desired. The man moves the it will be possible to find a more general piston up or down depending on whether the entropy law, which applies universally to all molecule is trapped in the upper or lower half measurements. Finally we shall consider a of the piston. In more detail, this motion very simple (of course, not living) device, may be caused by a weight, that is to be that is able to make measurements con- raised, through a mechanisni that transmits tinually and whose “biological phenomena” the force from the piston to the weight, in we can easily follow. By direct calculation, such a way that the latter is always dis- one finds in fact a continual entropy produc- placed upwards. In this way the potential tion of the magnitude required by the above- energy of the weight certainly increases mentioned more general entropy law de- constantly. (The transmission of force to the rived from the validity of the Second Law. weight is best arranged so that the force The first example, which we are going to exerted by the weight on the piston at any consider more closely as a typical one, is the position of the latter equals the average following. A standing hollow cylinder, closed of the gas.) It is clear that in this at both ends, can be separated into two manner energy is constantly gained at the possibly unequal sections of V,and expense of heat, insofar as the biological V,respectively by inserting a partition from phenomena of the intervening nian are ig- the side at an arbitrarily fixed height. This nored in the calculation. partition forms a piston that can be moved In order to understand the essence of the up and down in the cylinder. An infinitely man’s effect on the system, one best imagines large heat reservoir of a given temperature T that the movement of the piston is performed insures that any gas present in the cylinder niechanically and that the man’s activity undergoes isothermal expansion as the consists only in determining the altitude of piston moves. This gas shall consist of a the molecule and in pushing a lever (which single molecule which, as long as the piston steers the piston) to the right or left, depend- is not inserted into the cylinder, tumbles ing on whether the molecule’s height requires about in the whole cylinder by virtue of its a down- or upward movement. This nieans thermal motion. that the intervention of the huniari being Imagine, specifically, a man who at a given consists only in the coupling of two position time inserts the piston into the cylinder and co-ordinates, namely a co-ordinate x, which somehow notes whether the molecule is determines the altitude of the molecule, with caught in the upper or lower part of the cyl- another co-ordinate y, which determines the inder, that is, in volume V1 or V,. If he should find that the former is the case, then position of the lever and therefore also he would move the piston slowly downward whether an upward or downward motion is until it reaches the bottom of the cylinder. imparted to the piston. It is best to imagine During this slow movement of the piston the the of the piston as large and its speed molecule stays, of course, above the piston. sufficiently great, so that the thermal agita- 304 LEO SZILARD

tion of the piston at the temperature in ques- mental amount, but not smaller. To put it tion can be neglected. precisely: we have to distinguish here be- In the typical example presented here, we tween two entropy values. One of them, &, wish to distinguish two periods, namely: is produced when during t,he measurement y 1. The period of measuTeinent when the assumes the value 1, and the other, 8, , when piston has just been inserted in the middle of y assunies the value - 1. We cannot expect the cylinder and the niolecule is trapped to get general information about S1 or Sz either in the 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, the 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 x < 0; must always hold good. 2. The period of utilization of the measure- ment, “the period of decrease of entropy,” during which the piston is moving up or down. During this period the x-co-ordinate One sees from this formula that one can of the molecule is certainly not restricted to make one of the values, for instance 81, as the original interval x > 0 or x < 0. Rather, small as one wishes, but then the other value if the molecule was in the upper half of the A!?, becomes correspondingly greater. Fur- cylinder during the period of measurement, thermore, one can notice that the magnitude i.e., when x > 0, the molecule must bounce of the interval under consideration is of no on the downward-moving piston in the lower consequence. One can also easily understand part of the cylinder, if it is to transmit that it cannot be otherwise. energy to the piston; that is, the co-ordinate Conversely, as long as the 8,and , produced by the measurements, satisfy R: has to enter the interval x < 0. The lever, on the contrary, retains during the whole pe- the inequality (l),we can be sure that the riod its position toward the right, corre- expected decrease of entropy caused by the sponding to downward motion. If the posi- later utilization of the measurement will be tion of the lever toward the right is desig- fully compensated. Before we proceed with the proof of in- nated by y = 1 (and correspondingly the equality (l), let us see in the light of the position toward the left by y = - 1) we see that during the period of measurement, the above mechanical example, how all this fits together. For the entropies & and & pro- position > 0 corresponds to y = 1; but x duced by the measurements, we make the afterwards y = 1 stays on, even though x passes into the other interval x < 0. We see following Ansatz: that in the utilization of the measurement 8, = 8, = k log 2 (2) the coupling of the two paraineters x and y disappears. This ansatz satisfies inequality (1) and We shall say, quite generally, that a pa- the mean value of the quantity of entropy rameter y “nieasures” a parameter x (which produced by a measurement is (of course in varies according to a probability law), if the this special case independent of the fre- value of y is directed by the value of param- quencies w1 , w2 of the two events) : eter x at a given moment. A measurement procedure underlies the entropy decrease 8 = k log 2 (3) effected by the intervention of intelligent In this example one achieves a decrease of beings. entropy by the isothermal expansion? One may reasonably assuiiie that a meas- urement procedure is fundamentally asso- - 31 = --klog ____v1 . ciated with a certain definite average entropy vl+ v2 ’ production, and that this restores concord- (4) ance with the Second Law. The amount of entropy generated by the measurenient may, of course, always be greater than this funda- The entropy generated is denoted by s,, ss. ON THE DECREASEOF ENTROPY 305 depending on whether the molecule was also shall look upon them as chemically dif- found in volume Vl or VZwhen the piston ferent, if they differ only in that the y co- was inserted. (The decrease of entropy ordinate is +1 for one and - 1 for the other. equals the ratio of the quantity of heat taken We should like to give the box in which the froin the heat reservoir during the isothermal “molecules” are stored the form of a hollow expansion, to the temperature of the heat cylinder containing four pistons. Pistons A reservoir in question). Since in the above and A’ are fixed while the other two are inov- case the frequencies w1 , WP are in the ratio of able, so that the distance BB’ always equals the volumes Vl , Vz, the mean value of the the distance AA’, as is indicated in Figure 1 entropy generated is (a negative number) : by the two brackets. A‘, the bottom, and B, the cover of the container, are impermeable 3 =w1.(+31) + WY(+ s,) = for all “molecules,” while A and B’ are semi-

“1 permeable; namely, A is permeable only for k log ___ those “molecules” for which the parameter vl+ VP Vl+VZv1 + (5) x is in the preassigned interval, i.e., (xl , x2), B’ V1 is only permeable for the rest. V2 k log ____ VI+ vz Vl+VZ As one can see, we have, indeed

v1 v1 V2 klog ~ + ___ T’, + v, v1+ v2 v1+ vz Tr (6) aklog- VP + klog2 2 0 v, + v2 and therefore : S++g-Q. (7) In the special case considered, we would A’ actually have a full compensation for the de- crease of entropy achieved by the utilization FIQ.1 of the measurement. We shall not examine more special cases, In the beginning the piston B is at A and but instead try to clarify the matter by a therefore B’ at A’, and all “molecules” are general argument, and to derive formula (1). in the space between. A certain fraction of We shall therefore imagine the whole sys- the molecules have their co-ordinate x in the tem-in which the co-ordinate x, exposed to preassigned interval. We shall designate by some kind of thermal fluctuations, can be w1 the probability that this is the case for a measured by the parameter y in the way just randomly selected molecule and by wP the explained-as a multitude of particles, all probability that x is outside the interval. enclosed in one box. Every one of these par- Then w1+ WP = 1. ticles can move freely, so that they may be Let the distribution of the parameter y be considered as the molecules of an , over the values + 1 and - 1 in any propor- which, because of thernml agitation, wander tion but in any event independent of the about in the common box independently of x-values. We imagine an intervention by an each other and exert a certain pressure on the intelligent being, who imparts to y the value walls of the box-the pressure being deter- 1 for all “molecules” whose x at that moment mined by the temperature. We shall now is in the selected interval. Otherwise the consider two of these molecules as chemi- value -1 is assigned. If then, because of cally different and, in principle, separable by thermal fluctuation, for any “molecule,” the semipermeable walls, if the co-ordinate x for parameter x should come out of the preas- one molecule is in a preassigned interval signed interval or, as we also may put it, if while the corresponding co-ordinate of the the “molecule” suffers a monomolecular other molecule falls outside that interval. We with regard to x (by which 306 LEO SZILARD it is transformed from a species that can pass long as we do not use the fact that the molecules the seniipernieable piston A into a species for in the container BB’, by virtue of their co- which the piston is imperiiieable), then the ordinate y, “remember” that the r-co-ordinate parameter y retains its value 1 for the time for the molecules of this container originally being, so that the “molecule,” because of the was in the preassigned inlerval, full conapensa- value of the parameter y, “remembers” tion existsfor the calculated decrease of entropy, during the whole following process that x by virtue of the fact that the partial pres- originally was in the preassigned interval. sures in the two containers are snialler than We shall see immediately what part this in the original mixture. memory may play. After the intervention But now we can use the fact that all naole- just discussed, we niove the piston, so that cules in the container BB‘ hatie the y-co-ordi- we separate the two kinds of inolecules with- nate I, and in the other accordingly -1, to out doing work. This results in two con- bring all molecules baclc again to the original tainers, of which the first contains only the volume. To accomplish this we only need to one modification and the second only the replace the seniipermeable wall -4 by a wall other. Each modification now occupies the A *, which is seniipermeable not with regard same volume as the mixture did previously. to x but with regard to y, naniely so that it is In one of these containers, if considered by permeable for the molecules with the y-co- itself, there is now no equilibrium with re- ordinate 1 and impermeable for the others. gard to the two “modifications in x.)’ Of Correspondingly we replace B’ by a piston course the ratio of the two modifications has B’*, which is impermeable for the molecules remained w1: wz . If we allow t,his equilibrium with y = -1 and permeable for the others. to be achieved in both containers independ- Then both containers can be put into each ently and at constant volume arid tempera- other again without expenditure of energy. ture, then the entropy of the systeni cer- The distribution of the y-co-ordinate with tainly has increased. For the total heat regard to 1 and -1 now has become sta- release is 0, since the ratio of the two “niodifi- tistically independent of the r-values and be- cations in x’)w1:w2 does not change. If we sides we are able to re-establish the original acconiplish the equilibrium distribution in distribution over 1 and - 1. Thus we would both containers in a reversible fashion then have gone through a coniplete cycle. The the entropy of the rest of the world will de- only change that we have to register is the crease by the same amount. Thcrefore the resulting decrease of entropy given by (9) : entropy increases by a negative value, and, the value of the entropy increase per mole- 3 = Ic(w1 log w1 + wt log wz). (10) cule is exactly: If we do not wish to admit that the Second Law has been violated, we must conclude s = Ic(w1 log w1 + w2 log wp). (9) that the intervention which establishes the (The entropy constants that we must as- coupling between y and x, the iiieasurenzent of sign to the two “modifications in 5’’ do not x by y, must be accompanied by a production occur here explicitly, as the process leaves the of entropy. If a definite way of achieving this total number of molecules belonging to the coupling is adopted and if the quantity of one or the other species unchanged.) entropy that is inevitably produced is desig- Yow of course we cannot bring the two nated by Sl and Sz , where S1 stands for the back to the original volume without mean increase in entropy that occurs when y expenditure of work by simply moving the acquires the value 1, and accordingly Sz for piston hack, as there are now in the con- the increase that occurs when y acquires the tainer- which is bounded by the pistons value - 1, we arrive at the equation: BB’-also niolecules whose x-co-ordinate lies outside of the preassigned interval and for WlSl i-w2s2 = s“ (11) which the piston A is not permeable any In order for the Second Law to remain in longer. Thus one can see that the calculated force, this quantity of entropy must be decrease of entropy (Equation 191) does not greater than the decrease of entropy s, which mean a contradiction of the Second Law. As according to (9) is produced by the utiliza- ON THE DECREASEOF ENTROPY 307 tioii of the measurement. Therefore the fol- not possible. But we can try to describe sini- lowing unequality must be valid : ple nonliving devices that effect such cou- pling, and see if indeed entropy is generated S+SlO and in what quantity. Having already recog- WlSl -I- WZS‘B (12) nized that the only important factor is a certain characteristic type of coupling, a + k(w, log w1 + w2 log WB) 2 0 “measurement,” we need not construct any This equation must be valid for any values complicated models which imitate the inter- of w1 and w2,3 and of course the constraint vention of living beings in detail. We can be w2 + wz= 1 cannot be violated. We ask, in satisfied with t,he construction of this par- particular, for which w1 and w2 and given ticular type of coupling which is accoin- 8-values the expression becomes a minimum. panied by memory. For the two minimizing values w1 and wz the In our next example, the position co-or- inequality (12) must still be valid. Under the dinate of an oscillating pointer is “measured” above constraint, the minimum occurs when by the energy content of a body K. The the following equation holds : pointer is supposed to connect, in a purely mechanical way, the body K-by whose energy content the position of the pointer is to be measured--by heat conduction with one of two intermediate pieces, A or B. The body is connected with A as long as the co- ordinate-which determines the position of the pointer-falls into a certain preassigned, This is easily seen if one introduces the no- but otherwise arbitrarily large or small inter- tation val a, and ot,herwise if the co-ordinate is in 81 SZ the interval b, with B. Up to a certain mo- - log w1 = - log ~2 = X; (15) k + k + ment, namely the moment of the “measure- ment,” both intermediate pieces will be then : thermally connected with a heat reservoir at A -Silk w1 = e .e ; w2 = eA.e--s2’k. (16) temperature To. At this moment the inser- tion A will be cooled reversibly to the tem- If one substitutes these values into the in- perature TA, e.g., by a periodically function- equality (12) one gets: ing mechanical device. That is, after e-Sl’k + e--szik ) 20. (17) successive contacts with heat reservoirs of intermediate , A will be brought Therefore the following also holds: into contact with a heat reservoir of the temperature Ta . At the same time the inser- X 2 0. (18) tion B will be heated in the same way to If one puts the values w1 and wz from (16) temperature Tg . Then the intermediate into the equation w1 + w2 = 1, one gets pieces will again be isolated from the corre- e-S1/k + e--Ss/k - -A sponding heat reservoirs. - e . (19) We assume that the position of the pointer And because X 2 0, the following holds: changes so slowly that all the operations that we have sketched take place while the e-Sl/k + e-‘Zik -.5 1 (20) position of the pointer remains unchanged. This equation must be universally valid, if If the position co-ordinate of the pointer fell thermodynamics is not to be violated. in the preassigned interval, then the body As long as we allow intelligent beings to was connected with the insertion A during perforni the intervention, a direct test is the above-mentioned operation, and conse- quently is now cooled to temperature Ta . 3 The increase in entropy can depend only on In the opposite case, the body is now the types of measurement and their results but not on how many systems of one or the other type heated to temperature T, . Its energy con- were present. tent becomes-according to the position of 308 LEO SZILARD the pointer at the time of “measurement”- However, following this, the entropy with- small at temperature Ta or great at tempera- drawn from the reservoir To by direct con- ture TB and will retain its value, even if the tact with it was pointer eventually leaves the preassigned $To) - G(TA) interval or enters into it. After some time, (22) 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 increased by gard to the 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, the entropy will increase 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 piece B at the ally functioning mechanical device should time of the “measurenient”: connect the thermally isolated insertions A and R with the heat reservoir To. This has the purpose of bringing the body K-which is now also connected with one of the two intermediate pieces-back into its original We shall now evaluate Ihese expressions state. The direct connection of the intermedi- for the very simple case, where the body ate pieces and hence of the body K-which which we use has only two energy states, a has been either cooled to TA or heated to lower and a higher state. If such a body is in TB-to the reservoir To consequently causes thermal contact with a heat reservoir at any an increase of entropy. This cannot possibly temperature T, the probability that it is in be avoided, because it would make no sense the lower or upper state is given by re- to heat the insertion A reversibly to the spectively: temperature To by successive contacts with the reservoirs of intermediate temperatures and to cool B in the same manner. After the measurement we do not know with which of the two insertions the body K is in contact at that moment; nor do we know whether it had been in connection with TA or TB in the Here u stands for the difference of energy end. Therefore neither do we know whether of the two states and g for the statistical we should use intermediate temperatures be- weight. We can set the energy of the tween TA and To or between To and Te. lower state equal to zero without loss of The mean value of the quantity of en- generality. Therefore :4 tropy S1 and Sz, per measurement, can be calculated, if the as a function of the temperature a(T) is known for the body I(, since the entropy can be calculated from the heat capacity. We have, of course, neglected the heat capacities of the inter- mediate pieces. If the position co-ordinate of the pointer was in the preassigned inter- val at the time of the “measurement,” and accordingly the body in connection with in- sertion A, then the entropy conveyed to the heat reservoirs during successive cooling was Here p and p are the functions of T given (21) See the Appendix. ON THE DECREASEOF ENTROPY 309

hy equation (25),which are here to be taken and the mean energy of the body is given by: for the argu1iient.s TO, TA, or TB. If (as is necessitated by the above concept of a “measurement”) we wish to draw a dependable conclusion from the energy con- tent of the body K as to the position co-or- the following identity is valid: dinate of the pointer, we have to see to it that the body surely gets into the lower energy state when it gets into contact with TB. In other words: Therefore we can also write the equat’ion:

a(TA) - “(To) I dii BA = +lT- - dT (33) TO TdT as This of course cannot be achieved, but may be arbitrarily approximated by allowing TA to approach and the statis- tical weight g to approach infinity. (In this limiting process, Tois also changed, in such a way that TO) and q(To) remain constant.) The equation (26) then becomes: and by substituting the limits we obtain:

If we write the latter equation according to and if we form the expression e--sAJk + (25) : e-SBlk , we find: 1 1 + ge-tdkT = __ e--S~/k + e--SB/k - (36) - 1. (29) PO‘) Our foregoing considerations have thus for TAand TO,then we obtain: just realized the smallest permissible limiting care. The use of semipermeable walls ae- cording to Figure 1 allows a complete utilization of the measurement: inequality and if we then write according to (31): (1) certainly cannot be sharpened. As we have seen in this example, a simple ~(TA)= udTA) (38) inanimate device can achieve the same we obtain: essential result as would be achieved by the intervention of intelligent beings. We have examined the “biological phenomena” of a nonliving device and have seen that it gen- If we finally write according to (25): erates exactly that quantity of entropy which is required by thermodynamics. (40) APPENDIX for TA and TO, then we obtain: In the case considered, when the frequency of the two states depends on the temperature ac- cording to the equations: 3 10 LEO SZILARD

We obtain the corresponding equation for SB , expand and collect terms, then we get

Formula (41) is identical with (26), given, for REFERENCES SA,in the text. We can bring the formula for SBinto a some- Smoluchowski, F. Vortraye UbeT die kinelkhe what different form, if we write: Theorie der Materie u. Elektrizitat. Leipzig: 1814. dTe) = 1 - PVd, (43) Szilard, L. Zeitschrift fur Physik, 1925, 32, 753.

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