Acta Biotheoretica 25, 2-3:103-110 (1976)

M A X W E L L ' S D E M O N IN B I O L O G I C A L S Y S T E M S I. W A L K E R Department of Zoology and Applied Entomology,Imperial College, London, SW7 2BB (Received 22-IX-1975)

SUMMARY Boltzmann's gas model representing the second law of thermodynamics is based on the improbability of certain molecular distributions in space. Maxwell argued that a hypothetical 'being' with the faculty of seeing individual molecules (Maxweli's Demon) could bring about such improbable distributions, thus violating the law of entropy. However, it appears that to render the molecules visible for any observer would increase the entropy more than the demon could decrease it, hence 'Maxweli's Demon cannot operate' (Brillouin, 1951). In the study presented here MaxweU'sDemon is interpreted in a general way as a biological observersystem within (possibly dosed) systems which can upset thermodynamicprobabilities provided that the relative magnitudes between observer system and observed system are appropriate.

Maxwell's Demon within Boltzmann's Gas Model thus appears only as a special case of inappropriate, relative magnitude between the two systems. 1.

T H E S E C O N D L A W OF T H E R M O D Y N A M I C S , M A X W E L L ' S DEMON AND BRILLOUIN'S ARGUMENT

Maxwell's D e m o n was born in 1871 in discussions on the second law of thermodynamics. This law is formulated as follows:

dS >_ aQ T

where dS is the change of entropy when heat Q is introduced into a system at temperature T. The equality d S = 8 Q / T is valid only for very slow processes in reversible, open systems. In closed systems the entropy cannot decrease. This is the formal statement of the law as we find it in all textbooks of Thermodynamics. In other words, dosed systems are irreversible. Boltzmann's statistical presentation explains entropy in terms of molecular movement and energies. As this whole study is based on Boltzmann's model, it will be explained in more detail. In this model, the entropy S =k.lnW

k is Boltzmann's constant = R / N ; R being the universal gas constant ( = 8.31 erg/°K . Mol) and N is Avogadro's number ( = 6 • 1023 molecules per Mol). For our purpose we consider a system of an ideal gas at a certain macroscopic state of temperature and pressure. W is the so called " t h e r m o d y n a m i c pro-

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bability'. It gives the number of complexions, Le. phase-space distributions of the molecules which the gas can assume whilst maintaining this given state. Each molecule has a certain probability to find itself in a certain space cell and to belong to a certain kinetic energy class (mvV2) at a particular instant. Within any given volume, these molecular states may be distributed more or less evenly. As molecules of the same energy class can be exchanged between their space cells without the macroscopic state of the system being affected, a very large number W of such equivalent complexions results in the same temperature and pressure pattern within the gas. The larger this number 14I, the greater is the probability that the gas will find itself in these conditions. W increases with more equitably distributed energy patterns. In this statistical model the second law of thermodynamics states that a closed system moves from the less probable state to the more probable one, i.e. that W can only increase and that the energy distribution within the given volume approaches a state of equilibrium or randomness. Traditionally the law of entropy is visualized by the behaviour of an ideal gas in a thermally isolated container which is sub-divided into two equal compartments A and B. In an initial State I all molecules are confined to compartment A. Removing the separating wall we now allow the gas to escape into B. According to the law of entropy and to experience the gas will eventually be distributed equally between the two compartments in an equilibrium State 2 of highest entropy; the probability that a large part or all the molecules return ever again into A is practically nil, the process is irreversible. While expanding from A into B the gas could deliver work, it could push a piston for example. However, in order to get the molecules back into compartment A, we would have to push the piston in the reverse direction by applying energy from the outside; if we open the system, we can reverse it to State 1. As Atkins (1968) points out, the probability to find even a modest number of molecules in one compartment only, after their previous distribution between both (A + B) is remote. Neglecting precise complexions and merely deciding whether or not a molecule occupies compartment A, the probability for each molecule is ½and for n molecules (½)n. It is in the context of Boltzmann's probability model that Maxwell (1871) begot his disturbing demon. He writes: 'One of the best established facts in thermodynamics is that it is impossible in a system enclosed in an envelope which permits neither change of volume nor passage of heat, and in which both the temperature and the pressure are everywhere the same, to produce any ,inequality of temperature or of pressure without the expenditure of work. This is the second law of thermodynamics, and it is undoubtedly true so long as we can deal with bodies only in mass and have no power of perceiving or handling the separate molecules of which they are made up. But if we con-

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ceive a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us. For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics." Jeans (1925), who quotes Maxwell, continues: 'Thus, MaxweU's sorting demon ~ could effect in a very short time what would probably take a very long time to come about if left to the play of chance.' Brillouin (1951) exorcises this demon in a paper with the title 'Maxwell's Demon cannot operate... '. He argues that the demon would have to operate under conditions of black-body radiation in his closed box and could therefore not see anything at all. In order to receive visible light reflected from specific molecules he would need 'a strong torch' to illuminate them. Brillouin shows that the minimum required energy dispersed by the torch in order to make a single observation is larger than the energy differential gained by manoeuvering one molecule into the specified compartment. Hence, the entropy still increases; Maxwell's Demon cannot operate.

2. THE SIGNIFICANCE OF MAXWELL'S DEMON At MaxweU's time statistical treatment of radiation, on which Brillouin (1951) bases his argument, was not yet in existence. As matters stand today, Maxwell would hardly quarrel with BriUouin. Still, he might conceivably retort that he had to invent the demon because in his period, too, it was physically impossible to achieve the trick to monitor individual molecules. By inventing the demon, Maxwell explicitly separated the concepts 'knowledge' and 'energy' and thus anticipated modern Information Theory. Brillouin's calculations which show that the demon cannot operate are based on this distinction between 'information' and energy. Information itself emerges as an energy-requiring process and Brillouin's equations relate the energy dissipated in making the observation to the free energy gained by applying this information to the model situation. Italics by present author.

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The invention of the demon has yet another, portentious significance which, however, is so far hardly acknowledged: Maxwell committed the heresy of placing the observer inside his own system and included this observer's subjective experience into his statements on the physical behaviour of the system. In view of the fact that demons of all sorts and description ran the show in modern science's ancestral disciplines of Magic and Alchemy, objectivation of the scientific method was a historical 'must' in the evolution of the modern natural sciences: the observer is placed outside the system and is not allowed to interfere with the observed phenomena. This is the corner-stone of scientific method and Maxwell violated this corner-stone. Or did he, perhaps, have the notion that this stone obstructs his vision? Would a 'demonological autopsy' of the second law of thermodynamics reveal new vistas? We will explore this question in more detail. In order to determine the thermodynamic probability W the 'Demon' (observer, experimenter) has to measure the macroscopic parameters of his system such as volume, weight, temperature, pressure, as well as shape and arrangement of his container. It is conveniently neglected that, in order to do so, the 'Demon' has to open his box to introduce the meters and to obtain their readings. But, as long as this interference is relatively small we allow the neglection. The observer then shrinks into the position of Maxwell's Demon where he measures and counts space cells, specifies their position, determines the energy classes of individual molecules and assigns to each molecule a possible position in this multi-dimensional phase-space. Having gone through all this trouble he emerges from his lilliput universe like the imp released from his bottle, reassumes size and objectivity (not to say respectability) and declares the individual molecular constellations (complexions) as irrelevant because he, in his macroscopic state, cannot perceive them! After the neat formulation S = k . In W, this dual spirit recedes into obscurity: Maxwell's Demon lurks in W whilst his human counterpart hides behind the shield of scientific objectivity. From this secluded position he watches in gloom how his gas is drifting towards the uneventful state of lukewarm chaos. Meanwhile his microscopic twin is having decidedly a more exciting time. He operates on the level of individual molecular states which, as we imagine, he can perceive and manipulate. He calculates and measures mass, velocities, positions and collisions and he explores his universe by placing himself at appropriate moments onto specific molecules. His universe is a perpetuum mobile, the energy locked up with him is there to stay. It is, in fact, merely to bedevil his human counterpart that Maxwell's Demon takes the trouble to operate the shutter and to sort the molecules. There is one important aspect in this sorting activity: he needs no energy to transport the molecules to their respective compartments, they move

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there by their own energy. Thanks to his observations he merely traps them once they are in position. This is MaxweU's point in a nutshell. The external observer, lacking detailed information on individual molecules, will have to transport them wholesale against their actual direction of movement, if he wants to assemble them all in one compartment. Thus, he will have to introduce energy from the outside. He can force them to and fro with a piston or he can heat and cool the two compartments alternately in order to accumulate energy repeatedly on one side only. It is usually not stated, that all the energy he can get hold of is unlikely to achieve this task if it is not applied in an informed manner. Without the informed action of the experimenter, the system is improbable to reverse whether it is open or closed. In fact he needs all the information he gathered previously for the formulation of the entropy equation in order to reverse his macroscopic, open system. Thus the observer, too, finds himself inside his, presumably open, macro-universe, scheming and manipulating. Therefore, to fully appreciate our model system, we must consider both, the internal (Maxwell's Demon) and the external observers in their respective micro- and macro-environments. Maxwell's scheme cannot work because the necessary observational energy is so powerful as to interfere with the mechanical task; the human experimenter is unsuccessful because his observation apparatus does not have the necessary power of resolution to obtain the information. The limitation as expressed in the second law of thermodynamics, namely that entropy will never decrease in a closed system, i.e. that the system will not reassume an initial improbable state, seems thus to be the result of improper relative magnitude between observer and observed system rather than the result of 'elosedness per se'. The law of entropy is intrinsically subjective. This does not mean to say that it is invalid, but it implies that we expect specifications. By placing his demon inside the system, Maxwell uncovered the necessity that, in order to learn the whole story, the observer has to include his own activity into his observations. In other words, the observer has to objectivate his own role within his observed system. Where observations refer to two structural, hierarchical levels (microscopic and macroscopic) and to their interrelations, the observer has to assume an explicit triple role in order to understand the two observed levels: observations on the two hierarchical levels and of his own experimental interference. Once the necessary, multiple roles of the observer are explicitly understood by himself, he then can decide which ones may be irrelevant for the laws and statements he wants to formulate. This is not philosophical finickery. How, if not by the dogmatic belief of his own exclusion from his experimental set-up, can it be explained that The Physicist is not puzzled at the mere fact that he repeats the most 'improbable' experiments over and over again?

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If, then, the incapability of Maxwell's Demon to operate is the result of the unfortunate relation between his own magnitude and that of his observed system; and we cannot fit a 'Demon' to this micro-universe, could we fit an experimental universe to a functional 'Demon'? Some conditions for successful function of an observer within his system can be qualified. Reception of information must leave the observed function largely intact and must not alter the mechanism of information processing within the observer. If the first condition is not met the observation destroys the event to be observed before information is extracted; in the second case the process of observation would annihilate the information received. (This refers to the immediate event to be observed not to indirect observation methods where events have first to be made observable by specific interactions, as for instance staining in Histology or tracks in bubble chambers after interaction between elementary particles.) It follows, firstly, that the observer system per se is always an open system (Walker, 1972); secondly, that the energies involved in information transfer and processing must be relatively small as compared to the energy bound up in the perceived event and in the perception mechanism; and thirdly that an observer is always of a composite, complex structure. This complexity is necessarily greater than the information within the observation (not necessarily greater than within the observed object !). If the observer anticipates to react physically to the observation or even to monitor the observed event, the velocities in the observation process must be much larger than the velocities in the observed event. Where series of identical events occur this difficulty can be overcome as information of earlier events may serve to monitor later ones. Conditions of time relations and energy interactions are mathematically treated in Cybernetics and Communication Engineering. Quite generally, it may be stated that interfering processes of the same order of magnitude cannot observe each other, and that there are limits of complexity below which observations cannot be made. Note that the discussion on Heisenberg's uncertainty principle in relation to observation methods as well as Brillouin's minimal observation energy k (for both problems see Brillouin 1951 and 1961) refer to indirect methods to render an event observable (see above), not to the actual process of observation by the observer. As in the case of Maxwell's Demon, the observation energies were too large (Brillouin, 1951) - and as similarly precise, mathematical treatments would probably also reveal phase limitations - we magnify the observed system to observable proportions. We expand the container with the two compartments and the shutter in the separating wall; the molecules will be simply shaped objects of the size and speed of footballs for instance. We now place the Human observer at the shutter with his strong torch. Provided that the density of the

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objects is properly adjusted to the time interval to pass the shutter, the observer can now separate the fast moving objects from the slow moving ones in the two compartments A and B, or he can manoeuver all objects to one side after an initial random distribution between both sides. The probability for this process occurring by chance would be (½)" for n objects as indicated earlier (p. 16) for n molecules. However, with the operator at the shutter this probability could approach 1, as is the rule for most experiments repeated according to design. The operator, by the way, could replace himself by an electronic device or simply by an asymmetrical trap door opening towards one compartment only. Thus, on the macroscopic scale, the statement on which the entropy law is based, namely that a closed system cannot move from the more probable to appreciably less probable states, cannot be upheld. What, in this process, happens to the thermodynamic entropy is not trivial to ascertain. In other words an explicit separation between the probability argument and molecular energy evaluations is essential in order to calculate the change of entropy. The complexions are no longer a function of the individual molecule's properties only, but also of the macroscopic arrangements monitored by the observer. In our experimental system the entropy presumably decreases. The 'demon', inside his closed environment, can interfere with thermodynamic probabilities; he plays havoc with thermodynamic probabilities in open systems. Ultimately, the law of entropy will always take over as more and more energy is dissipated into the microscopic substructures of the respective systems. But living systems and the physical empires monitored by them are a far cry from 'the ultimate' at both extremes, the macrocosmic and the microscopic. For the here and now of biological systems on this planet it is irrelevant whether the Universe is open or closed, and on the elementary level of matter they don't exist; thus they require specifications of physical laws which apply to the appropriate levels of complex structure and relative magnitudes. In an earlier study (Walker, 1972) concentrating largely on the symmetry conditions which allow for the repetition of improbable, complex states, Maxwelrs Demon was referred to as 'Biological Memory'. Maxwell's imaginary, experimental set-up shows the living, information-generating system interfering with thermodynamic probabilities. His demoniacal pair, the outside and inside observer, stands as a symbol for the most intriguing, complementary aspects of Man's physical environment. Over the last century, the outside observer has established, beyond a shadow of doubt, that live and lifeless matter are the same; thus, Biology became Physics and Chemistry. The inside observer, objectivated by his external twin, reveals that living systems are different from lifeless systems, that Physics and Chemistry as a result of biological scheming are Biology. The computer is a better argument for the view that physics is biology than for the far more common view that neural systems are 'nothing but' physics. The experimental method itself, relying on innumer-

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able repetitions of similar arrangements, with occasional replacement of earlier methods by later, more sophisticated ones, follows the pattern of reproduction, mutation and selection. And trapping molecules, indeed ! Is trapping not one of the oldest and at all times commonest methods of biological systems to accumulate desirable events without doing work? By recognizing the 'Demon"s rightful position inside systems it must be possible to formulate biophysical laws which are specifically relevant for complex systems. It is, after all, in the widely branched Trees ofLife andKnowledge that the Demons dwell, since time immemorial.

ACKNOWLEDGEMENT I wish to thank Professor T. W. B. Kibble for reading the manuscript and for helpful comments and discussion.

REFERENCES Adkins, C. J. (1968). Equilibrium dynamics. - London, McGraw Hill, 283 pp. Brillouin, L. (1931). Maxwdl's Demon cannot operate: information and entropy. - J . appl. Phys. 22, p. 334-337. Brillouin, L. (1961). Thermodynamics, statistics and information. - Amer. J. Phys. 29, p. 318-328. Jeans, J. (1925). The dynamical theory of gases. 4th ¢d. - London, Dover Publ. Inc., 444 pp. Walker, I. (1972). Biological Memory.-Acta biotheor., Leiden, 21, p. 203-235.

Maxwell's demon in biological systems.

Acta Biotheoretica 25, 2-3:103-110 (1976) M A X W E L L ' S D E M O N IN B I O L O G I C A L S Y S T E M S I. W A L K E R Department of Zoology and A...
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