Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 38
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He developed the Berginprocess to combine coal and hydrogen at high pressure and hightemperature. Huge hydrogenation plants were built in Germany to supplythe Wehrmacht, the German armed forces. Strangely enough the AlliedBomber Command overlooked the strategic importance of these vulnerableplants – 54 of them – until well into 1944. Then they were bombed anddestroyed in May 1944.42Fuel became very scarce indeed after that, and soon the vehicles of theGerman army were converted for the use of wood-gas, a comparativelylow-tech application of mass action: Wood was burned with an insufficient airsupply in a barrel-shaped furnace – that was loaded into the trunk –, andthe resulting carbon monoxide was fed into the motor. I remember from mychildhood that, half-way up even moderate hills, the drivers had to stop andstoke before they could proceed. Obviously this would not do for airplanes.Socio-thermodynamicsOn several occasions in previous chapters I have hinted at the usefulness ofthermodynamic concepts in remote areas, i.e.
fields that have little ornothing to do with thermodynamics at first sight. Those hints would bewanton remarks unless I corroborated them somehow, in order to acquaintthe reader with the spirit of extrapolation away from thermodynamicsproper. To be sure, most such subjects belong more to the future ofthermodynamics rather than to its history. They are struggling to be takenseriously, and to obtain admission into the field. But anyway, let usconsider the non-trivial proposition which has been called sociothermodynamics. It extends the concepts described above for theconstruction of phase diagrams in binary solutions to a mixed population ofhawks and doves with a choice of different contest strategies.We let ourselves be motivated by an often discussed model of gametheory43 for a mixed population of hawks and doves who compete for the414243Bergius shared the 1931 Nobel prize with Karl Bosch, Haber’s colleague and assistant inthe Haber-Bosch synthesis of ammonia.According to A.
Galland: “Die Ersten und die Letzten, die Jagdflieger im ZweitenWeltkrieg.” [The first and the last, fighter pilots in World War II] Verlag Schneekluth,Augsburg (1953).Adolf Galland was himself a highly decorated fighter pilot before he was given an officejob; he became the last inspector of the Luftwaffe in the war and then the first inspector ofthe after-war Luftwaffe in 1956.J. Maynard-Smith, G.R. Price: “The logic of animal conflict.” Nature 246 (1973).P.D. Straffin: “Game Theory and Strategy.” New Mathematical Library. TheMathematical Association of America 36 (1993).1605 Chemical Potentialssame resource, whose value, or price, is denoted by IJ.
Prices are out ofcontrol for the birds, but they must be taken into account by them. Indeed,in their competition the birds may assume different strategies A or B whichwe define as follows.Strategy AIf two hawks meet over the resource, they fight until one is injured. The winnergains the value IJ, while the loser, being injured, needs time for healing his wounds.Let that time be such that the hawk must buy 2 resources, worth 2IJ to feed himselfduring convalescence. Two doves do not fight. They merely engage in a symbolicconflict, posturing and threatening, but not actually fighting. One of them willeventually win the resource – always with the value IJ – but on average both losetime such that after every dove-dove encounter they need to catch up by buyingpart of a resource, worth 0.2IJ.
If a hawk meets a dove, the dove walks away, whilethe hawk wins the resource; there is no injury, nor is any time lost.Assuming that winning and losing the fights or the posturing game isequally probable, we conclude that the elementary expectation values forthe gain per encounter are given by the arithmetic mean values of the gainsin winning and losing, i.e.eAHH = 0.5 (IJ – 2 IJ) = - 0.5 IJeAHD = IJeADH = 0eADD = 0.5 IJ – 0.2 IJ = 0.3 IJfor the four possible encounters HH, HD, DH, and DD.Note that both, the fighting of the hawks and the posturing of the doves,are irrational acts, or luxuries.
Indeed both species would do better, if theycut down in these activities, or abandoned them altogether. Also themeekness of the doves confronted with a hawk may be regarded asovercautious. Such observations have let to the formulation of strategy B.Strategy BThe hawks adjust the severity of the fighting – and thus the gravity of the injury –to the prevailing price IJ. If the price of the resource is higher than 1, they fight less,so that the time of convalescence in case of a defeat is shorter and the value to bebought during convalescence is reduced from 2IJ to 2IJ(1-0.2(IJ – 1)).
Likewise theThe issue in these presentations is the proof that a mixed population of two species may beevolutionarily stable, if the species follow the proper contest strategy. In the presentaccount of socio-thermodynamics the objective is different: No evolution is allowed buttwo different strategies may be chosen which both depend on the price of the contestedresource.Socio-thermodynamics161doves adjust the duration of the posturing, so that the payment for lost time isreduced from 0.2IJ to 0.2IJ (1 – 0.3(IJ – 1)).
But that is not all: To be sure, in strategyB the doves will still not fight when they find themselves competing with a hawk,but they will try to grab the resource and run. Let them be successful 4 out of 10times. However, if unsuccessful, they risk injury from the enraged hawk and mayneed a period of convalescence at the cost 2IJ (1 + 0.5(IJ – 1)).Thus the elementary expectation values for gains under strategy B maybe written aseBHH = 0.5(IJ – 2 IJ(1 – 0.2(IJ – 1)))= (0.2 IJ – 0.7) IJeBHD = 0.6 IJeBDH = 0.4 IJ – 0.6·2 IJ (1 + 0.5(IJ – 1)) = –(0.6 IJ + 0.2) IJeBDD = 0.5 IJ – 0.2 IJ (1 – 0.3(IJ – 1)) = (0.06 IJ + 0.24) IJ .The assignment of numbers is always a problem in game theory. Here thenumbers have been chosen so as to fit a conceivable idea of the behaviourof the species. Let us consider this:The grab-and-run policy is clearly not a wise one for the doves, because they getpunished for it.
So, why do they adopt that policy? We may explain that byassuming, that doves are no wiser than people, who start a war with the expectationof a quick gain and then meet disaster. This has happened often enough in history.Note that for IJ >1 the intra-species penalties for either fighting or posturing becomesmaller, because we have assumed that these activities are reduced when theirexecution becomes more expensive. However, the interspecies penalty – the injuryof the doves – increases, because the hawks will exert more violence against theimpertinent doves when the stolen resource is more valuable.IJ = 1 is a reference price in which both strategies coincide, except for the grab-andrun feature of strategy B.
Penalties for either fighting or posturing should never turninto rewards for whatever permissible value of IJ. This condition imposes aconstraint on the permissible values of IJ: 0< IJ<4.33.44Now, let zH and zD = 1 – zH be the fractions of hawks and doves, and let allhawks and doves either employ strategy A or B. Therefore the gainexpectations eiH and eiD (i = A,B) of a hawk and a dove per encounter withanother bird may be written aseiH = zH eiHH + (1 – zH) eiHDandeiD = zH eiDH + (1 – zH) eiDDin terms of the elementary expectation values.
And the gain expectations eifor strategy i per bird and per encounter reads44Such a constraint could be avoided, if we allowed non-linear penalty reductions which, forsimplicity, we do not.1625 Chemical Potentialsei = zH eiH + (1 – zH) eiD or explicitlyei = zH2 (eiHH + eiDD – eiHD – eiDH) + zH (eiHD + eiDH – 2 eiDD) + eiDD.Specifically we haveeA= –1,2 IJ zH2 + 0.4 IJ zH + 0.3eB = 0.86 IJ (IJ – 1) zH2 – (0.72IJ + 0.08)IJ zH + (0.06 IJ + 0.24)IJ.The graphs of these functions are parabolae which – for some values ofIJ – are plotted in Fig. 5.9.a–e.Fig.
5.9. Expectation values as functions of zH for some values of the price IJ.Concavification. Strategy diagramThe interpretation of those graphs is contingent on the reasonableassumption that the population chooses the strategy that provides themaximal gain expectation. Obviously for IJ = 0.6 and IJ = 1 that strategy isstrategy A.
At that price level the hawks and doves will therefore all choosestrategy A irrespective of the hawk fraction zH in the population.For higher price levels the situation is more subtle, because the graphmax[eA, eB] is not concave. This provides the possibility of concavification,cf. Fig. 5.9.c–e. There are intervals of zH where the concave envelope ofmax[eA, eB] lies higher than that graph itself. The population then has thepossibility to increase the expected gain by un-mixing; it segregates intoSocio-thermodynamics163homogeneous colonies with hawk fractions corresponding to the end-pointsof the concavifying straight lines, which are dashed in the figures. InFigs.
5.9c,d the adopted strategies are A and B and the species are mixed inthe colony with strategy A, whereas the colony with strategy B is pure-doveor pure-hawk, depending on whether the extant overall hawk fraction liesbelow the left, or right tangent respectively. For IJ > 3.505 the concaveenvelope connects the end-points of the parabolae eB so that hawks anddoves are fully segregated in two colonies, both employing strategy B.Mutatis mutandis all this is strongly reminiscent of the considerations ofphase diagrams of solutions or alloys with a miscibility gap, see above atFig. 5.6. To be sure, there we minimized Gibbs free energies whereas herewe maximize gain. Accordingly in solutions we convexify the graphmax[Gƍ,GƎ] whereas her we concavify the graph max[eA, eB], but those aresuperficial differences.
And just as we constructed phase diagrams before,we may now construct a strategy diagram by projecting the concavifyinglines unto the appropriate horizontal line in a (price, hawk fraction)-diagram, cf. Fig. 5.9f. We recognize four regions in that diagram.x I: Full integration of species employing strategy A.x II: Colony of pure doves with strategy B and integrated colony ofhawks and doves with strategy A. Partial segregation.x III: Colony of pure hawks with strategy B and integrated colony withstrategy A.
Partial segregation.x IV: Colonies of pure doves and pure hawks. Full segregation.The curves separating the regions II and III from region I can easily becalculated:and IJ = 6 zH2 – 12 zH + 7IJ = 20 zH2 + 1respectively. Those two curves intersect in the eutectic point E, so called inanalogy to thermodynamics.Although the analogy between our sociological model and thermodynamics of solutions is fairly striking, there are differences.
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