# Models and the dynamics of theory-building in physics. Part II—Case studies

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Studies in History and Philosophy ofModern Physics 38 (2007) 6837231. IntroductionDenition 1.1.1. H- modeling aims to explore the connections to be established between thecoreespecially the physical postulatesand its physical semantic relevance: in particularARTICLE IN PRESSwww.elsevier.com/locate/shpsb1355-2198/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.shpsb.2006.10.005E-mail address: gge@math.u.edu.1Permanent address: Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA.1.1. Summary of Part IConsidering the view that scientic theories are developing as the nodes of Quines webof knowledge, I proposed that these nodes have a structure which I compared to that of acell, composed as they are of a core surrounded by experiments and models. I focused onthe role of modeling strategies as agents of change. I claimed they provide much of thedynamics governing the cells evolution, especially before it becomes rmly embedded inQuines web as a closed theory. For this process to happen, I specied two mainmodeling strategies from which the models proceed. I advanced the following denitions.Models and the dynamics of theory-building inphysics. Part IICase studiesGerard G. Emch1All Souls College, University of Oxford OX1 4AL, UKReceived 4 April 2005; received in revised form 5 October 2006; accepted 10 October 2006AbstractIn Part I, it was argued that models are best explained by considering the strategies from which theyissue. A distinction was proposed between two classes of modeling that contribute to theory-building: H-modeling and L-modeling. Case studies are presented in this Part II to illustrate the characteristic featuresof these modeling strategies; examples are drawn from classical statistical mechanics and quantum physics.r 2007 Elsevier Ltd. All rights reserved.Keywords: Epistemology; Models; Theory constructionthe observability of its concepts and their adequacy to describe the world as apprehendedby laboratory experiments.Denition 1.1.2. L- modeling is designed to test the correctness and economy of the syntax,the logical consistency and independence of its axioms, the formal value of its assertionsARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723684and theorems.Denition 1.1.3. The products of these two modeling activities are called respectivelyH-models and L-models.Denition 1.1.4. By default, a mere model designates an H-model or an L-model separatedpurposefully or notfrom the modeling strategy that was followed to construct it.1.2. Methodology for Part IIIn this second part of the paper, I present case studies that show how the different typesof modeling strategies effectively contribute to the evolution of a cell. In choosing whichmodels to include, I kept in mind three constraints; namely, the illustrations ought to be:(a) familiar and accessible; (b) detailed enough for their roles in H- and L-modelings to beconvincing; (c) varied enough to indicate how extended my project is intended to be.I collect my illustrations in four sequences belonging each to a different eld of physics.More models will be briey listed in the Appendix according to the strategy from whichthey proceed.2. From equilibrium thermodynamics to statistical mechanicsThis section might also be entitled: roles that modeling played in the beginnings of thereduction of thermodynamics to statistical mechanics.2.1. Empirical presentation of the ideal gasI designate by ideal gas what is also called the perfect gas, to emphasize that it is anidealized description of a gas in thermodynamical equilibrium, namelypV NmoleRT , (2.1.1)where p; V ; T label pressure, volume and temperature; Nmole measures the quantity of thegas,2 and R is an universal constant.3 In particular, note the proportionality of incrementsa along isotherms i:e: at constant temperature : Dp aDr with r 1Vb along isochores i:e: at constant volume : Dp aDT9=;.(2.1.2)2The modern denition is: a mole is the amount of a substance that contains as many elementary particles(atoms, molecules, ions, etc.) as the number of carbon atoms in 12 g of C12 (carbon-12). The number of carbonatoms contained in 12 g of C12 is approximately 6:02 1023; this is called Avogadros number. A mole, then, is anamount of any substance that weighs, in grams, as much as the numerically equivalent atomic weight of thatsubstance.3The modern value of R is 8:3143 107 erg deg1 mol1.Empirical background: Boyle (1660, 1662) and Mariotte (1679) performed crucialexperiments on air, the results of which they described by writing that underotherwise similar circumstances the pressure and volume of a gas vary only in such awh2.2The rst facet is its purpose as an explanatory activity: explaining the observable inARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 685terms of the unobservable, the observable phenomena being described by relations (2.1.1)or (2.1.2), while the unobservable is the putative existence of the particles constitutingthe gas. The other facet is the exploration of the unobserved itself, through itsconsequences on the observed. As these specic models unfold, I will point outthe contributions each makes to the development of a core, the microscopic viewof matter.4This was valid until signicant deviations were systematically observed by Andrews (1869); see Section 2.5.5This arbitrariness later found a justication in the extensive observations of Gay-Lussac (1802, 1807) and thetheoretical considerations of Fourier (1822) and Thomson (1848). Mentioning these here would mask thegashisis idealization.. Microscopic modelings for the ideal gasThe following three modelings explore possible microscopic structures of the idealin terms of the mechanics of material points. They exhibit two facets of H-modeling.prothontons thermometer is linear in T . This completes the description of the relation 2.1.1ich gives the empirical, macroscopic description of the ideal gas.The purposes of this section are: (i) to present the microscopic modeling strategiesmpted by this empirical description; and (ii) to indicate how real gases may depart fromconAmway that the product pV remains constant.4 To specify the operational meaning of whatBoyle and Mariotte perceived as similar circumstances requires one to postulate thatthese could be characterized by a parameter having to do with what one calls todaytemperature. The delineation of this notion of temperature involves several issues in thehistory and philosophy of science (e.g. Chang, 2004). Here, I focus on the followingdevelopments.1. The construction of thermometers, measuring the expansion of diverse substances,gaseous air with Amontons (1702) and liquid mercury with Fahrenheit (1724); and thecalibration of these thermometers, resulting from Fahrenheits remark that Amontonshad discovered that water boils at a xed degree of heat (Magie, 1969, p. 131) andthat the same holds for other liquids as well, each with its characteristic boilingtemperature.2. The discoveryor was it an invention?of Newtons (1701) cooling equation_T / T Tu, the universality of which is predicated on the assumption that thetemperature is dened up to a change of scale T ! aT Tob where a; b and To areuniversal constants.3. The empirical fact that Nmole can be accounted for in units adapted to the gasconsidered in such a way that R becomes a universal constant.Once one accepts these observations/idealizations, one can arbitrarily choose5 thevenience of a temperature scale in which a 1; and To 0 with b 1, so that thetorical context of discovery for the modeling strategies to be described in the next subsection.2.2.1.ThPrincdoesARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723686that the density of air is either exactly or at least very nearly as the compressing force. (p.696) This is indeed (2.1.2(a)). As a microscopic model for this empirical relation, Newtonoffers an assembly of particles that are repelled from one another by forces that areinversely proportional to the distances between their center. (p. 697) He then purports toshow that these particles constitute a uid whose density is proportional to thecompression. (p. 697)Remarks 2.2.1. (1) As detailed in Guicciardini (2003), Newton argues in a heuristicmanner, characteristic of Euclid: Newton states rst what he wants to prove; then hemakes the assumptions that would allow him to prove it; and nally he derives what hesaid he would prove.(2) Note the semantic distinctionquite remarkable for his timethat Newton makesbetween the proposition he has just demonstrated and the world out there: he commentsthat he is satised with his mathematical argument, and that . . . whether elastic uidsconsist of particles that repel one another is, however, a question for physics. (Newton,1726, p. 699) Hence, Newton knows when he does what I call H-modeling, and he warns usof the limits such an exploration necessarily involves.(3) This model marks a rst systematic step in what became later the core of aprogramme: reducing thermodynamics to some form of mechanics.(4) Yet, Newton models a gas constituted of particles at rest, i.e. in a state of staticequilibrium. This strategic option did not survive subsequent modelings of the phenomena.(5) Moreover, Newtons hypothesis on the space-dependence of the interparticle forcesdid not survive either. Sometimes a models value lies less in its immediate explanatorypower than in its suggestive call for better constructs. Although Newton does recover themacroscopic Boyle formula for the ideal gas, Newtons microscopic model is not that of afree gas; compare with the Bernoulli and Maxwell modelings of the ideal gas discussedbelow; for real gases, see Section 2.5.(6) Furthermore, the modern reader will notice that Newton assumes throughout a xedtemperature, although he does not make explicitly this hypothesis. He does not evenmention temperature here, although he was aware of this notion, as witnessed by his owncooling equation (see (2) in Section 2.1).2.2.2. Bernoullis modeling of the ideal gasIn contrast to Newtons static model (see above), Bernoulli (1738) in the 10th chapter ofhis Hydrodynamica offers an attempt towards a kinetic theory of gases. Indeed he exploresthere the hypotheses on the motion of particles that would provide the underlyingatomistic structure of the ideal gas. He proceeds then to establish:. . . a theorem . . . in which it is shown that in air of any density but at xedtemperature, the elasticities[6] are proportional to the densities, and further that theincrements of elasticity which are produced by equal changes of temperature are6AsNewtons modeling of the ideal gase modeling is proposed in Proposition 23, Book II, Section V of Newtons (1726)ipia. Newton is not forthcoming about the empirical sources of his motivation as henot refer explicitly to Boyle, only mentioning that . . . it is established by experimentsis still the habit in the xviiith century, the elasticity of the gas means its pressure.proportional to the density, this theorem, I say, Dr. Amontons discovered bymercuagreeARTICLE IN PRESS7Th(1702)G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 687ry, the other with alcohol, could not be calibrated in such a manner that they wouldwith one another over their entire range. This suggestedand was later found to bee translation is from (Magie, 1969, pp. 230231). As for the reference to the experiment, see Amontonsexperiment and presented it in the Memoirs of the Royal Academy of Sciences ofParis in 1702.7Compare with the empirical formulas (2.1.1) and (2.1.2).Instead of Bernoullis somewhat contrived description of his model [cf. e.g. (Magie,1969, pp. 247251)] where the gas is contained in a vertical cylindrical vessel closed by amovable weighted piston, and is constituted by a very large number of minute sphericalparticles, I present below a somewhat simplied version, the elegance of which I learned inHyllerhaas (1970, vol. 1, p. 398).The gas is construed as an assembly of N identical, independent, point-like particles ofmass m, enclosed in a volume V . One makes the following simplifying assumptions oridealizations: (i) the container is a cubic box of side L; (ii) the particles are uniformlydistributed in space among six beams, parallel to the edges of the box; (iii) the particlesmove with constant speed v kvk ; and (iv) the walls are perfectly elastic so that, uponeach collision of a particle with a wall, its vis viva is conserved, which implies that itsmomentum changes by 2mv.Hence the pressure p force=surface exercised by the gas on each wall is given byp 1t 2mv n 1L2with n 16NL2 vtV, (2.2.1)where n is the average number of particles that hit a wall during a small time t; vt is themaximal distance within which these particles are from the wall, and thus L2 vt is thevolume from which they come in the time t that precedes their collision with the wall.Hence, (2.2.1) gives immediatelypV N 2312mv2 . (2.2.2)The immediate interpretation of this formula is that the model explains the pressure of thegas in terms of the motion of the particles which are assumed to constitute the gas.Moreover, a comparison of the consequence (2.2.2) of this microsopic model and themacroscopic empirical observation (2.1.1) gives32NmoleNR T 12mv2. (2.2.3)Thus, the model suggests that temperature be interpreted in terms of the kinetic energy ofthe individual particles of the gas.Remarks 2.2.2. (1) The reference Bernoulli (1738) makes to the Amontons (1702)experimentssee the quote at the beginning of this subsectionreminds us that Amontonshad seen in the gas thermometer an empirical determination of an absolute thermometricscale. However, even by the time Bernoulli proposed his model, Boerhaave (1732, p. 87) hadalready noticed that two thermometers, built for him by no less than Fahrenheit, one with.empirically correctthat all substances (and even gases) may not be equivalent for theARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723688purpose of thermometry; the question was then raised as to whether there exists an ideal gasthat would serve this purpose better than other gases by satisfying the BoyleMariotte formula2.1.1. Thus, the model, and in particular the identication (2.2.3), may be viewed as a proposalfor a microscopic interpretation of such an empirically dened ideal gas, namely that it consistsof particles that do not interact among themselves and which, for this purpose, can beconsidered as point-like objects.(2) Incidentally, Bernoulli already proposes a preliminary discussion of the correctionsto be brought to (2.2.2) to account for the nite size d of the particles as compared to theinterparticle distance D. He nevertheless recognizes that testing these corrections is toodelicate for the current state of the measuring techniques, and thus, he considershenceforth only the limit where d=D ! 0.(3) As for the ambient web: a sensitive empirical determination of how close a real gas isfrom the ideal gas obtains when the concept of specic heat is introduced and isquantitatively mastered. This concept is usually attributed to Black (1803). Yet, Black(17281799) mentions that after starting his own thinking about the subject in 1760, herealized others had trodden along the same road, prompted by Boerhaaves observation(1732) reported in Remark 1. In particular, Black describes the experimental works ofFahrenheit and of a certain Dr. Martin.8 While Black uses a language predicated on thecaloric theory of heat, he argues cogently from these experiments that different substanceshave different capacities for heat, i.e. are heated or cooled at different rates. Blackproposes a denition, the modern version of which reads: the specic heat CV of a gas atconstant volume is the amount of heating necessary to elevate one unit of mass of the gasby one degree of temperature. While Bernoullis Hydrodynamica (1738) is far anterior tothis modern denition, and even to Blacks penetrating conclusions (published in 1803),Hydrodynamica is essentially contemporaneous to the motivating empirical evidencesadvanced by Boerhaave y1738, by Fahrenheit y1736 and, especially, by Martiney1741.(4) Consider nowa posteriorithe relevance of the concept of specic heat toascertain the scope of the Bernoulli model. In this model there are, by denition, nointeractions between the constituent particles of an ideal gas. Thus, the sum of the kineticenergies of the particles gives the total internal energy of the ideal gasand thus according to 2:2:3 :UV ; T N12mv2UV ; T 32NmoleRT). (2.2.4)Hence the specic heat per mole, at constant volume isCV 1NmoleqT UV ; T 32R and thus in particular qT CV 0. (2.2.5)(5) For the syntactic role of this property, cf. e.g. the constitutive equation (2.3.3(ii))below.(6) Upon returning to the empirical adequacy of this H-model, note in relation with itsconsequence qT CV 0 in (2.2.5), that whether the specic heat CV of a real gas is constantor not can be tested in the laboratory. Yet, the methods of calorimetry had to be8Most likely, Martine (17021741). Indeed, Black gives for his source a paper entitled Essay on the heating andcooling of bodies; a paper with that title is reproduced in the posthumous volume (Martine, 1780, pp. 5390).iclesitionARTICLE IN PRESSR3dv1 dv2 dv3fv N=V i:e: A N=V . Finally, Maxwell adjusts the constant a bycomputing the pressure resulting from the collisions of the particles with the walls, verymuch as Bernoulli did, except that Maxwell now takes the averages with respect to thisdistribution, namely: p 1=V Rv140dv1RRR2dv2 dv3fv2mv1 v1. One obtains thena2 kT=m, with k Nmole=NR. Hence the result now known as the Maxwelldistributionfv NVm2pkT 3=2exp m2kTkvk2n o. (2.2.6)The consequences (2.2.4) and (2.2.5) of the Bernoulli model for the ideal gas carry over tothe Maxwell model, provided that Bernoullis v is reinterpreted as an average with respectto the distribution (2.2.6). In particular, 12kT is now to be identied with the mean kineticenergy per degree of freedom. Again, the specic heat CV is constant; and its value32Rremains the same; compare to (2.2.5).Remarks 2.2.3. (1) To place Maxwells modeling strategy in the wider context of nascentstatistical thinking note that, while still a student, Maxwell read John Herschels 1850review of Quetelets advocacy of the use of statistics in social matters. In a sweeping case oftheory transfer, Maxwell suggested that the true logic for this world is the Calculus ofProbabilities (Letter to Campbell, June 1850; Campbell & Garnett, 1882, p. 143). And,MaconRpaxwell then recognizes that for f to model the velocity distribution in a gas of N parttained in a box of volume V the constant A must satisfy the condThen fv A2pa23=2 expf12a2kvk2g where A and a are two free constants.considerably rened until Regnault (1853) was able to carry out the experiments withsufcient reliability to evaluate by how much CV fails to be constant in actual gases.(7) These measurements in turn proved to play an important role in the dismissal of thecaloric theory of heat (e.g. Emch & Liu, 2002, p. 53). Nevertheless, the concept of specicheatalbeit originally predicated on the caloric theorysurvives the dismissal of thistheory since it ultimately involves only the differential form of the theory of heat; cf. e.g.Section 2.3 below.2.2.3. Maxwells modeling of the ideal gasLike Bernoulli (see above), Maxwell (1860) models an ideal gas as a spatially uniformassembly of particles enclosed in a box with rigid, perfectly elastic walls; the shape of thebox, however, is not prescribed. The main difference from Bernoullis modeling is that thethree components of the velocity v of the particles are now assumed to bein the languageof modern probability theoryindependent, identically distributed, Gaussian randomvariables.Scholium 2.2.1. If f : v 2 R3 7!fv 2 R is a smooth function satisfying the following threeproperties:1. f is normalizable, i.e.RR3dv1 dv2 dv3fvo1;2. the three components of v are mutually independent, i.e. fv f1v1f2v2f3v3;3. f is isotropic, i.e. fv Fkvk2.G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 689raphrasing his result (here Scholium 2.2.1), Maxwell remarked in his 1860 paper:ARTICLE IN PRESSthe velocities are distributed among the particles, according to the same law as the errorsare distributed among the observations in the theory of the method of least squares.(Garber, Brush, & Everitt, 1986, p. 291) This exact same sentence can be found in a letterto Stokes, dated May 30, 1859 (Larmor, 1907, vol. 2, p. 10). Maxwell is right: hishypotheses and conclusion are the same as those Gauss (1809) gave in his pioneering workon the distribution of errors arising in repeated observations.(2) Maxwells a priori approachnamely Scholium 2.2.1, together with the naturaladjustment of constantsto the distribution (2.2.6) is corroborated by the role thisdistribution plays in Boltzmanns kinetic theory of gases; see e.g. Emch and Liu (2002,Section 3.3). Yet, Boltzmanns kinetic theory of gases notwithstanding, the phrasestatistical mechanics appears only later, namely in the treatise by Gibbs (1902).(3) In the 1950s, the statistical distribution (2.2.6) itself, not just its thermodynamicalconsequences, was tested directly in measurements of the Doppler shift in the radiationemitted by a hot plasma (e.g. Emch & Liu, 2002, p. 91).2.2.4. Consolidations emerging from the three modelsAll three modelsNewtons, Bernoullis, and Maxwellswere primarily conceived asexploratory: they are H-models. Newtons was certainly more tentative than the other two.Bernoullis brought in a working hypothesis on the collisions of individual particles withthe walls; however, his model involves too drastic an idealization on the velocitydistribution of the particles. Rening Bernoullis, Maxwells model was sufciently realisticfor its microscopic tenets to be tested empirically. Still, both Bernoullis and Maxwellsmodels neglect possible interactions between the particles themselves; compare withSection 2.5.While Maxwells distribution models an equilibrium situation, later on Maxwellsuggested and Boltzmann purported to prove that the approach to this equilibrium is dueto an ingredient not introduced in the above account, namely random collisions betweenextended particles. Clausius (1858) estimated that in a gas in equilibrium the mean-freepath between consecutive collisions is given by l rd21 where d is the diameter of theparticles, and r N=V with N the number of particles in the volume V . The mean-freepath l enters in the computation of directly observable macroscopic quantities, such asviscosity. Loschmidt (1865) pursued this to compute numerical values for d and N, andhence for the Avogadro Number NAv N=Nmoles, the number of molecules in a mole.Thus, the macroscopic consequences of this chain of H-modeling turned out to be anessential contribution to the core, namely the postulate that gases (later fluids, and ultimatelyall matter), have a statistical, atomistic description (e.g. Emch & Liu, 2002; or Truesdell,1980). Yet, this raised an ontological debate that lingered until Einsteins treatment ofBrownian motion (Einstein, 1905b) when quantitative, instantaneous and not onlystatistical, properties were assigned to individual particles; cf. Remark 3.1.1(4).2.3. Clausius axiomatization of thermodynamicsI argued in Part I that the appraisal of a model depends on its intended vocation, i.e.what it is intended to do: exploring the real world or establishing the internal consistency ofa system of axioms/postulates. The present subsection shows how the ideal gasG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723690independently of any of the microscopic interpretations associated to the H-modelsARTICLE IN PRESSdiscussed in the previous subsectionappears as an L-model for the Clausius axiomsregarding the thermodynamics of homogeneous media, gases in particular.2.3.1. Clausius axioms (Clausius, 1879)I use below the notation of modern real analysis, thus assuming that the rules ofdifferential calculus belong to the core of thermodynamics. Clausius actually knew thedifference between exact differentials and those that call for an integrating factor; cf.Remark 2.3.1(5).Axioms 2.3.1. Let Z and t be two differential forms defined on D fV ; T 2 R Rg byZ LV dV CV dT and t pdV , (2.3.1)where LV ; CV and p are smooth real-valued functions of V ; T; and require that Z and tsatisfy, along all smooth simple closed contours G D, the two conditionsiZGZ t 0 and iiZG1TZ 0. (2.3.2)This grammar allows one to build the theory without references to any interpretation ofits terms, for instance to establish immediately the following general consequences of theaxioms:Theorem 2.3.1. (1) There exist smooth functions U and S from D to R such that Z T dSand t dU Z;(2) qT LV p qV CV , LV TqTLV qV CV , LV TqT p.2.3.2. The ideal gas as an L-model for Clausius axiomsTo prove the internal consistency of the Axioms 2.3.1, an L-model is constructed: theideal gas (here with Nmole 1). It is dened by two constitutive relations, involving astrictly positive constant Ri pV ; T R TVand ii qT CV 0. (2.3.3)Scholium 2.3.1. The first of these two constitutive relations already entailsLV p; qV U 0 and qV CV 0. (2.3.4)And the two constitutive relations together entailthe function CV is constant, 2:3:5UT CV T To UTo, 2:3:6SV ; T CV ln TTo R ln VVo SV o; To. 2:3:7Remarks 2.3.1. (1) While the rst assertion in Theorem 2.3.1 only claims existence, nowthe two differentiable functions U and S have been explicitly constructed. From theirdifferentials dS and dU one constructs Z T dS and t dU Z and one veries thatthese differentials satisfy the conditions (2.3.1) and (2.3.2). Hence an L-model for AxiomsG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 6912.3.1 has been obtained.(2) Theorem 2.3.1 is a consequence of the syntax alone. Its derivation proceeds according onlysystformHilbmatthel(3the(4nottheof v(a)ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723692semantics, and it is only a special case of what is known in thermodynamics as aLegendre transformation to adapt the description to the purposes being pursued;compare this to the relation between the Lagrangian and Hamiltonian formalisms inclassical mechanics; see e.g. Arnold (1978, pp. 6166) and in the present contextWannier (1966, pp. 134138).(b) A semantic change of variables. The notations used in Axioms 2.3.1 are irrelevant to thesyntax: for instance, instead of the coordinates p and V we could have used H and M;proceeding consistently with this notation thereafter, one would only obtain a formalrewriting of Scholium 2.3.1. This change of variables would be syntactically irrelevant;nevertheless, at the semantic level, it conforms with the notation used to describeinsteadof an ideal gasa perfect, or ideal, paramagnet with magnetic moment M embedded in amagnetic eld H. As an illustration, the usual discussion of the Carnot cycle can befaithfully translated into this dialect (e.g. Wannier, 1966, pp. 122124).(5) As for the mathematical apparatus required by Clausius axiomatics, where I use thenotation t for the working differential, Clausius still writes dW , but he cautionsexplicitly and repeatedly (e.g. Clausius, 1879, p. 112) that his notation does not assume thatthere exists some function W of which dW would be the differential: t is not an exactdifferential. Ditto for the heating differential Z a.k.a. dQ. But dS 1=TZ and dU 9For the choice of the words heating and working, see remark (5).10itive constant R.) The question could be raised as to why the constitutive relations (2.3.3) shouldbe moved up into the core. I wish to argue against such a move. The context in whichideal gas appears here allows one to distinguish between two types of changesariables.A syntactic change of variables. The choice of the variables V and T to describe thestate-spacei.e. the domain D on which the functionals Z and t are dened in Axioms2.3.1is arbitrary; in particular, the functions (2.3.6) and (2.3.7) can be inverted toexpress T and V in terms of U and S. This change of variables does not affect thechoposematic axiomatization programme, while Clausius intentions here are plainly limited to aalization of thermodynamics of homogeneous media. It is also evident that, in contrast toert, Clausius uses rather indiscriminately Satz to signify various assertions thathematicians, even mathematical physicists, are now used to distinguish carefully. Never-ess, see Remark (5) below for one ontological distinction upon which Clausius does insist.) Note also that even the proof of Scholium 2.3.1 itself uses only the formal aspect ofconstitutive relations (2.3.3) and is not predicated on the semantics that guided theice of precisely those relations, nor does it depend on any specic value of the strictlyto the syntax expressed in Axioms 2.3.1; thus, its proof does not depend on any semantics thatwould identify V with volume, T with temperature, Z with heating, t with working,9 p withpressure, CV with specic heat at constant volume, U with internal energy, or S with entropy. Inthis sense, Clausius approach is akin to how Hilbert envisaged his Foundations of Geometry,(1899): One must be able to say at all timesinstead of points, straight lines, and planestables, chairs, and beer mugs.10 Evidently, Hilbert (1899) is but a prototype of HilbertsAs reported in Reid (1970, pp. 57, 60).Z t are exact, and can be integrated along any path to give the same functions (2.3.6) and(2.3.7). The names working and heating are systematically used here to emphasizethat these are intrinsic objects, while work or heat would only encumber thepresentation by referring to properties that are not intrinsic, but depend on paths ofintegration. With Clausius understanding of this distinction, a new mathematicalingredientthe notion of differentials that are not necessarily exactenters thesyntactic core of a specic physical theory. With hindsight, precursors can be traced backin other elds, e.g. in the works of Clairaut,11 Ampe`re and Green. The contributionClausius made here to the core of thermodynamics is to introduce a language in which one(b)parameter space of equilibrium states. The L-model provided by the ideal gasARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 693allows to compute explicitly the analytic form of this function. It is important to note thatin this context, to say that S increases along an isotherm may only mean that, when thevariable T is kept constant in (2.3.7), the numerical value of S is larger whenever the value of Vis larger.1211For instance, Clairaut (1740, p. 294) notes that: the smooth differential o Px; ydy Qx; ydx is exacthe says comple`tei.e. there exists a function F such that o dF iff qxP qyQ; for instance: xdy ydx is exact(it is the differential of xy); but xdy ydx is not exact; nevertheless, it admits an integrating factor, namely x2that makes it exact: x2 xdy ydx dy=x.12The notion of entropy has an amazing history. First introduced by Clausius, it was later extended when Sappears as the limiting equilibrium value of Boltzmanns H-functional dened on the space of time-dependentdistributions f on the microscopic phase space. Deeper justications for the nature and role of entropy in thetheory of non-equilibrium thermodynamics were proposed still later, from Caratheodory (1909) right throughLieb & Yngvason (1998, 1999, 2000); one of the achievements of the latter is to cleanse general thermodynamicsfrom heretofore pervasive intrusions of quasi static transformations. Collateral philosophical issues are pursued(7in Ubringing to bear on fundamental issues a structurethe ideal gasthat is onlyapproximatively met in the actual world. This empirical discrepancy cannot detractfrom Clausius having provided hereby a proof that his axioms are mutually consistent.) So far, the entropy S does not depend on time; it is a function dened on thean L-model, i.e. a mathematical structure for which the mathematical Axioms 2.3.1 aresatised. And they are. Hence, the axioms at the core of Clausius theory are mutuallyconsistent. The distinction between the intentions of L- and H-modeling strategies wasapparently not perceived by those of Clausius contemporaries who reproached him for(c)Clausius own thinking, to which he arrived after many earlier attempts, some datingback to 1850, and including a conceptually different book with the same title, namelyClausius (1859).In the last quarter of the xixth century, the ideal gas had a bad press, exemplied by:an ideal substance called perfect gas, with none of its properties realized rigorously byany real substance . . . (Thomson, 1880, Section 46, p. 47); for the context of thislordly sneer, see Remarks 2.2.2, and Chang (2004, pp. 202219).In his 1879 presentation, Clausius appeal to the ideal gas is plainly limited to offeringcould describe such things as a heat engine or a Carnot cycle without having to appealto the mythical substance called the caloric.(6) Yet, some confusion long persisted about Clausius axiomatization; I see threereasons for this.(a) The presentation is original with Clausius (1879, Chapter V). It is a new departure infnk (1995, 2001), and in Brown and Ufnk (2001). See also Part I, Section 3.2.2.4. Isothermal vs. adiabatic modelings for the speed of soundThe rst analytic understanding of the speed of sound in uids is due to Newton (1726,Book II, pp. 769779), particularly Propositions 4750, and the last Scholium of Section 8.ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723694The most immediate parts of the ambient landscape are Newtons own theory of harmonicmotion and his mechanics of uids. Yet, it must be realized that when he comes to thespeed of sound Newton is stepping into totally new territory.Newton postulates that sound is an oscillatory phenomenon, specically a harmoniccompression wave in an elastic fluid. To explore the empirical consequences of thistheoretical proposal, he calls upon a modeling strategyan H-modelingthat consists ofthree steps. First, he proposes the following relations between the speed v of the soundwave, the elastic force k of the uid, its density r, and the pressure p, namelyv krrwith k r qpqr. (2.4.1)The second step in Newtons H-modeling is to posit that the pressure is proportional to thedensityp / r, (2.4.2)that is Boyles observation, for which Newton had his own model; see Section 2.2. Thus:Scholium 2.4.1. Upon inserting (2.4.2) in (2.4.1), v becomes:vN prr. (2.4.3)Newton then proceeds to confront this prediction with experiments on the propagationof sound in air, for which he measures separately the two sides of (2.4.3). After varioustrials, he settles onvN 979 f=s andprr 1142 f=s (2.4.4)a discrepancy he attempts to correct with two hypotheses.13 Here is the third step inNewtons modeling strategy. The rst hypothesis is meant to account for the crassitudeof air, namely the fact that the particles of air are of nite size; to this, Newton attributesan increment of 109 English feet per second. The second hypothesis is similarly alleged toaccount for the water vapor suspended in the air, which Newton estimates to contribute amultiplicative factor 2120. Newton does not justify separately either of these corrections; theyjust combine to give13Despite his famous, but often misinterpreted, dictum hypotheses non fingo (Newton, 1726, p. 764), Newtonalso admitted that the best and safest method of philosophizing seems to be, rst diligently to investigate theproperties of things and establish them by experiments, and then to seek hypotheses to explain them (Newton,1730, p. xxiv); and whatever is not deduced from the phenomena is to be called an hypothesis. (Newton, 1730,p. xxxv); for a modern illustration, compare this to the role of the isotopic effect in the construction of the BCSmodel, Section 5.1.ARTICLE IN PRESSScholium 2.4.2.v vN 109 2120i.e. with vN 979 f=s : v 1142 f=s. (2.4.5)The agreement between prediction and experiment is veried now within one part in athousand! There was no way for Newton to justify with such a precision either theextemporaneous estimates he enters in his two corrections (2.4.5), or the experimentalvalues of the pressure and density he uses to obtain (2.4.4).Several ways could be imagined to disentangle the knot left by Newton: to explain thecorrections entering (2.4.5); to question the experimental results (2.4.4); to revise the core(2.4.1) of the theory; to question the original model (2.4.2) and its consequence (2.4.3).Various combinations of these suggestions were tried unsuccessfully, e.g. by Euler (1727,1759), or Lagrange (1759). The solution was found by Laplace (1816) and Poisson (1823).While accepting the core of Newtons theory, namely (2.4.1), Laplace and Poissonupdate Newtons assumption (2.4.2) to read as follows:Scholium 2.4.3. Ifp / rg with g CpCV, (2.4.6)where, since air is a diatomic gas, CV 52 R and Cp CV R 72 R; then (2.4.3) becomesv prr gp vN gp with gp 1:4p 1:183. (2.4.7)Within the probable precision of the experiments, (2.4.7) can be taken to resolve theempirical discrepancy 1142C9791:166see (2.4.4)that plagued Newtons originalmodel.Remarks 2.4.1. (1) Newtons (2.4.5) is a clear a posteriori revision of a prediction so as tomatch experiment. That such a heuristic reconstruction may be ill-fated comes with theterritory of H-modeling.(2) The physical justication for the LaplacePoisson model is that the phenomenon istoo rapid to be isothermal: the heat generated by compression has no time to go anywhere,and the process is adiabatic, i.e. occurs without dissipation nor absorption of heat. Eq.(2.4.6) gives the phasespace trajectory of such a process, and it is known today as theLaplace Poisson equation for adiabatic compression; the modern reader will derive it uponcomputing dS 0 from (2.3.7).(3) A major change in the ambient network presided over the passage from the earlymodeling in Newton (1726) to the modeling options chosen in Laplace (1816) and Poisson(1823) amid active concerns with the nature of temperature and heat, in the Paris whereFourier (1822) and Carnot (1824) were published.(4) The current reading of Scholium 2.4.3 is: (i) it eliminates the idiosyncrasy ofNewtons two ad hoc hypotheses summarized in (2.4.5); (ii) it renes the interpretation ofthe core (2.4.1) of Newtons theory as it adds to the core the distinction between anisothermal process and an adiabatic one, i.e. between a process that occurs at constantG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 695temperature and one that occurs at constant entropy (i.e. without exchange of heat).ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723696(5) Since Newtons general theory, as summarized in (2.4.1), was not in doubt whenLaplacePoisson proposed Eq. (2.4.6), and since the direct empirical testing of the latterwas not then a routine experiment, another reading is that Scholium 2.4.3 provides anH-model Laplace and Poisson would have devised to demonstrate the empirical validity ofthe adiabatic formula (2.4.6).(6) In the LaplacePoisson model, the observable k in Newtons relations (2.4.1)now is the adiabaticrather than the isothermalcompressibility. This gives anexperimental method to obtain the numerical value of k by measuring the speed ofsound. The lasting exploratory value of this H-model is enhanced by the reliability ofthese measurements even far beyond the validity of the ideal gas approximation; see e.g.Levelt-Sengers (1966).2.5. Modelings of real gases: from van der Waals to Lenz IsingBy the mid-19th century it was becoming empirically clear that the ideal gas did not offera faithful description of real gases. One crucial line of experiments was started by Andrews(1869) who observed: (i) the existence of a critical point pc; V c; Tc, which he characterizedby the appearance of an opalescent glow; (ii) the attening of isotherms in the vicinity ofthe critical point; and (iii) the nearly horizontal portion of the isotherms in the regionwhere the liquid and gaseous phases coexist.These phenomena have an analogue in the discovery by Curie (1895) that magneticsubstances such as iron loose their magnetism above a materialspecic temperature, nowreferred to as the Curie temperature. Compare to the analogy between the ideal gas and theperfect paramagnet mentioned in Remark 2.3.1 (4b). The transfer of this analogy to realgas and ferromagnets will be pursued in Section 2.5.2.On the microscopic front: if the models devised by Bernoulli and Maxwell are to betaken literally, namely that a gas consists of particles, then real particles do have anextension in space, and should be expected to interact with one another. The beauty of thenext modeling strategy is that it takes its clue in these theoretical expectations to explainthe above mentioned empirical phenomena.2.5.1. The van der Waals gasvan der Waals (1873) proposed an H-model to improve on the ideal gas, namelyp a NV 2" #V Nb NkT , (2.5.1)where a and b are substance-specic positive constants which model two effects: a toaccount for the long-range attractive interaction between the molecules; and b for theshort-range repulsive hard-core that prevents the molecules from penetrating each other.Note that when the particle density N=V becomes vanishingly small in (2.5.1), one recoversthe ideal gas relation (2.1.1).14The new fact is that (2.5.1) exhibits a critical temperature Tc. Above T c, no big surprise:the isotherms are stable, in the sense that qV po0. Below T c, however, the isothermspresent an unstable interval where qV p40, i.e. where increasing the pressure would14With the notational substitution Nmole=NR k.ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 697increase the volume. On the isotherm T Tc, qV po0 except for exactly one critical valueof V V c at which qV p 0.Scholium 2.5.1. With the notations pc pvc; T c, vc V c=N, the critical values of thespecific volume, the temperature and the pressure are given by vc 3b, kT c 8a=27b,pc a=27b2; and thus satisfy, for all values of a; b; c the universal relationpcvckT c 38. (2.5.2)Remarks 2.5.1. (1) The prediction (2.5.2) is fairly well veried for real substances, even asthe coefcients a, and b to a lesser extent, vary wildly from one substance to another (e.g.Emch & Liu, 2002, p. 382; Stanley, 1971, p. 69).(2) Scholium (2.5.1) implies that upon rewriting the van der Waals equation (2.5.1) interms of the reduced variables v v=vc (with v V=N ), T T=T c, and p p=pc, oneobtains the law of corresponding statesp 3 1v2 v 13 83T (2.5.3)proposed in 1880 by van der Waals. This law played a motivating role15 in the successfulexperimental quest for the liquefaction of helium, achieved in 1908 by KammerlinghOnnes.(3) The justication for the model given by van der Waals, later rened by Ornstein (e.g.Emch & Liu, 2002, Section 11.3.2) obtains upon simulating local interactions by anaveraging over the whole volume occupied by the uid and a simultaneous limit where thestrength of the interaction vanishes. Weiss (1907) transferred van der Waals modelingfrom uids to ferromagnets: each spin is viewed as being submitted to the mean magneticeld generated by all the other spins. This modeling strategy was later extended beyondthese two cases, cf. e.g. the BCS model in Section 5; it is then generally known under thename Weiss gave it: the molecular eld approximation. All H-models constructed alongsuch lines explore the putative existence of a collective behavior that was later provenanalytically to pertain to a wide range of phase transitions, namely long-range order.(4) Eq. (2.5.1) entails the instability qV p40 along a segment of each isotherm below T c.This physically unacceptable prediction is palliated by an ad hoc thermodynamicalargumentnamely the Maxwell construction (e.g. Emch & Liu, 2002, pp. 382384)toobtain an isothermal coexistence plateau which is attached to the corresponding isothermthrough a rst-order discontinuity. As temperature varies, these discontinuities trace asmooth coexistence curve in the plane V ; p, with its maximum at the critical point. Thists qualitatively well with the empirical behavior of real uids.(5) The construction of the Maxwell plateau is a theoretical artifact that can be avoided.This was demonstrated by the KacBaker L-model (e.g. Emch & Liu, 2002, pp. 388389).The four main features making that L-model exactly solvable are: (a) it is one-dimensional;(b) a hard-core repulsive potential ensures that the particles are kept in order; (c) a two-body attractive potential of the form: cattrjrj aol expljrj; and (d) the model issolved in the limit l! 0 thus mimicking asymptotically van der Waals hypothesis of avery weak, very long-range potential. Taking this limit is necessary in order to bypassgeneral theorems asserting that analyticity is ineluctable in one-dimensional systems with15This role is documented in Kammerlingh Onnes Nobel lecture (1913).behavior of the uid in the immediate neighborhood of this critical point; and (iii) itARTICLE IN PRESSanticipates the emergence of long-range order in condensed matter.2.5.2. L-modeling for phase transitions: Lenz IsingThe successes and shortcomings of the van der Waals gas call for some L-modelingthat would sort out the essentials from the accidental. The lattice gas model is a stepin this direction, although the path ahead is impeded by no-go theorems, andserious computational difculties. The modeling strategy is to consider particlesthat are distributed on the sites of a cubic lattice Zd with the restrictions that (i) each site isoccupied by at most one particle; (ii) energy favors congurations where neighboring sitesare simultaneously occupied; and (iii) the system is invariant under lattice translations.Specically, one associates to every nite cube L Zd the conguration space f0; 1gL fn : i 2 L7!ni 2 f0; 1gg and the partition function ZL Pn2f0;1gL expb mNLn HLn where b 1=kT is the natural temperature; m is the chemical potential;NLn is the number of occupied sites in L; HLn is the energy of the congurationn 2 f0; 1gL.The change of variables s 2n 1 : i 2 Zd 7!si 2ni 1 2 f1;1g associates toevery site i 2 Zd a classical spin si pointing either up si 1 or down si 1. Theproblem of modeling the Maxwell plateau is thus equivalent to modeling a magneticsystem that exhibits a non-vanishing spontaneous magnetization. Lenz suggested thefollowing problem to Ising: to compute, within the strict connes of statistical mechanics,the magnetizationMLb; B; J jLj1XLhsii, (2.5.5)reasonably nite-range interactions: (Ruelle, 1968, 1969; Takahashi, 1942; van Hove,1950).(6) The van der Waals model gives an empirically poor description of the uid in theimmediate vicinity of the critical point where various divergences are observed,among them the critical opalescence discovered by Andrews (1869). To these divergencesare associated critical coefcients a;b; g and dfor their denitions, cf. e.g. Emchand Liu (2002, pp. 385386)the individual values of which, when computed fromthe van der Waals model, are grossly falsied by laboratory experiments. Yet, when takentogether, the predicted values still satisfy relations that are borne out by experiments,namelya 2b g 2 and a b1 d 2, (2.5.4)thus pointing to some possible universality that wouldand actually does (cf. Cardy,1996; Kadanoff et al., 1967 )transcend the particular H-model just considered.Hence, the van der Waals H-modeling goes even beyond its initial purpose to exploremicroscopic structures compatible with the observed deviations from the ideal gasisotherms. It suggests specically several directions in which the core ought to bedeveloped. Indeed, (i) it reveals some seemingly universal macroscopic features of realuids: (2.5.3), the law of corresponding states, and (2.5.2) which puts an empirically wellsupported constraint on the location of the critical point; (ii) it warns about the dgetyG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723698s2f1;1gARTICLE IN PRESSwhere jLj is the number of sites in L and hi is the canonical equilibrium averagecorresponding to the partition functionQLB; J;b Xs21;1Lexpb HLs with HLs BXi2Lsi JXhi;jisisj, (2.5.6)where B is an external magnetic eld, J40, andPhi;ji sums over pairs of neighboring sitesin L.Ising (1925) showed that when d 1 this model is exactly solvable and that the resultingmagnetization shows no singularity, even when the system becomes very large. Specically,QLB; J;b ljLj ljLj with l ebJ cosh bB e2JB sinh2 bB e2bJ 1=2 from which onegets:Scholium 2.5.2. At fixed J and b, MLB; J;b is analytic in B and ML0; J; b 0.Moreover, MB; J;b limjLj!1 MLB; J;b tanh bB1 1 e4bJ cosh2 bB1=2and thus MB; J;b is analytic, with M0; J;b limB!0MB; J;b 0.Hence, even in the innite volume limit L " Z, this model does not exhibit any phasetransition.To go beyond this negative result, one might want to consider interactions with longer,but nite range; this however would not succeed; cf. Remark 2.5.1(5).Despite the conjecture Ising imprudently wrote at the end of his otherwise perfectly correctpaper, the next step is to go up in dimension where collective behavior may be easier toachieve since each site then interacts with 2d neighbouring sites. While going up in dimensionthus seems to be necessary, it is still not sufcient. Indeed, with d 2, Yang and Lee (1952)proved that as long as the regions L remain nite, analyticity of the isotherms persists for theLenzIsing model, and thus for the corresponding two-dimensional lattice gas.After much hesitations epitomized (e.g. Dresden, 1987, p. 321) by the then inconclusivediscussions at the 1937 van der Waals Congress in Amsterdam, it was recognized later thatit is legitimate physics to consider the so-called thermodynamical limit where the system isallowed to become innite. In this limit, one has nally:Scholium 2.5.3. The two-dimensional Lenz Ising model exhibits a critical temperature T c,given bysinh 2J1kT c sinh 2 J2kT c 1below which the model exhibits a non-vanishing spontaneous magnetization satisfyingMOJ;b 1 sinh 2J1kT sinh 2 J2kT 2" #1=2for ToT c,where J1 and J2 are the transverse coupling constants between nearest neighbor spins.Remarks 2.5.2. (1) The simplest proof I know for Scholium 2.5.2 proceeds by following thetransfer matrix technique due to Kramers and Wannier; see e.g. Emch and Liu (2002,Section 12.1). The initial derivation of Scholium 2.5.3 (Onsager, 1944; Kaufman, 1949;Kaufman & Onsager, 1949) was somewhat cryptic. An accessible proof is provided inSchultz, Mattis, and Lieb (1964); for a description of the main steps of their proof; see e.g.G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 699Emch and Liu (2002, Sections 12.1, 12.2).ARTICLE IN PRESSThe above L-modeling establishes the logical consistency of the syntax of statisticalmechanics inasmuch as its ability to describe phase transitions is concerned. It establishesthe role of the thermodynamical limit in the emergence of a class of cooperativephenomena, such as long-range order and the concomitant occurrence of phasetransitions. In addition, this L-model shows how the onset of cooperative behaviourdepends on the interconnectedness between microscopic parts of the system considered,and most remarkably on dimension.3. Non-equilibrium and ergodic theory3.1. Stochastic modeling of the diffusion equationThe diffusion equation governs the ow of energy or matter from higher to lowerconcentrations in a wide range of macroscopic phenomena observed in gases, liquids, andsolids.In order to construct a microscopic model for the onedimensional diffusion equationqt f vqx f Dq2x f with x; t 2 R R, (3.1.1)it is useful to note that the distributionf x; t 14pDtp exp 14Dtx vt2 (3.1.2)solves (3.1.1) subject to the initial condition that at time t 0 the distribution f isconcentrated at the origin. For all t40, the average position and mean square-deviationhxit ZRdx f x; tx and hx hxit2it ZRdx f x; tx hxit2, (3.1.3)when computed with respect to the distribution f in (3.1.2), are proportional to the elapsedtime t:hxit vt and hx hxit2it 2Dt. (3.1.4)These two properties will play an essential role in the construction of the forthcomingmodel. For their empirical relevance, see Remark 3.1.1(3).The modeling strategy to be followed here starts from a situation in which space is a one-dimensional lattice fx sa j s 2 Zg; time is a discrete variable t nt with n 2 Z; and pand q are positive numbers satisfying p q 1. Assume now that a particle moving on Zmay, at every time t nt, only either jump to the site to its right with probability p or jumpto the site to its left with probability q. Then, the probability f sa; nt that the particle be atsite sa at time nt satises the equationf x; t t pf x a; t qf x a; t. (3.1.5)With the initial condition that the particle be at the origin at time t 0, the solution of (3.1.5) isf sa; nt n12n s0@1A pns=2qns=2 when jsjpn;8>>>: (3.1.6)G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 6837237000 otherwise:With respect to this distribution, the average position of the particle and the mean-squarea 1 vthediffRemlimiis s(2(a) If one is satised with a statistical explanation, the model belongs to L-modeling.ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 701(b) If one requires an explanation in terms of the emergence of an irreversible macroscopicapproach to equilibrium from a reversible microscopic Hamiltonian theory, one hashere an H-model pointing to the role of mutually consistent rescalings of space andtime in the pursuit of the elusive arrow of time. More sophisticated quantum, as well asclassical, H-model strategies were conceived to conrm the importance of the choice ofa proper time scale (e.g. Martin & Emch, 1975; Martin, 1979; or Spohn, 1977, 1990,1991); in particular, these models rened the techniques proposed in the pioneeringpapers by van Hove (1955, 1957, 1959) on the so-called long-time/weak-coupling l2t-limit.(3) The Brownian motion, observed when colloidal particles are suspended in a uid,satisfies the characteristic diffusion relations (3.1.4). The best early experimental set-upcf. Perrin (1909)scanned the positions of such particles at successive and equal timeintervals. The erratic patterns obtained in this manner were the same over several orders ofmagnitude, provided time and space be rescaled according to tx2, thus most elegantlyconrming the mutual consistency of the two relations (3.1.4).(4) Upon considering the collisions between the few heavy colloidal particles and themanypresumablyvery light molecules of the uid in which they are suspended,Einstein (1905b) and Smolukowski (1906) produced a model16 that anticipates/reproducesthe basic features of Brownian motion. In addition, they proposed for the diffusion16answlima!0 t 2D and lima!0 ap q 2D(3.1.9)distribution (3.1.6) converges to the distribution (3.1.2); and Eq. (3.1.5) converges to theusion equation (3.1.1).arks 3.1.1. (1) The mathematical trimmings necessary to prove the existence of thesets are discussed in Kac (1946, 1947a, 1947b); the extension to higher dimensions d41traightforward.) Does this model proceed from L-modeling or from H-modeling? Here again, theer depends on the context.deviation at time t are proportional to thxit p qath it and hx hxit2it 4pqa2t t. (3.1.7)The comparison between (3.1.4) and (3.1.7) suggests the following modeling:Scholium 3.1.1. If the transition probabilities p and q are chosen to satisfyp q at v and 4pq a2t 2D, (3.1.8)then, in the joint limit2For further references cf. Emch & Liu (2002, pp. 8687).pxt1 n Pm;n pxt m with Pm;n N m for n m 1;>ARTICLE IN PRESSm1 N0 otherwise:>>>:(3.2.1)Exact properties of the model:Scholium 3.2.1 (Uniqueness of equilibrium). The evolution equation (3.2.1) admits exactlyone stationary solution, namely: pn 2N Nnwhere Nnis the binomial coefficientN!=N n!n!.Scholium 3.2.2 (Tendency to approach equilibrium). For any value of n larger (resp. smaller)constant the interpretation D kT=6pr Z where k R Nmole=N is the Boltzmannconstant; T is the temperature; r is the radius of the suspended particles; and Z is theviscosity of the uid. Thus the EinsteinSmoluchowski modeling combined with the Perrinlaboratory measurements led to trustworthy estimates of the Avogadro number NAv andconsequently of the number of molecules in a uid. In the judgement of thecontemporaries, the EinsteinSmolukowski model qualied as an L-model that could beused conclusively against a persistent minority refractory to the existence of atoms asactual particles with measurable nite dimensions. Henceforth, the EinsteinSmolukowskimodel also contributed to the growing acceptability of stochastic modeling in physicaltheories.3.2. The dogs-and-fleas modelTheoretical background of the model: This stochastic model was conceived in response totwo objections against Boltzmanns kinetic theory of gases, namely the reversibilityobjection, or Umkehreinwand, of Loschmidt (1876, 1877), and the recurrence objection, orWiederkehreinwand, of Zermelo (1896a, 1896b). Boltzmanns responses to these twoobjectionsrespectively, Boltzmann (1877, 1878a, 1878b) and Boltzmann (1896, 1897)provide much of the motivation for the dogs-and-eas model constructed by Ehrenfest(1907, 1911); cf. also Kac (1946, 1947a, 1947b) for the necessary mathematical tightenings,and Emch and Liu (2002, Section 3.4) for some recent computer simulations.The heuristic formulation of the model: Two dogs A and B share a constant population ofN eas labeled from 1 to N; for notational simplicity N is taken to be even. Every second,one integer between 1 and N is called at random, and the corresponding ea jumps fromone dog to the other.Mathematical formulation of the problem: To nd out the time-dependence of thenumber xt of eas on dog A, knowing its initial value xo at time t 0. Since exactly oneea jumps at every call, the transition probability Pm;n Probfxt1 n j xt mg vanishesunless jn mj 1; nally, the probability that the ea jumps from the dog on which it sitsis proportional to the number of eas on that dog. Consequently, for all tX0, theprobability pxt1 n that there are n eas on dog A at time t 1 satisesXn1Nm for n m 1;18>>>>>>>>>>:9>>>>=>>>>;. (3.3.3)The Hadamard property is now written precisely as the following hyperbolic relations:tG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723706at ct at ce t. (3.3.4)The second ingredient of the model is a discrete subgroup G G that acts freely andproperly discontinuously on the Poincare half-plane and is co-compact17 which ensure thatthe quotient spaces F GnG=K GnH and M GnG T1F are compact differentiablemanifolds.18 G plays here the role played by Z2 to get the at torus T2Z2nR2 ( R2nZ2since R2 is abelian). In particular, the Poincare half-plane H is tiled by translated images ofARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 707F H, and F is to be pictured as one of these tiles where opposite edges are identied byan element of G; hence a curve in H, upon reaching an edge, re-enters the tile by theopposite edge, and F is to be viewed as a manifold without boundary.The beauty of the model resides in the fact that G acts on G, and thus on H, from theleft, whereas the geodesic ow a and the transversal actions c act from the right; see(3.3.2) and (3.3.3) above. This is precisely the reason why these group actions may as wellbe considered as acting on the quotient manifold M GnG, where the hyperbolic property(3.3.4) thus still holds.Remarks 3.3.2 (Properties of the HedlungHopf model). (1) The model was proven to beergodic, and even strongly mixing, by Hedlund and Hopf. This already places it higherthan the Boltzmann model.(2) Since the HedlundHopf model is a geodesic ow on a Riemannian surface F , it is agenuine Hamiltonian system (there, M T1F is a three-dimensional constant-energymanifold). Therefore, this L-model establishes that ergodic behavior is compatible withHamiltonian dynamics.(3) Eq. (3.3.4) is a stronger (integral) form of a (differential) condition that denesAnasov flows, which are dynamical systems sitting almost at the top of the ergodichierarchy19A K L X E. (3.3.5)Here A stands for the Anosov condition which, by virtue of (3.3.4), is satised by theHedlundHopf model. K stands for the Kolmogorov condition, which roughly states thatin the course of the evolution, there exists a partition of the phasespace M which becomesner in the future, to the point of generating all measurable subsets of M, and which in thepast becomes so coarse that in the limit it contains no more than sets of measure either zeroor 1; Kolmogorov ows have positive dynamical entropy, i.e. again roughly speaking, nomatter how far in the nite past one has collected information on the system, anysubsequent observation brings genuinely new information. In (3.3.5) L stands here forLebesgue, and is a spectral condition on the Liouville operator, the generator of thedynamics. X stands for mixing, of which there are several kinds, besides that of Denition3.3.1(iii); and E stands for ergodic.Remarks 3.3.3 (Beyond classical Hamiltonian systems). (1) The Boltzmann and HedlundHopf modelings discussed above involve continuous times, i.e. t 2 R; ergodic theory is17The existence of such subgroups G G was established by Poincare in 1882, and he extended their study in aseries of papers that occupy much of Tome II of his Oeuvres.18The groups H and G are not abelian, so that one must distinguish, as done here, left- from right-actions.19I mention here only the main levels of this hierarchye.g. Arnold & Avez (1968)and I use the logiciansnotation (e.g. Shapiro, 2005) g j to denote that j is a logical consequence of g, i.e. that j can be deduced from gby a chain of inferences the rules of which are codied in the syntax. Theoretical physicists and manymathematicians would use, instead of , symbols such as !;) or 7! for which I have other usages.ARTICLE IN PRESSmuch richer in models involving discrete times t 2 Z, such as successive coin tossings(extending to innity) or the Arnold CAT (e.g. Arnold & Avez, 1968, Chapter 3).(2) Each implication in (3.3.5) is strict, i.e. the direction is a theorem, and an L-modelhas been exhibited that disproves the opposite implication (e.g. Arnold & Avez, 1968,Chapter 2).(3) The HedlundHopf model served as a stimulus toward the creation of symbolicdynamics by Hedlund (1935, 1939, 1940, 1969) and by Morse (1921), Morse and Hedlund(1938); the classic text is (Morse, 1966). Sufce it to say here that symbolic dynamics allowsto reduce the study of many complex dynamical systems, in particular chaotic ones, to thestudy of shift maps on spaces of sequences. For an elementary introduction, see Devaney(1992, Chapter 9 & 10); and for a rich overview of the scope of recent research, see Badiiand Politi (1997). Already from these two books, the reader will notice that modelingcontinues to provide much of the ferment for modern developments.(4) Classical ergodic theory has been extended to the quantum realm: e.g. Emch (1972,Theorem II.2.8), Emch and Radin (1971), Emch (1976), Emch, Narnhofer, Thirring, andSewell (1994), Jauslin, Sapin, Guerin, and Wreszinski (2004).(5) Beyond models where t is interpreted as time, ergodic theory has been extended toamenable groups, i.e. groups that possess invariant means. Mathematical physicists havebeen interested in the n-dimensional groups Zn and Rn interpreted as translations onlattices or Euclidean conguration spaces for models in statistical mechanics (e.g. Sewell,2002). Due to widely valid causality conditions, strong ergodic properties with respect tospace, rather than time, are much more prevalent among systems of physical interest. Inparticular, time-ergodicity is not a generic property, even among classical Hamiltoniansystems (e.g. Emch & Liu, 2002, Chapter 9); for the original result, see Markus and Meyer(1974).4. Modelings in early quantum theoryThe wealth of widely available materials and personal reminiscences makes the birth ofquantum theory one of the best documented events in the history of science. This sectionfocuses on the role modeling played in the invention and early developments of the theory.4.1. Modelings of the atom: from Balmer to Schrodinger, and beyondFor the study of modeling strategies, the Bohr atom offers an instructive succession ofphases.The experimental data collected in atomic spectroscopy were already quite extensive bythe beginning of the 20th century. By choosing judiciously the simplest case, Balmer (1885)had noted that the spectral lines of hydrogen obey the rule nmnn2 m2 withn; m 1; 2; 3; . . .; this is a purely numerical model, with no built-in explanatoryjustication; in fact, hardly a model at all.Bohr (1913) proposed a model for the hydrogen atom in which he pictures one electroncircling around a nucleus on discrete orbits of energy Enn2; n 1; 2; 3; . . .; electro-magnetic transitions between these discrete energy levels then absorb or emit lightaccording to Balmers formula: a potent idea but not yet a full explanation. At this stage,G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723708the Bohr model is an H-model.4.2. Plancks modeling for black-body radiationARTICLE IN PRESSThe experimental evidence available to Planck was the spectral density rT n of theenergy per unit volume of electromagnetic radiation, as a function of its frequency n, whenthe radiation is in equilibrium with a black-body at temperature T . Two qualitativeformulas had been identied by Wien (1894), Stefan (1879) and Boltzmann (1884). Inaddition, specic analytic expressions had been proposed: one, due to Wien (1896), wasfound to be empirically valid only when n=T is large; in contrast the other, due to Rayleigh(1900, 1902), see also Jeans (1905a, 1905b), was found empirically valid only when n=T issmall.At rst, Planck (1900a, 1900b) proposes in a lucky guesshis own words (Jammer,1966, p. 19)an ad hoc formula:rT n Ahnehn=kT 1 with A 8pn2c3. (4.2.1)(i) This expression satises the Wien and StephanBoltzmann qualitative expressions;(ii) it interpolates analytically between the Wien and RayleighJeans formulas; (iii) it tsexperimental results so well that it allows the experimental determination of the value of c,the speed of light and k, the thermodynamical Boltzmann constant. Plancks guessintroduces a new constant, h. The Planck constant h, together with the Boltzmann constantk, gives quantitative meaning to the conditions that n=T be large (resp. small), namelyhn=kTb1 (resp. hn=kT51).To explain the success of his interpolation, Planck resorted to an act of desperationIn the same way as the ideal gas serves as an L-model for Clausius thermodynamicssee Section 2.3 abovethe Bohr model provides also an L-model on which Schrodinger(1926a, 1926b, 1926c) could test the differential equation that now bears his name: heshowed that in the case of one non-relativistic charged particle moving in a centralCoulomb potential V xjxj1, the eigenvalues of H P3i1q2xi V x are the Bohrenergy levels En.In turn, the Schrodinger wave equation, together with Heisenbergs uncertainty relation,is one of the two most genuinely creative H-models that led to the forthcoming quantumtheory, the mathematical core of which was formulated by von Neumann (1932).Beyond this, however, for modeling atoms higher up in the Mendeleyev table, themutual interactions between the electrons prevent an exact solution of the Schrodingerequation. To bypass the difculty, Thomas (1927) and Fermi (1927, 1928) proposed totreat the electrons as a cloud of charged particles satisfying the classical Poisson equationfor electrostatics, with however a charge density restricted by the quantum statistics rulethat Fermi had formulated just a few months earlier. This H-model, by its being semi-classical, was inconsistent; yet, up to the middle of the Mendeleyev table, it still providescorrect estimates for the radius of the atom as a function of the charge of its nucleus. In thecourse of the next 75 years, the model has been a fruitful scheme in directions for which itwas not initially intended, in particular in the control of the asymptotics governing thestability of matter, from molecules to stars (Lieb, 1997) or along a different vein (Catto,Le Bris, & Lions, 1998).G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 709again his own words (Jammer, 1966, p. 22)the construction of a heuristic model in whichstaPlatheeffect (see Section 4.3), these difculties led Einstein to advance an alternate ontologicalARTICLE IN PRESS710postulate, namely that the quantization is to be looked for in the radiation eld itself(rather than in a dubious mechanism of interaction with the walls). As Einstein lifted againthe question as to whether light is wave-like or particle-like, his response in Einstein (1909)was that light ought to be viewed simultaneously as both particle and wave; specically:Scholium 4.2.1. If the average energy huT ni of quantum oscillators of frequency n inthermal equilibrium at temperature T is computed from Plancks distribution (4.2.1), then forall values of hn=kT the energy fluctuation hDu2i kT2qT huT ni is the sum of two termshDu2i hDu2ip hDu2iw withhDu2ip huT nihn;hDu2iw huT ni2c38pn28>: . (4.2.2)Hence, the particle-like contribution hDu2ip dominates when hn=kTb1, and the wave-likecontribution hDu2ip dominates when hn=kT51. In this interpretation, the particlewaveduality is thus a matter of degree, not of essence: quantum behavior occurs mostly at lowtemperatures, whereas classical behavior emerges at high temperatures. This observationinitially made on an H-modelwas extended later to more general circumstances, so muchso that it is now seen as part of the core of quantum mechanics.4.3. Einsteins modeling for the photoelectric effectThe syntax of classical theories (mechanics and electromagnetism) provided all the termsnecessary to describe the photoelectric effect, and yet, it failed to provide any satisfactoryexplanation for the phenomenon. Einstein broke through to propose his quantum H-modelin Einstein (1905a, Section 8), a paper the title of which, the reader will note, refersexplicitly to a heuristic point of view. Let me specify rst the sense in which the empiricalbackground was classical.1. On the experimental front: between 1830 and the end of the 19th century, the followingexperimental facts were established. Faraday postulated the existence of indivisiblegrains of electricity. Plucker discovered cathode rays which J. J. Thomson identied asbeams of particles with properties (mass and electric charge) that were soon measured.Rontgen discovered X-rays, high-energy electromagnetic waves generated by cathoderays impinging on some anode targets.2. On the theoretical front: Maxwell had predicted the existence of electromagnetic waveswhich, in a certain range of frequencies, could be interpreted as rays of visible light.raddisrmodynamical equilibrium.Asides from his noticing that Plancks account did not conform to Boltzmannstistical counting as closely as Planck made it appear, Einstein (1906) objected thatncks treatment of his model involved a theoretical inconsistency between: (a) the use ofMaxwell theory of electromagnetism to compute the average energy of a resonator in aiation eld; and (b) the assumption that the energy of a resonator must changecontinuously. Together with other empirical problems, among which the photoelectricthetheradiation exchanges energy in discrete quantas with putative resonators in the walls inG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723Accordingly, monochromatic light was viewed as a wave Cx; t Aeiotkx whereof electrons ejected increases with the intensity of the light; (iii) unless the frequency ofmeisdeARTICLE IN PRESSDulong and Petit (1819) had proposed an argument to the effect that the specic heatasured in calories per mole per degreeought to be the same for all solids: 3R where Rthe universal gas constant. However, it later became apparent that the specic heat couldthelight reaches a certain threshold, no electron is ejected; and this threshold depends onthe nature of the metal that is irradiated by the light; (iv) the velocity of the ejectedelectrons increases with the frequency of the incident light.Here is Einsteins explanation.Scholium 4.3.1. If light can only be absorbed in discrete light quanta of energy hn, then theconservation of energy readsE hn Eo, (4.3.1)where E 12mv2 is the kinetic energy of the electrons emitted in the photoelectric effect; Eo isthe energy to extract an electron from the metal, and thus depends on the constitution of thelatter.The experimental data (i)(iv) listed above are then immediate consequences of (4.3.1).Property (iv) is even made more specic: the square of the velocity of the ejected electronsincreases linearly with the frequency n of the incident light. Moreover, a directdetermination of the Planck constant h obtains by measuring the slope of the straightline(4.3.1) (Millikan, 1916).Einstein saw in this H-model an indication that a light-ray of frequency n ought to bethought of as a beam of particles of energy hn which he referred to as a light quanta.With this interpretation of his model, Einstein was surely extending the domain of Plancksquantum hypothesis to a much wider scope than Planck had originally imagined. Insupport of the general argument of my paper, I note that the Nobel committee for 1921chose to single outfrom the wealth of Einsteins contributionsthe explanation of thephotoelectric effect, which at rst sight could have been regarded as a relatively minorepisode; on the contrary, I see here a clear-cut case where an exploratory H-modeldrastically contributes to the evolution of a theory.4.4. Debyes modeling for the specific heat of solidsFor the purpose of the present paper, one of the main interests of Debyes model is thatit involves interactions between several nodes of the ambient web: quantum theory,rmodynamics, statistical mechanics, crystallography and Rontgen rays diffraction.n o=2p is the frequency of the light ray, l 2p=jkj is the corresponding wave length,and I jCj2 A2 is the light intensity.3. The photoelectric effect itself was discovered by Hertz (1887), and soon conrmed byseveral physicists, among them Hallwachs (1888) after whom the effect was oftennamed; the experiments of Lenard (1902) and Ladenburg (1903) established rmly thatwhen an electromagnetic wave of short enough wave-length impinges on a metalsurface, electrons are ejected exhibiting the following properties: (i) the velocity of theejected electrons is independent of the intensity of the incident light; (ii) only the numberG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 711crease dramatically with temperature. By the end of the 19th century, the experimentalARTICLE IN PRESSdata became sufciently systematic to lead to the conjecture that the specic heat of solidswould become vanishingly small as the temperature approaches 0K.On the wider network, Bravais speculations (1850) on crystalline solids having themicroscopic structure of regular lattices at the vertices of which sit the atoms, led von Laueto conjecture that crystals may serve as diffraction gratings for X-rays, a prediction almostimmediately conrmed in the laboratory (Friedrich, Knipping, & von Laue, 1912, 1913).While von Laues rst intention had been to explore the nature of X-rays, the paper marksthe birth of X-ray crystallography. For the purpose of the present subsection, it providesrmer quantitative parameters for the crystalline structure on which Debyes H-modelingis predicated.As no classical explanation of the observed temperature-dependence of the specic heatseemed forthcoming, Debye (1912) offered the following heuristic model, improving on(Einstein, 1907, 1911).The Planck formula for black-body radiation (4.2.1) is rst given a new interpretation interms of the vibrational energy of a solid at temperature TUT ZdngnUn; T with Un; T hnehn=kT 1 andZ 10dngn 3N,(4.4.1)where N is the number of the nodes in the crystalline solid, each node being occupied by avibrating atom. Where Einstein had assumed that g is concentrated on a xed frequency no,Debye takes for g the simplest vibrational distribution that takes into account that in acrystal the vibrations have a minimal wave-length of the order of the average interatomicdistance in the lattice:gn G1 if 0pnpno0 if n4no( )with G 12pn2s3V . (4.4.2)G takes into account that vibrations are now sound waves in a volume V rather thanelectromagnetic waves; compare with A 8pn2=c3 in (4.2.1); thus s is now the speed ofsound, instead of the speed c of light; and the replacement of 8p 2 4p by 12p 2 1 4p reects the fact that sound-waves in solids have, in addition to the twotransverse polarizations also present in light, a third degree of freedom, namelylongitudinal modes. These hypotheses entail the following consequence.Scholium 4.4.1. There exists a temperature Y, such that the specific heat satisfiesCV 3R for TbY;125p4RTY 3for T5Y:8>: (4.4.3)Hence, Debyes H-model describes two extreme regimes: at high temperatures it recoversthe DulongPetit value; and, at low temperatures, it predicts that as the temperatureapproaches 0K, the specic heat vanishes according to CVT3. The temperature Y, nowcalled the Debye temperature, depends on the density N=V of the solid considered, and onthe cut-off frequency no, and thus on the speed of sound in that solid.In addition, Debye gives an exact interpolation formula from which one obtains that CVG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723712decreases monotonically and continuously over the whole range of temperatures T 2 R.ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 713Note that this and the preceding subsection witness a considerable enlargement of theconcept of particle: the electrons as grains of electricity, the photons as the quanta of light,the phonons as the elementary sound vibrations in solids; a clean example of the dynamicsby which models act as agents for concept-transfer.5. Superconductivity: from H- to L-Modeling and backIn this section, I review one episode in the history of superconductivity, an episode in thecourse of which a model became for a while the commanding model in the eld. I thensketch a few of its subsequent avatars.5.1. Immediate experimental backgroundWhen the BCS model was advanced in 1957 as a Theory of superconductivity the eldhad a long history, dating back to 1911 when in Kammerlingh Onnes laboratoryexperiments showed that below 4.2K the electric resistivity of mercury vanishes abruptlyor at least had become less than a thousand-millionth part of that at normaltemperature. (Kammerlingh Onnes, 1913, p. 333). The phenomenon was then observedto be accompanied by a great diversity of manifestations. The physics community came toagree that this bundle of phenomena were to nd their explanation in the quantum theoryof solids, and more specically as a manifestation of an electron electron interactionmediated by phonons, i.e. by the vibration modes of the crystalline solid (Frohlich, 1952).The latter hypothesis 20 had been supported by the prediction and almost simultaneousdiscovery of a new phenomenon, the isotope effect, namely that the critical temperature T cbelow which a solid becomes superconducting is proportional to the frequency of thephonons, and thus to the inverse square-root of the mass M of the atoms in the solid:T cMp C, where C was observed to be nearly a constant across the four isotopes ofmercury. This is where I pick up the story, which I recount in three tableaux: H-modeling,L-modeling, and theory-transfer.5.2. H-modeling: the original BCS modelIn the original modeling phase (Bardeen, Cooper, & Schrieffer, 1957), the above featuresinform the conjecture that the following Hamiltonian describes the relevant system ofelectrons enclosed in a nite cubic box L of edge 2L V 1=3HL Xp;spaspasp Xp;qbp ~vp; qbq. (5.2.1)Here p is the momentum of an electron; p is its free-energy; asp is the creationoperator for an electron of momentum p and spin s up " or down #; and aspis the corresponding annihilation operator. Similarly, bp a"pa#p is thecreation operator for a pair of two electrons of opposite momentum and spin, a so-called Cooper pair. And bp ~vp; qbq describes the energy transfer during a collisionbetween Cooper pairs which models the electronphonon interaction. In the molecular field20Compare to footnote 13 above.ARTICLE IN PRESSapproximation,cf. Section 2.5.1the Hamiltonian becomesHL Xp;sEpgspgsp, (5.2.2)where the quasi-particles created and annihilated by gsp and gsp are related to the originalelectron creation and annihilation operators by the BogoliubovValatin transformationg"p upa"p vpa#pg#p vpa"p upa#p). (5.2.3)The one-particle energy of these quasi-particles isEp p2 DpDp1=2, (5.2.4)where the energy-gap Dp satises the self-consistency equationDp Xq~vp; q Dq2Eq tanh12bEq . (5.2.5)There exists then a critical temperature T c such that: (i) for all T4Tc;Dp 0 is the onlysolution of (5.2.5); (ii) for all ToTc, (5.2.5) admits a non-zero solution (unique up to a phase).The dependence of Dp on the temperature b 1=kT compares very well with empirical data;see Schrieffer (1964, Figs. 13). Thus, so far, the H-modeling seems empirically successful.For completeness, I should add that in the course of the derivation of (5.2.2) from(5.2.1), one obtains that the coefcients of the BogoliubovValatin transformation (5.2.3)satisfyupup 121 p=Epvpvp 121 p=Epupvp 12Dp=Ep9>=>;. (5.2.6)In particular, this entails upup vpvp 1 which ensures that the quasi-particles gare fermions; notice that for T4Tc these quasi-particles are nothing but the free electrons.5.3. L-modeling: the BCS Haag modelThere are two conceptual difculties with the BCS model. First, the Hamiltonian (5.2.1)is invariant under the gauge transformations dened byasp ! eiyasp; as p ! eiyas p (5.3.1)whereas the Hamiltonian (5.2.2) is not. Second, the energy spectrum Ep depends ontemperature whereas no temperature dependence had been put in the originalHamiltonian.The way out of these difculties was found by Haag (1962), and polished in a series ofpapers; for references later than Emch and Guenin (1966), see Emch and Liu (2002,G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723714Section 14.1). This solution came from the treasure chest of the mathematical theory ofC-algebras and their representations.21 For presentations of this theory, tailored to theneeds of the quantum physics of systems with an innite number of degrees of freedom, seeEmch (1972) or Haag (1996).The argument goes essentially as follows. The BCS modeling involves a passage to theARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 715thermodynamical limit; Bardeen et al. had assumed tacitly that in this limit the algebra ofobservables is still irreducible, so that any observable that commutes with all observablesmust be a multiple of the identity. This tacit assumption is wrong below the criticaltemperature Tc. Then, with A denoting the algebra of quasi-local observables, therepresentation prA corresponding, at natural temperature b 1=kT , to the thermo-dynamical limit of the equilibrium state r, has a non-trivial center Zr prA00 \ prA0,i.e. ZraCI , where prA00 is the weak-operator closure of prA, and prA0 is the set ofall operators that commute with all observables in prA or equivalently in prA00.In this limit, the energy gap is still an element of the center Zr, but it is not ac-number, i.e. is not a scalar multiple of the identity, nor is it even a quasi-localobservable, for it belongs to the weak-operator closure of a representation that depends onthe temperature through the canonical equilibrium state r. The gauge group S1 actstransitively on Zr, with Dp ! ei2y Dp, up ! ei2y up, vp ! vp andgsp ! eiygsp. The gauge symmetry of the theory is thus preserved. In addition,prA00 decomposes uniquely as a direct integral, over S1, of primary representations, eachof which is interpreted as a pure thermodynamical phase; in each of these, the energy gap isa non-zero c-number satisfying the self-consistency condition (5.2.5). It is the decomposi-tion that breaks the gauge invariance of the theory: gauge transformations carry purethermodynamical phases onto one another, in such a way that the set of all the pure phasescorresponding to the same temperature ToTc is gauge-invariant.Hence the thermodynamical limit can be controlled: and the BCSHaag model becomesnow an L-model through which the temperature-dependence of the energy gap and thespontaneous symmetry breaking of gauge invariance are proven to be consistent with thebasic interaction mechanism (5.2.1).Liu and Emch (2005) discuss in a philosophical perspective the identication of purethermodynamical phases in quantum statistical mechanics as extremal KMS states, thoseequilibrium states that generate primary representations. This identication was formalizedfrom the lessons of a variety of models, among which the BCS model as discussedin this subsection: thus, in the course of more than a quarter-century, this characteri-zation of pure thermodynamical phases has become part of the core of the quantumtheory of phase transitions; for a review of the reasons for this consolidation; see Emch(2006, Section 5.7).5.4. Beyond low-temperature superconductivity: theory-transferThis would have been the occasion to declare the theory of superconductivity a closedtheory, if only a new experimental discovery had not be made in 1987, namely hightemperature superconductivity in special ceramics, the perovskites. Along this line of21The usage of the word representation here coincides with its use in model theory: the Hilbert spacerepresentations are realizations of the abstract objects called C-algebras. In fact, these objects were rst calledB-algebra (B for Banach), whereas the C in C-algebras was reserved for those concrete B-algebras that were-algebras of operators acting on a Hilbert space; it was later recognized that every B-algebra can be faithfullyrealized as a C-algebra, and the notational distinction faded out.ARTICLE IN PRESSlaboratory investigations, the experimental upper limit for Tc has been pushed up to about130K. It was immediately recognized that this phenomenon may go beyond the purview ofthe BCS model. In yet another case of model-induced inter-theory transfer, Bednorz andMuller, the discoverers of this new phenomenon, had turned to the perovskites becausethese exhibit a very strong distortionknown in molecular chemistry as the JahnTellereffectand thus they conjectured that these materials could also exhibit an essentialingredient of the BCS model, namely strong electron bindings.Yet, another high Tc superconducting materialMgB2was discovered recently, thebehaviour of which could be accounted for directly by the BCS model. The immediatesignicance of this discovery is that, in technological applications, metals are much easierto handle than ceramics; this evidently stimulated much empirical activity (e.g. Caneld &Budko, 2005). In the scope of this paper, note that this experimental discovery alsorenewed theoretical interest in the BCS modeling of superconductivity. Finally, the BCSmodel seems to have found new powers in H-modeling through the analogies it allows toapprehend in yet several other phenomena (Anderson, 1987).Hence, this section illustrates how an H-model, even when is not mathematically correct,may suggest: (a) explanations for the primary phenomenon; and (b) L-models for moregeneral phenomenahere, spontaneous symmetry breakingthat emerge in the repair ofan initial formal shortcut.6. ConclusionsTwo intimately related theses were proposed in Part I. (1) I argued that a deeperunderstanding of models obtains if, instead of considering what models are, one search forwhat they do, and for what purpose. (2) I proposed that it is essential to this understandingthat one observes how most models issue from modeling strategies that fall neatly into twosharply distinct classes; I labelled these classes H-modeling and L-modeling. The conclusionto Part I summarized the general implications of these proposals. In the present Part II, Ishowed how these claims are supported by specic case studies.Here, H-modeling is explicitly demonstrated in more examples than are presented toillustrate L-modeling. This is not accidental, but it reects the very nature of thesestrategies. Indeed H-modeling appeals more immediately to what physicists commonly callintuition, a feature difcult to command, but easier to share when one sets to report onan exploratory mission; moreover, while the computations carried out when solvingH-models are often quite complex, the results are usually straightforward to describe. Onthe contrary, L-modeling necessarily involves mathematical rigour and often abstractconstructions; going into the often esoteric technical details of these exertions is essential totheir convincing power; this is particularly true when L-modeling uncovers inconsistenciesin the canonical core.For each model, I sketched the ambient landscapetheoretical, experimental, or boththat guided the stategy to be developed. I also described the contributions the specicmodeling strategies made to the core of the theory and its eventual insertion into the widerweb of knowledge. Finally, I noted that for a few of the models I discussed, the evolutionof the core caused a shift in the perceived purpose of the model, helping the transfer ofconcepts or theories.I hold that the diversity of the case studies presented in this paper is necessary to clarifyG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723716the issues that modeling strategies are designed to address; and that an awareness of theseARTICLE IN PRESSissues is essential to the resolution of controversies on what a given model doesaccomplish.AcknowledgmentsI wish to record here my appreciation for the Visiting Fellowship tended to me by AllSouls College, for the stimulating hospitality of the Faculty of Philosophy at the Universityof Oxford, and for helpful comments, questions and suggestions from referees.AppendixAs I surely omitted some of the readers favorite models, I list below a few more resultsof each of the two classes of modeling strategies, some of which I studied in earlier papers(Emch, 1993, 1995, 2002, 2003, 2004); see also Emch and Liu (2002) and Emch (2006)where I examined the contributions to modeling made by Wigner, C. F. von Weizsaecker,and E. H. Lieb.H-modeling: In quantum mechanics: the Schrodinger cat; some of the BohrEinsteinGedanken experiments; early explorations of the quantum measurement process. Innuclear physics: the tunnel effect and the emission of a-particles; the HeisenbergWignerisospin; the compound nucleus; the shell model. In condensed matter physics: the imperfectBose gas; the jellium model; the scaling hierarchy (KadanoffWilsonFisher); theGrossPitaevski equation. In special relativity: the longitudinal and transversal Dopplershifts; the adjustments for stellar aberration. In quantum field theory: Feynman pathintegrals. In high energy physics: the Okubo mass formula; the quark model; thestandard model. In cosmology: the Chandrasekhar bound on the mass of white dwarfs;the big bang; Hawking radiation.L-modeling: In nuclear physics: the random matrix model for distribution of energylevels. 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Part II--Case studiesIntroductionSummary of Part IMethodology for Part IIFrom equilibrium thermodynamics to statistical mechanicsEmpirical presentation of the ideal gasMicroscopic modelings for the ideal gasNewtons modeling of the ideal gasBernoullis modeling of the ideal gasMaxwells modeling of the ideal gasConsolidations emerging from the three modelsClausius axiomatization of thermodynamicsClausius axioms (Clausius, 1879)The ideal gas as an L-model for Clausius axiomsIsothermal vs. adiabatic modelings for the speed of soundModelings of real gases: from van der Waals to Lenz-IsingThe van der Waals gasL-modeling for phase transitions: Lenz-IsingNon-equilibrium and ergodic theoryStochastic modeling of the diffusion equationThe dogs-and-fleas modelTwo modelings for ergodicity: Boltzmann and Hedlund-HopfThe Boltzmann modelThe Hedlund-Hopf model (Hedlund, 1935, 1939, 1940; Hopf, 1939; Gelfand & Fomin, 1952, 1955)Modelings in early quantum theoryModelings of the atom: from Balmer to Schrdinger, and beyondPlancks modeling for black-body radiationEinsteins modeling for the photoelectric effectDebyes modeling for the specific heat of solidsSuperconductivity: from H- to L-Modeling and backImmediate experimental backgroundH-modeling: the original BCS modelL-modeling: the BCS-Haag modelBeyond low-temperature superconductivity: theory-transferConclusionsAcknowledgmentsAppendixReferences

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