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Studies in History and Philosophy of

Modern Physics 38 (2007) 683–723

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Models and the dynamics of theory-building inphysics. Part II—Case studies

Gerard G. Emch1

All Souls College, University of Oxford OX1 4AL, UK

Received 4 April 2005; received in revised form 5 October 2006; accepted 10 October 2006

Abstract

In Part I, it was argued that models are best explained by considering the strategies from which they

issue. 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 features

of these modeling strategies; examples are drawn from classical statistical mechanics and quantum physics.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Epistemology; Models; Theory construction

1. Introduction

1.1. Summary of Part I

Considering the view that scientific theories are developing as the nodes of Quine’s 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 cell’s evolution, especially before it becomes firmly embedded inQuine’s web as a ‘‘closed theory’’. For this process to happen, I specified two mainmodeling strategies from which the models proceed. I advanced the following definitions.

Definition 1.1.1. H- modeling aims to explore the connections to be established between thecore—especially the physical postulates—and its physical semantic relevance: in particular

see front matter r 2007 Elsevier Ltd. All rights reserved.

.shpsb.2006.10.005

dress: [email protected].

t address: Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA.

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dx.doi.org/10.1016/j.shpsb.2006.10.005

mailto:[email protected]

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the observability of its concepts and their adequacy to describe the world as apprehendedby laboratory experiments.

Definition 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 assertionsand theorems.

Definition 1.1.3. The products of these two modeling activities are called respectivelyH-models and L-models.

Definition 1.1.4. By default, a mere model designates an H-model or an L-model separated—purposefully or not—from the modeling strategy that was followed to construct it.

1.2. Methodology for Part II

In 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 field of physics.

More models will be briefly listed in the Appendix according to the strategy from whichthey proceed.

2. From equilibrium thermodynamics to statistical mechanics

This 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 gas

I designate by ideal gas what is also called the perfect gas, to emphasize that it is anidealized description of a gas in thermodynamical equilibrium, namely

pV ¼ 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 increments

ðaÞ along isotherms ði:e: at constant temperatureÞ : Dp aDr with r ¼1

VðbÞ along isochores ði:e: at constant volumeÞ : Dp aDT

9=;.

(2.1.2)

2The modern definition 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 carbon

atoms contained in 12 g of C12 is approximately 6:02� 1023; this is called Avogadro’s number. A mole, then, is an

amount of any substance that weighs, in grams, as much as the numerically equivalent atomic weight of that

substance.3The modern value of R is 8:3143� 107 erg deg�1 mol�1.

ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683–723 685

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 away 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.

4

5

the

his

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 Fahrenheit’s remark that Amontons‘‘had discovered that water boils at a fixed degree of heat’’ (Magie, 1969, p. 131) andthat the same holds for other liquids as well, each with its characteristic boilingtemperature.

2.
The discovery—or was it an invention?—of Newton’s (1701) cooling equation_T / ðT � TuÞ, the universality of which is predicated on the assumption that thetemperature is defined up to a change of scale T ! aðT � ToÞ
b 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 theconvenience of a temperature scale in which a ¼ 1; and To ¼ 0 with b ¼ 1, so that theAmontons thermometer is linear in T . This completes the description of the relation 2.1.1which gives the empirical, macroscopic description of the ideal gas.

The purposes of this section are: (i) to present the microscopic modeling strategiesprompted by this empirical description; and (ii) to indicate how real gases may depart fromthis idealization.

2.2. Microscopic modelings for the ideal gas

The following three modelings explore possible microscopic structures of the idealgas in terms of the mechanics of material points. They exhibit two facets of H-modeling.The first facet is its purpose as an explanatory activity: explaining the observable interms 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 specific models unfold, I will point outthe contributions each makes to the development of a core, the microscopic viewof matter.

This was valid until significant deviations were systematically observed by Andrews (1869); see Section 2.5.

This arbitrariness later found a justification in the extensive observations of Gay-Lussac (1802, 1807) and the

oretical considerations of Fourier (1822) and Thomson (1848). Mentioning these here would mask the

torical context of discovery for the modeling strategies to be described in the next subsection.


2.2.1. Newton’s modeling of the ideal gas

The modeling is proposed in Proposition 23, Book II, Section V of Newton’s (1726)Principia. Newton is not forthcoming about the empirical sources of his motivation as hedoes not refer explicitly to Boyle, only mentioning that ‘‘. . . it is established by experimentsthat 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 fluid whose density is proportional to thecompression.’’ (p. 697)

Remarks 2.2.1. (1) As detailed in Guicciardini (2003), Newton argues in a heuristic

manner, characteristic of Euclid: Newton states first what he wants to prove; then hemakes the assumptions that would allow him to prove it; and finally he derives what hesaid he would prove.(2) Note the semantic distinction—quite remarkable for his time—that Newton makes

between the proposition he has just ‘‘demonstrated’’ and the world out there: he commentsthat he is satisfied with his mathematical argument, and that ‘‘. . . whether elastic fluidsconsist 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 first systematic step in what became later the core of a

programme: reducing thermodynamics to some form of mechanics.(4) Yet, Newton models a gas constituted of particles at rest, i.e. in a state of static

equilibrium. This strategic option did not survive subsequent modelings of the phenomena.(5) Moreover, Newton’s hypothesis on the space-dependence of the interparticle forces

did not survive either. Sometimes a model’s 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, Newton’s 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 fixed

temperature, 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. Bernoulli’s modeling of the ideal gas

In contrast to Newton’s 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:

6As

. . . a theorem . . . in which it is shown that in air of any density but at fixedtemperature, the elasticities[6] are proportional to the densities, and further that theincrements of elasticity which are produced by equal changes of temperature are

is still the habit in the xviiith century, the elasticity of the gas means its pressure.

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7Th

(1702)

G.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683–723 687

proportional to the density, this theorem, I say, Dr. Amontons discovered byexperiment and presented it in the Memoirs of the Royal Academy of Sciences ofParis in 1702.7

Compare with the empirical formulas (2.1.1) and (2.1.2).Instead of Bernoulli’s somewhat contrived description of his model [cf. e.g. (Magie,

1969, pp. 247–251)] 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 simplified 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 ¼ 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 ¼ force=surface exercised by the gas on each wall is given by

p ¼1

t� 2mv � n �

1

L2with n ¼

1

6N

L2 � 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 immediately

pV ¼ N2

3

1

2mv2

� �. (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) gives

3

2

Nmole

NR

� �T ¼

1

2mv2. (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)experiments—see the quote at the beginning of this subsection—reminds 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 withmercury, the other with alcohol, could not be calibrated in such a manner that they wouldagree with one another over their entire range. This suggested—and was later found to be

e translation is from (Magie, 1969, pp. 230–231). As for the reference to the ‘‘experiment’’, see Amontons

.


empirically correct—that all substances (and even gases) may not be equivalent for thepurpose 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 Boyle–Mariotte formula2.1.1. Thus, the model, and in particular the identification (2.2.3), may be viewed as a proposalfor a microscopic interpretation of such an empirically defined 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 corrections

to be brought to (2.2.2) to account for the finite 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 is

from the ideal gas obtains when the concept of specific heat is introduced and isquantitatively mastered. This concept is usually attributed to Black (1803). Yet, Black(1728–1799) mentions that after starting his own thinking about the subject in 1760, herealized others had trodden along the same road, prompted by Boerhaave’s 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 definition, the modern version of which reads: the specific 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 Bernoulli’s Hydrodynamica (1738) is far anterior tothis modern definition, and even to Black’s penetrating conclusions (published in 1803),Hydrodynamica is essentially contemporaneous to the motivating empirical evidencesadvanced by Boerhaave ðy1738Þ, by Fahrenheit ðy1736Þ and, especially, by Martineðy1741Þ.(4) Consider now—a posteriori—the relevance of the concept of specific heat to

ascertain the scope of the Bernoulli model. In this model there are, by definition, 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 gas

and thus according to ð2:2:3Þ :UðV ;TÞ � N1

2mv2

UðV ;TÞ ¼ 32

NmoleRT

). (2.2.4)

Hence the specific heat per mole, at constant volume is

CV �1

NmoleqT UðV ;TÞ ¼

3

2R and thus in particular qT CV ¼ 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 its

consequence qT CV ¼ 0 in (2.2.5), that whether the specific heat CV of a real gas is constantor not can be tested in the laboratory. Yet, the methods of calorimetry had to be

8Most likely, Martine (1702–1741). Indeed, Black gives for his source a paper entitled Essay on the heating and

cooling of bodies; a paper with that title is reproduced in the posthumous volume (Martine, 1780, pp. 53–90).


considerably refined until Regnault (1853) was able to carry out the experiments withsufficient 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 specificheat—albeit originally predicated on the caloric theory—survives 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. Maxwell’s modeling of the ideal gas

Like 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 Bernoulli’s modeling is that thethree components of the velocity v of the particles are now assumed to be—in the languageof modern probability theory—independent, identically distributed, Gaussian randomvariables.

Scholium 2.2.1. If f : v 2 R3 7!fðvÞ 2 Rþ is a smooth function satisfying the following three

properties:

1.
f is normalizable, i.e.R
R3 dv1 dv2 dv3fðvÞo1;
2. the three components of v are mutually independent, i.e. fðvÞ ¼ f1ðv1Þf2ðv2Þf3ðv3Þ; 3. f is isotropic, i.e. fðvÞ ¼ Fðkvk2Þ.
Then fðvÞ ¼ Að2pa2Þ�3=2 expf�12a�2kvk2g where A and a are two free constants.

Maxwell then recognizes that for f to model the velocity distribution in a gas of N particlescontained in a box of volume V the constant A must satisfy the conditionR

R3 dv1 dv2 dv3fðvÞ ¼ N=V i:e: A ¼ 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 ¼ ð1=V Þ

Rv140 dv1

RRR2 dv2 dv3fðvÞð2mv1Þ � v1. One obtains then

a2 ¼ kT=m, with k ¼ ½Nmole=N�R. Hence the result now known as the Maxwelldistribution

fðvÞ ¼N

V

m

2pkT

� �3=2exp �

m

2kTkvk2

n 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 Bernoulli’s v is reinterpreted as an average with respectto the distribution (2.2.6). In particular, 1

2kT is now to be identified with the mean kinetic

energy per degree of freedom. Again, the specific heat CV is constant; and its value 32

R

remains the same; compare to (2.2.5).

Remarks 2.2.3. (1) To place Maxwell’s modeling strategy in the wider context of nascentstatistical thinking note that, while still a student, Maxwell read John Herschel’s 1850review of Quetelet’s 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,paraphrasing his result (here Scholium 2.2.1), Maxwell remarked in his 1860 paper:


‘‘the 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) Maxwell’s a priori approach—namely Scholium 2.2.1, together with the natural

adjustment of constants—to the distribution (2.2.6) is corroborated by the role thisdistribution plays in Boltzmann’s kinetic theory of gases; see e.g. Emch and Liu (2002,Section 3.3). Yet, Boltzmann’s 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 thermodynamical

consequences, 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 models

All three models—Newton’s, Bernoulli’s, and Maxwell’s—were primarily conceived asexploratory: they are H-models. Newton’s was certainly more tentative than the other two.Bernoulli’s 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. Refining Bernoulli’s, Maxwell’s model was sufficiently realisticfor its microscopic tenets to be tested empirically. Still, both Bernoulli’s and Maxwell’smodels neglect possible interactions between the particles themselves; compare withSection 2.5.While Maxwell’s distribution models an equilibrium situation, later on Maxwell

suggested 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 ’ ðrd2

Þ�1 where d is the diameter of the

particles, and r ¼ 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 ¼ N=Nmoles, the number of molecules in a mole.Thus, the macroscopic consequences of this chain of H-modeling turned out to be an

essential contribution to the core, namely the postulate that gases (later fluids, and ultimately

all matter), have a statistical, atomistic description (e.g. Emch & Liu, 2002; or Truesdell,1980). Yet, this raised an ontological debate that lingered until Einstein’s 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 thermodynamics

I 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 gas—independently of any of the microscopic interpretations associated to the H-models


discussed in the previous subsection—appears 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 ¼ fðV ;TÞ 2 Rþ � Rþg by

Z ¼ LV dV þ CV dT and t ¼ pdV , (2.3.1)

where LV ;CV and p are smooth real-valued functions of ðV ;TÞ; and require that Z and tsatisfy, along all smooth simple closed contours G � D, the two conditions

ðiÞ

ZGðZ� tÞ ¼ 0 and ðiiÞ

ZG

1

TZ ¼ 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 ¼ T dS

and t ¼ �dU þ Z;(2) qT ðLV � pÞ ¼ qV CV , LV ¼ TðqTLV � qV CV Þ, LV ¼ TqT p.

2.3.2. The ideal gas as an L-model for Clausius’ axioms

To prove the internal consistency of the Axioms 2.3.1, an L-model is constructed: theideal gas (here with Nmole ¼ 1). It is defined by two constitutive relations, involving astrictly positive constant R

ðiÞ pðV ;TÞ ¼ RT

Vand ðiiÞ qT CV ¼ 0. (2.3.3)

Scholium 2.3.1. The first of these two constitutive relations already entails

LV ¼ p; qV U ¼ 0 and qV CV ¼ 0. (2.3.4)

And the two constitutive relations together entail

the function CV is constant, ð2:3:5Þ

UðTÞ ¼ CV ðT � ToÞ þUðToÞ, ð2:3:6Þ

SðV ;TÞ ¼ CV lnT

To

� �þ R ln

V

Vo

� �þ SðV o;ToÞ. ð2:3:7Þ

Remarks 2.3.1. (1) While the first 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 ¼ T dS and t ¼ �dU þ Z and one verifies thatthese differentials satisfy the conditions (2.3.1) and (2.3.2). Hence an L-model for Axioms2.3.1 has been obtained.


(2) Theorem 2.3.1 is a consequence of the syntax alone. Its derivation proceeds according onlyto 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 specific 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 times—instead of points, straight lines, and planes—tables, chairs, and beer mugs’’.10 Evidently, Hilbert (1899) is but a prototype of Hilbert’ssystematic axiomatization programme, while Clausius’ intentions here are plainly limited to aformalization of thermodynamics of homogeneous media. It is also evident that, in contrast toHilbert, Clausius uses rather indiscriminately Satz to signify various assertions thatmathematicians, even mathematical physicists, are now used to distinguish carefully. Never-theless, see Remark (5) below for one ontological distinction upon which Clausius does insist.(3) Note also that even the proof of Scholium 2.3.1 itself uses only the formal aspect of

the constitutive relations (2.3.3) and is not predicated on the semantics that guided thechoice of precisely those relations, nor does it depend on any specific value of the strictlypositive constant R.(4) The question could be raised as to why the constitutive relations (2.3.3) should

not be moved up into the core. I wish to argue against such a move. The context in whichthe ideal gas appears here allows one to distinguish between two types of changes

of variables.

(a)

9F10

A syntactic change of variables. The choice of the variables V and T to describe thestate-space—i.e. the domain D on which the functionals Z and t are defined in Axioms2.3.1—is 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 thesemantics, 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. 61–66) and in the present contextWannier (1966, pp. 134–138).

(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 describe—insteadof an ideal gas—a perfect, or ideal, paramagnet with magnetic moment M embedded in amagnetic field H. As an illustration, the usual discussion of the Carnot cycle can befaithfully translated into this dialect (e.g. Wannier, 1966, pp. 122–124).
(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 ¼ ð1=TÞZ and dU ¼

or the choice of the words heating and working, see remark (5).

As 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 mathematicalingredient—the notion of differentials that are not necessarily ‘‘exact’’—enters thesyntactic core of a specific physical theory. With hindsight, precursors can be traced backin other fields, e.g. in the works of Clairaut,11 Ampere and Green. The contributionClausius made here to the core of thermodynamics is to introduce a language in which onecould 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)

11

he sa

(it is

that12

appe

distr

theo

Lieb

from

in U

The presentation is original with Clausius (1879, Chapter V). It is a new departure inClausius’ 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).

(b)
In the last quarter of the xixth century, the ideal gas had a bad press, exemplified 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. 202–219).
(c)
In his 1879 presentation, Clausius’ appeal to the ideal gas is plainly limited to offeringan L-model, i.e. a mathematical structure for which the mathematical Axioms 2.3.1 aresatisfied. 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 forbringing to bear on fundamental issues a structure—the ideal gas—that is onlyapproximatively met in the actual world. This empirical discrepancy cannot detractfrom Clausius having provided hereby a proof that his axioms are mutually consistent.
(7) So far, the entropy S does not depend on time; it is a function defined on theparameter space of equilibrium states. The L-model provided by the ideal gasallows 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 V

is larger.12

For instance, Clairaut (1740, p. 294) notes that: the smooth differential o ¼ Pðx; yÞdyþQðx; yÞdx is exact—

ys complete—i.e. there exists a function F such that o ¼ dF iff qxP ¼ qyQ; for instance: xdyþ ydx is exact

the differential of xy); but xdy� ydx is not exact; nevertheless, it admits an integrating factor, namely x�2

makes it exact: x�2 � ðxdy� ydxÞ ¼ dðy=xÞ.

The notion of entropy has an amazing history. First introduced by Clausius, it was later extended when ð�SÞ

ars as the limiting equilibrium value of Boltzmann’s H-functional defined on the space of time-dependent

ibutions f on the microscopic phase space. Deeper justifications for the nature and role of entropy in the

ry of non-equilibrium thermodynamics were proposed still later, from Caratheodory (1909) right through

& Yngvason (1998, 1999, 2000); one of the achievements of the latter is to cleanse general thermodynamics

heretofore pervasive intrusions of quasi– static transformations. Collateral philosophical issues are pursued

ffink (1995, 2001), and in Brown and Uffink (2001). See also Part I, Section 3.2.


2.4. Isothermal vs. adiabatic modelings for the speed of sound

The first analytic understanding of the speed of sound in fluids is due to Newton (1726,Book II, pp. 769–779), particularly Propositions 47–50, and the last Scholium of Section 8.The most immediate parts of the ambient landscape are Newton’s own theory of harmonicmotion and his mechanics of fluids. 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, specifically a harmonic

compression wave in an elastic fluid. To explore the empirical consequences of thistheoretical proposal, he calls upon a modeling strategy—an H-modeling—that consists ofthree steps. First, he proposes the following relations between the speed v of the soundwave, the elastic force k of the fluid, its density r, and the pressure p, namely

v ¼

ffiffiffikr

rwith k � r

qp

qr. (2.4.1)

The second step in Newton’s H-modeling is to posit that the pressure is proportional to thedensity

p / r, (2.4.2)

that is Boyle’s 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 ¼

ffiffiffip

r

r. (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 on

vN ¼ 979 f=s and

ffiffiffip

r

r¼ 1142 f=s (2.4.4)

a discrepancy he attempts to correct with two hypotheses.13 Here is the third step inNewton’s modeling strategy. The first hypothesis is meant to account for the ‘‘crassitude’’of air, namely the fact that the particles of air are of finite 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 21

20. Newton does not justify separately either of these corrections; they

just combine to give

13Despite his famous, but often misinterpreted, dictum hypotheses non fingo (Newton, 1726, p. 764), Newton

also admitted that ‘‘the best and safest method of philosophizing seems to be, first diligently to investigate the

properties 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 BCS

model, Section 5.1.


Scholium 2.4.2.

v ¼ ðvN þ 109Þ �21

20i.e. with vN ¼ 979 f=s : v ¼ 1142 f=s. (2.4.5)

The agreement between prediction and experiment is verified now within one part in a

thousand! 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 Newton’s theory, namely (2.4.1), Laplace and Poissonupdate Newton’s assumption (2.4.2) to read as follows:

Scholium 2.4.3. If

p / rg with g ¼Cp

CV

, (2.4.6)

where, since air is a diatomic gas, CV ¼52

R and Cp ¼ CV þ R ¼ 72

R; then (2.4.3) becomes

v ¼

ffiffiffip

r

r ffiffiffigp¼ vN �

ffiffiffigp

withffiffiffigp¼

ffiffiffiffiffiffiffi1:4p

¼ 1:183. (2.4.7)

Within the probable precision of the experiments, (2.4.7) can be taken to resolve theempirical discrepancy 1142C979�1:166—see (2.4.4)—that plagued Newton’s originalmodel.

Remarks 2.4.1. (1) Newton’s (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 justification for the Laplace–Poisson 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 phase–space 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 ¼ 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 ofNewton’s two ad hoc hypotheses summarized in (2.4.5); (ii) it refines the interpretation ofthe core (2.4.1) of Newton’s theory as it adds to the core the distinction between anisothermal process and an adiabatic one, i.e. between a process that occurs at constanttemperature and one that occurs at constant entropy (i.e. without exchange of heat).


(5) Since Newton’s general theory, as summarized in (2.4.1), was not in doubt whenLaplace–Poisson 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 Laplace–Poisson model, the observable k in Newton’s relations (2.4.1)

now is the adiabatic—rather than the isothermal—compressibility. 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– Ising

By 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 ðpc;V c;TcÞ, which he characterizedby the appearance of an opalescent ‘‘glow’’; (ii) the flattening 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 magnetic

substances such as iron loose their magnetism above a material–specific 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 be

taken 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 gas

van der Waals (1873) proposed an H-model to improve on the ideal gas, namely

pþ aN

V

� �2" #

½V �Nb� ¼ NkT , (2.5.1)

where a and b are substance-specific 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).14

The 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 would

14With the notational substitution ðNmole=NÞR ¼ k.


increase the volume. On the isotherm T ¼ Tc, qV po0 except for exactly one critical valueof V ¼ V c at which qV p ¼ 0.

Scholium 2.5.1. With the notations pc ¼ pðvc;T cÞ, vc ¼ V c=N, the critical values of the

specific volume, the temperature and the pressure are given by vc ¼ 3b, kT c ¼ 8a=27b,pc ¼ a=27b2; and thus satisfy, for all values of a; b; c the universal relation

pcvc

kT c¼

3

8. (2.5.2)

Remarks 2.5.1. (1) The prediction (2.5.2) is fairly well verified for real substances, even asthe coefficients 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 ¼ v=vc (with v ¼ V=N ), T ¼ T=T c, and p ¼ p=pc, oneobtains the law of corresponding states

pþ 31

v2

� �v�

1

3

� �¼

8

3T (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 justification for the model given by van der Waals, later refined 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 fluid and a simultaneous limit where thestrength of the interaction vanishes. Weiss (1907) transferred van der Waals’ modelingfrom fluids to ferromagnets: each spin is viewed as being submitted to the mean magneticfield 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 field approximation.’’ All H-models constructed alongsuch lines explore the putative existence of a collective behavior that was later proven

analytically 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 thermodynamicalargument—namely the Maxwell construction (e.g. Emch & Liu, 2002, pp. 382–384)—toobtain an isothermal coexistence plateau which is attached to the corresponding isothermthrough a first-order discontinuity. As temperature varies, these discontinuities trace asmooth coexistence curve in the plane ðV ; pÞ, with its maximum at the critical point. Thisfits qualitatively well with the empirical behavior of real fluids.

(5) The construction of the Maxwell plateau is a theoretical artifact that can be avoided.This was demonstrated by the Kac–Baker L-model (e.g. Emch & Liu, 2002, pp. 388–389).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: cattr

ðjrjÞ ¼ �aol expð�ljrjÞ; 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 with

15This role is documented in Kammerlingh Onnes’ Nobel lecture (1913).


reasonably finite-range interactions: (Ruelle, 1968, 1969; Takahashi, 1942; van Hove,1950).(6) The van der Waals model gives an empirically poor description of the fluid in the

immediate vicinity of the critical point where various divergences are observed,among them the critical opalescence discovered by Andrews (1869). To these divergencesare associated critical coefficients a;b; g and d—for their definitions, cf. e.g. Emchand Liu (2002, pp. 385–386)—the individual values of which, when computed fromthe van der Waals model, are grossly falsified by laboratory experiments. Yet, when takentogether, the predicted values still satisfy relations that are borne out by experiments,namely

aþ 2bþ g ¼ 2 and aþ bð1þ dÞ ¼ 2, (2.5.4)

thus pointing to some possible universality that would—and 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 gas’isotherms. It suggests specifically several directions in which the core ought to bedeveloped. Indeed, (i) it reveals some seemingly universal macroscopic features of realfluids: (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 fidgetybehavior of the fluid in the immediate neighborhood of this critical point; and (iii) itanticipates the emergence of long-range order in condensed matter.

2.5.2. L-modeling for phase transitions: Lenz– Ising

The 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 difficulties. The modeling strategy is to consider ‘‘particles’’that 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 configurations where neighboring sitesare simultaneously occupied; and (iii) the system is invariant under lattice translations.Specifically, one associates to every finite cube L � Zd the configuration space f0; 1gL ¼fn : i 2 L7!ni 2 f0; 1gg and the partition function ZL ¼

Pn2f0;1gL expð�b ½�mNLðnÞ þ

HLðnÞ�Þ where b ¼ 1=kT is the natural temperature; m is the chemical potential;NLðnÞ is the number of occupied sites in L; HLðnÞ is the energy of the configurationn 2 f0; 1gL.The change of variables s ¼ 2n� 1 : i 2 Zd 7!si ¼ ð2ni � 1Þ 2 f�1;þ1g associates to

every site i 2 Zd a ‘‘classical spin’’ si pointing either up si ¼ þ1 or down si ¼ �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 confines of statistical mechanics,the magnetization

MLðb;B; JÞ ¼ jLj�1X

s2f�1;þ1gLhsii, (2.5.5)


where jLj is the number of sites in L and h�i is the canonical equilibrium averagecorresponding to the partition function

QLðB; J;bÞ ¼X

s2½�1;þ1�Lexp½�b HLðsÞ� with HLðsÞ ¼ �B

Xi2L

si � JXhi;ji

sisj, (2.5.6)

where B is an external magnetic field, J40, andPhi;ji sums over pairs of neighboring sites

in L.Ising (1925) showed that when d ¼ 1 this model is exactly solvable and that the resulting

magnetization shows no singularity, even when the system becomes very large. Specifically,QLðB; J;bÞ ¼ ljLjþ þ ljLj� with l ¼ ebJ cosh bB ðe2JB sinh2 bBþ e�2bJ Þ

1=2 from which onegets:

Scholium 2.5.2. At fixed J and b, MLðB; J;bÞ is analytic in B and MLð0; J; bÞ ¼ 0.Moreover, MðB; J;bÞ ¼ limjLj!1MLðB; J;bÞ ¼ tanh bB½1� ð1� e�4bJÞ cosh�2 bB��1=2

and thus MðB; J;bÞ is analytic, with Mð0; J;bÞ ¼ limB!0MðB; J;bÞ ¼ 0.

Hence, even in the infinite 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 finite 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 sufficient. Indeed, with d ¼ 2, Yang and Lee (1952)proved that as long as the regions L remain finite, analyticity of the isotherms persists for theLenz–Ising 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 infinite. In this limit, one has finally:

Scholium 2.5.3. The two-dimensional Lenz– Ising model exhibits a critical temperature T c,given by

sinh 2J1

kT c� sinh 2

J2

kT c¼ 1

below which the model exhibits a non-vanishing spontaneous magnetization satisfying

MOðJ;bÞ ¼ 1� sinh 2J1

kT� sinh 2

J2

kT

� ��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.Emch and Liu (2002, Sections 12.1, 12.2).


The 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 theory

3.1. Stochastic modeling of the diffusion equation

The diffusion equation governs the flow 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 one–dimensional diffusion equation

qt f ¼ vqx f þDq2x f with ðx; tÞ 2 R� Rþ, (3.1.1)

it is useful to note that the distribution

f ðx; tÞ ¼1ffiffiffiffiffiffiffiffiffiffiffi4pDtp exp �

1

4Dtðx� vtÞ2

� �(3.1.2)

solves (3.1.1) subject to the initial condition that at time t ¼ 0 the distribution f isconcentrated at the origin. For all t40, the average position and mean square-deviation

hxit �

ZR

dx f ðx; tÞx and hðx� hxitÞ2it �

ZR

dx f ðx; tÞðx� hxitÞ2, (3.1.3)

when computed with respect to the distribution f in (3.1.2), are proportional to the elapsedtime t:

hxit ¼ vt and hðx� hxitÞ2it ¼ 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 ¼ sa j s 2 Zg; time is a discrete variable t ¼ nt with n 2 Zþ; and p

and q are positive numbers satisfying pþ q ¼ 1. Assume now that a particle moving on Z

may, at every time t ¼ 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 ðsa; ntÞ that the particle be atsite sa at time nt satisfies the equation

f ðx; tþ tÞ ¼ pf ðx� a; tÞ þ qf ðxþ a; tÞ. (3.1.5)

With the initial condition that the particle be at the origin at time t ¼ 0, the solution of (3.1.5) is

f ðsa; ntÞ ¼

n1

2ðnþ sÞ

0@

1A pðnþsÞ=2qðn�sÞ=2 when jsjpn;

0 otherwise:

8>><>>: (3.1.6)


With respect to this distribution, the average position of the particle and the mean-squaredeviation at time t are proportional to t

hxit ¼ ðp� qÞa

t

h it and hðx� hxitÞ

2it ¼ 4pq

a2

t

� �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 satisfy

ðp� qÞa

t¼ v and 4pq

a2

t¼ 2D, (3.1.8)

then, in the joint limit

lima!0

a2

t¼ 2D and lim

a!0

1

aðp� qÞ ¼

v

2D(3.1.9)

the distribution (3.1.6) converges to the distribution (3.1.2); and Eq. (3.1.5) converges to the

diffusion equation (3.1.1).

Remarks 3.1.1. (1) The mathematical trimmings necessary to prove the existence of theselimits are discussed in Kac (1946, 1947a, 1947b); the extension to higher dimensions ðd41Þis straightforward.

(2) Does this model proceed from L-modeling or from H-modeling? Here again, theanswer depends on the context.

(a)

16

If one is satisfied with a statistical explanation, the model belongs to L-modeling.
(b) If one requires an explanation in terms of the emergence of an irreversible macroscopic
approach 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 confirm 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 refined 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 fluid,satisfies the characteristic diffusion relations (3.1.4). The best early experimental set-up—cf. 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 of

magnitude, provided time and space be rescaled according to t�x2, thus most elegantlyconfirming the mutual consistency of the two relations (3.1.4).

(4) Upon considering the collisions between the few heavy colloidal particles and themany—presumably—very light molecules of the fluid 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 diffusion

For further references cf. Emch & Liu (2002, pp. 86–87).


constant the interpretation D ¼ kT=6pr Z where k ¼ R Nmole=N is the Boltzmannconstant; T is the temperature; r is the radius of the suspended particles; and Z is theviscosity of the fluid. Thus the Einstein–Smoluchowski modeling combined with the Perrinlaboratory measurements led to trustworthy estimates of the Avogadro number NAv andconsequently of the number of molecules in a fluid. In the judgement of thecontemporaries, the Einstein–Smolukowski model qualified as an L-model that could beused conclusively against a persistent minority refractory to the existence of atoms asactual particles with measurable finite dimensions. Henceforth, the Einstein–Smolukowskimodel also contributed to the growing acceptability of stochastic modeling in physicaltheories.

3.2. The dogs-and-fleas model

Theoretical background of the model: This stochastic model was conceived in response totwo objections against Boltzmann’s kinetic theory of gases, namely the reversibilityobjection, or Umkehreinwand, of Loschmidt (1876, 1877), and the recurrence objection, orWiederkehreinwand, of Zermelo (1896a, 1896b). Boltzmann’s responses to these twoobjections—respectively, Boltzmann (1877, 1878a, 1878b) and Boltzmann (1896, 1897)—provide much of the motivation for the dogs-and-fleas 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 fleas 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 flea jumps fromone dog to the other.

Mathematical formulation of the problem: To find out the time-dependence of thenumber xt of fleas on dog A, knowing its initial value xo at time t ¼ 0. Since exactly oneflea jumps at every call, the transition probability Pm;n ¼ Probfxtþ1 ¼ n j xt ¼ mg vanishesunless jn�mj ¼ 1; finally, the probability that the flea jumps from the dog on which it sitsis proportional to the number of fleas on that dog. Consequently, for all tX0, theprobability pðxtþ1 ¼ nÞ that there are n fleas on dog A at time ðtþ 1Þ satisfies

pðxtþ1 ¼ nÞ ¼Xn

m¼1

Pm;n pðxt ¼ mÞ with Pm;n ¼

1

Nm for n ¼ m� 1;

1

NðN �mÞ for n ¼ mþ 1;

0 otherwise:

8>>>><>>>>:

(3.2.1)

Exact properties of the model:

Scholium 3.2.1 (Uniqueness of equilibrium). The evolution equation (3.2.1) admits exactly

one stationary solution, namely: pðnÞ ¼ 2�N Nn

where N

n

is the binomial coefficient

N!=ðN � nÞ!n!.

Scholium 3.2.2 (Tendency to approach equilibrium). For any value of n larger (resp. smaller)than the equilibrium N=2, the probability Pmax

n for the string fn� 1; n; n� 1g to occur in the


course of the evolution is larger (resp. smaller) than the probability Pminn that the string

fnþ 1; n; nþ 1g occurs.

Scholium 3.2.3 (Exponential approach to equilibrium). Let hxti �PN

n¼1 n pðxt ¼ nÞ be the

average number of fleas on dog A at time t. Then hxti satisfies the difference equation

hxtþ1i ¼ 1þ ð1� 2=NÞhxti which, for any initial condition hxt¼0i ¼ xo with 1pxopN,admits the unique solution jhxti �N=2j ¼ jxo �N=2j expð�gtÞ where g ¼ � lnð1� 2=NÞ40.

Scholium 3.2.4 (Time-reversal symmetry). The backward transition probability Pfxt�1 ¼

n j xt ¼ mg � pðnÞ=pðmÞ � Pfxt ¼ m j xt�1 ¼ ng is equal to the forward transition probability

Pfxtþ1 ¼ n j xt ¼ mg. Moreover, the probability Pupn for the string fn� 1; n; nþ 1g to occur

in the course of the evolution is equal to the probability Pdownn that the string fnþ 1; n; n� 1g

occurs.

Scholium 3.2.5 (Recurrence). Let f kðnÞ be the probability that a semi-infinite sequence fxt j

t 2 Zþg starting with xo ¼ n comes back to n for the first time at time t ¼ k40. Let Yn �P1k¼1kf kðnÞ be the mean recurrence time. Then

1.
P1
k¼1 f kðnÞ ¼ 1, i.e. every initial condition surely recurs;
2. Yn ¼ 2Nn!ðN � nÞ!=N!; hence for any n moderately away from the equilibrium value N=2
the recurrence time rapidly becomes huge as N gets large.

Scholium 3.2.6 (Fluctuations in recurrence times). Let Ln �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP1k¼1ðk �YnÞ

2 f kðnÞ

qbe the

variance of the recurrence time. Then, for N large and n ’ N: Ln=Yn ’ 1, i.e. in this regime

the average recurrence time loses much of its empirical meaning.

Computer implementation: Individual trajectories governed by (3.2.1) had beencomputed, so to speak by hand (e.g. Kohlrausch & Schrodinger, 1926). The tediousnessof the task surely discouraged investigators to go beyond a few hundred iterations,and thus prevented them from confronting the predictions about very long-time behaviour.For N ¼ 100, Rick Smith and I implemented computer runs using a random numbergenerator to govern the drawing of the fleas. The digital observations of long strings ofsuccessive values of x are reported graphically and tabulated in Emch and Liu (2002,pp. 109, 111): (i) for runs of only a few hundred steps, a distinct exponential approach toequilibrium is observed, and the decay coefficient g matches the value predicted inScholium 3.2.3; (ii) in contrast, for runs of the order of 105 steps, the process looks likerandom noise; yet, our digital recordings of all the events allowed us to track recurrences;the observed recurrence-times corroborate Scholium 3.2.5. Hence, in these computerexperiments, N ¼ 100 is large enough to discriminate between two time scales: one, inwhich the process shows an approach to equilibrium; and the other, in which recurrencesbecome manifest. Yet N ¼ 100 is small enough to avoid the model’s weakness identified inScholium 3.2.6. The same conclusions obtain when we read the records backward, inagreement with Scholium 3.2.4.

Remarks 3.2.1. Here again, we have a specific case that shows the importance of thecontext in deciding a posteriori whether a model is to be understood as proceeding fromL- or H-modeling


1.
In the wide context of the theory of stochastic dynamical systems, the dogs-and-fleasmodel is an L-model establishing that the following properties are compatible:(a) an intrinsically time–symmetric evolution (Scholium 3.2.4) with a unique time–
invariant (or equilibrium ) state (Scholium 3.2.1); the latter plays here the role playedby the Liouville measure of the Poincare Hamiltonian dynamics in the argument ofZermelo (1896a, 1896b);

(b) a tendency to approach equilibrium (Scholium 3.2.2) confirmed by a monotonic,exponential decay of mean quantities (Scholium 3.2.3);

(c) individual trajectories are surely recurrent (Scholium 3.2.5).
2. With regards to the much more demanding context of statistical mechanics, the dogs-
and-fleas model deliberately ignores one postulate of this theory, namely the existenceof an underlying deterministic Hamiltonian mechanism. It could thus only beconceived—and it was indeed—as an H-model to explore issues in the kinetic theoryof gases. Its rigorously derived properties suggest that:(d) the objections of Loschmidt and Zermelo involve more than just classical

mechanics;(e) Boltzmann’s rejection of these objections correctly advocates their empirical

inaccessibility, in particular calling, as Boltzmann does, for distinctions betweentime-scales.

3.3. Two modelings for ergodicity: Boltzmann and Hedlund– Hopf

The early history of ergodicity in Hamiltonian systems is framed by two L-models: oneproposed by Boltzmann, the other being the geodesic flow on a compact surface ofconstant negative curvature. I will refer to them in this subsection as the Boltzmann model

and the Hedlund– Hopf model.

3.3.1. The Boltzmann model

This model was advanced by Boltzmann (1871) to argue that what is now known as thestrict ergodicity hypothesis is compatible with an underlying Hamiltonian evolution.Consider a pair of uncoupled harmonic oscillators with irrational relative frequencies;

the total Hamiltonian of this system is H ¼ H1 þH2 with Hk ¼12

p2k þ

12okq2

k whereðk ¼ 1; 2Þ; the condition that there are no resonances is that there are no pairs ðn1; n2Þ ofnon-zero integers such that

Pknkok ¼ 0. Since Hk are separately constants of the motion,

the total energy surface OE ¼ fðp1; q1; p2; q2Þ 2 R4 j Hðp1; q1; p2; q2Þ ¼ Eg foliates into time-invariant two-dimensional surfaces

OðE1;E2Þ ¼ fðp1; q1; p2; q2Þ 2 R4 j H1ðp1; q1Þ ¼ E1;H2ðp2; q2Þ ¼ E2 with E1 þ E2 ¼ Eg.

Restricted to any of these surfaces the Hamiltonian evolution generates a flow isomorphicto the parallel flow on the two-dimensional torus T2 ¼ R2=Z2

t 2 R7!jðtÞ ¼ ðj1ðtÞ;j2ðtÞÞ 2 T2 with jkðtÞ � jkð0Þ þ1

2pokt mod 1. (3.3.1)

Remarks 3.3.1. (1) Boltzmann (1871) infers from the condition that o1=o2 is irrationalthat this flow is ‘‘ergodic’’ in the sense that each trajectory (3.3.1) covers the whole torusT2. Hertz (1910) and Plancherel (1912a, 1912b) prove this to be false: the failure of thisputative L-model for ergodicity is not accidental, and actually uncovers an intrinsic


inconsistency in Boltzmann’s ergodic hypothesis. Indeed, non-periodic trajectories inphase–space are continuous non-intersecting curves (i.e. they are injective topologicalmaps from R to T2); and to assume that one of these curves covers the space is to assumethat, in addition, these maps are surjective. But a general theorem proven by Brouwer(1910, 1912a, 1912b) asserts that dimension is a topological invariant under bijection.Since dimðRÞ ¼ 1 and dimðT2Þ ¼ 2, the assumption that any one-dimensional trajectorycovers T2 cannot be correct. Note also that Cantor (1878), Peano (1890), and Hilbert(1891) had constructed L-models showing that both continuity and bijectivity arenecessary assumptions for Brouwer’s result. These arguments extend to all cases where theenergy surface is of dimension 2 or larger.

(2) The above mathematical refutation of Boltzmann’s strict version of the ergodichypothesis may have been bypassed on account of Boltzmann’s finitism (cf. Dugas, 1959);or on account of putative reservations on the empirical relevance of sets of measure zero(cf. von Plato, 1994). These and similar philosophical options—discussed e.g. in Gallavotti(1994) and Uffink (2006)—were mostly ignored by mathematicians and physicists. Historychose instead to examine more closely the actual properties that the model does satisfy.

(3) It turned out that Boltzmann’s analysis was not far off the mark. The condition thatthe oscillators have irrational relative frequencies entails that orbits fjðtÞ j t 2 Rg are denseproper subsets of T2; i.e. in the course of time, the orbit will come arbitrarily close to everyc 2 T2, although it is not true in general that there exists a time T such that jðTÞ ¼ cholds exactly. This is to say that, in a probabilistic sense, j acts transitively on T2; i.e. thatthere is no measurable subset S � T such that: (i) 8t 2 R : S ¼ jt½S�, wherejt½S� ¼ fjtðxÞ j x 2 Sg; and (ii) mðSÞa0 or 1, with dm ¼ dx1 dx2. Thus, the rehabilitationof Boltzmann’s construct as a L-model required some mathematical sophistication thatmatured only in the 1930s.

Definition 3.3.1. (i) A triple ðM ;m; aÞ is said to be a dynamical system whenever a : ðt; xÞ 2R�M 7! at½x� 2M is a measure preserving action on a probability space ðM ;mÞ.

(ii) ðM ;m; aÞ is said to be ergodic whenever the only measurable subsets A M which arestable under a—i.e. satisfy at½A� ¼ A for all t—are of measure 0 or 1.

(iii) ðM; m; aÞ is said to be mixing whenever limt!1 mðA \ at½B�Þ ¼ mðAÞmðBÞ for all pairsðA;BÞ of measurable subsets of M, with measure not 0 or 1.

(iv) A Hamiltonian system is said to be ergodic whenever each orbit of the Hamiltonianflow is dense on its energy surface.

(4) Now, ðT2; dx1 dx2;jÞ is an ergodic dynamical system in the sense of definition (ii)above. However, the original Hamiltonian system—two uncoupled harmonic oscillators withirrational relative frequencies—is not ergodic, since each energy shell with E40 splits intostable leaves, due to the existence of the additional constants of the motion H1 and H2.

(5) Moreover the model ðT2; dx1 dx2;jÞ, although ergodic, fails to be mixing. Indeed, withk ¼ 1; 2; let ðS1Þk ’ R=Z and IA; IB be two small intervals in ðS1Þ1; consider the two thinstrips across T2�ðS1Þ1 � ðS

1Þ2: A ¼ IA � ðS1Þ2 and B ¼ IB � ðS

1Þ2; then mðA \ jt½B�Þ isperiodic in t, thus violating condition 3.3.1(iii) above. Hence, this L-model serves a furtherpurpose in the theory of dynamical system: it establishes that ergodicity does not imply mixing.

To sum up: while ðT2;dx1 dx2;jÞ is a bona fide L-model for ergodicity in the generalframework of dynamical system, it fails to establish the compatibility of ergodic hypothesis


with the basic mechanical tenets of Boltzmann’s kinetic theory of gases. This is still a hardproblem (e.g. Sinai, 1963, 1970).

3.3.2. The Hedlund– Hopf model (Hedlund, 1935, 1939, 1940; Hopf, 1939; Gel’fand &

Fomin, 1952, 1955)

This is a bona fide L-model of a Hamiltonian system that satisfies the ergodic hypothesis.It presents two essential features which involve mathematical developments that were notavailable to Boltzmann: the differential geometry of Riemann and the group theory of Lie;see Emch and Liu (2002, pp. 287–293).The first ingredient is an observation made by Hadamard (1898), namely that whereas

neighbouring straightlines in a flat manifold, say the Euclidean plane R2, diverge from oneanother linearly in time, neighbouring geodesics in a Riemannian manifold of negativecurvature, say the Poincare half-plane P, diverge exponentially, thus entailing very strongsensitivity to initial conditions. To formalize this remark, consider the bijection

p : abþ ia2 2 P 7!a b

0 a�1

� �2 H,

where P ¼ fz ¼ xþ iy j ðx; yÞ 2 R2; y40g denotes the Poincare half-plane with metricg ¼ y�2ðdx2 þ dy2Þ, and where

H ¼a b

0 a�1

� ��a40; b 2 R

� .

This identification lifts to the identification of the unit tangent bundle ~M � T1H with theunimodular group

G ¼a b

c d

� �j a; b; c; d 2 R and ad � bc ¼ 1

� ,

with unique Iwasawa decomposition G ¼ fh � k j h 2 H; k 2 Kg where

K ¼cos y sin y

� sin y cos y

� ��y 2 ½0; 2pÞ�

is viewed as the fiber of the unit tangent bundle T1H. Thus, the geodesic flow a on~M ¼ T1H is described by

8t 2 R : at

a b

c d

� �� ¼

a b

c d

� �e�t=2 0

0 eþt=2

!. (3.3.2)

Two ancillary actions c, transverse to the geodesic flow, are then defined by

8t 2 R :

cþðtÞa b

c d

� �� ¼

a b

c d

� �1 t

0 1

� �

c�ðtÞa b

c d

� �� ¼

a b

c d

� �1 0

t 1

� �8>>>><>>>>:

9>>>>=>>>>;. (3.3.3)

The Hadamard property is now written precisely as the following hyperbolic relations:

at � cðtÞ � a�t ¼ cðettÞ. (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 ¼ GnG=K ¼ GnH and M ¼ GnG ¼ T1F are compact differentiablemanifolds.18 G plays here the role played by Z2 to get the flat torus T2�Z2nR2 ( ’ R2nZ2

since R2 is abelian). In particular, the Poincare half-plane H is tiled by translated images ofF � H, and F is to be pictured as one of these tiles where opposite edges are identified 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 flow 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 ¼ GnG, where the hyperbolic property(3.3.4) thus still holds.

Remarks 3.3.2 (Properties of the Hedlung–Hopf 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 Hedlund–Hopf model is a geodesic flow on a Riemannian surface F , it is agenuine Hamiltonian system (there, M ¼ T1F is a three-dimensional constant-energymanifold). Therefore, this L-model establishes that ergodic behavior is compatible with

Hamiltonian dynamics.

(3) Eq. (3.3.4) is a stronger (integral) form of a (differential) condition that definesAnasov flows, which are dynamical systems sitting almost at the top of the ergodichierarchy19

A ‘ K ‘ L ‘ X ‘ E. (3.3.5)

Here A stands for the Anosov condition which, by virtue of (3.3.4), is satisfied by theHedlund–Hopf model. K stands for the Kolmogorov condition, which roughly states thatin the course of the evolution, there exists a partition of the phase–space M which becomesfiner 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 flows have positive dynamical entropy, i.e. again roughly speaking, nomatter how far in the finite 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 Definition3.3.1(iii); and E stands for ergodic.

Remarks 3.3.3 (Beyond classical Hamiltonian systems). (1) The Boltzmann and Hedlund–Hopf modelings discussed above involve continuous times, i.e. t 2 R; ergodic theory is

17The existence of such subgroups G � G was established by Poincare in 1882, and he extended their study in a

series 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 hierarchy—e.g. Arnold & Avez (1968)—and I use the logicians’

notation (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 codified in the syntax. Theoretical physicists and many

mathematicians would use, instead of ‘, symbols such as !;) or 7! for which I have other usages.


much richer in models involving discrete times t 2 Z, such as successive coin tossings(extending to infinity) 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-model

has been exhibited that disproves the opposite implication (e.g. Arnold & Avez, 1968,Chapter 2).(3) The Hedlund–Hopf model served as a stimulus toward the creation of symbolic

dynamics by Hedlund (1935, 1939, 1940, 1969) and by Morse (1921), Morse and Hedlund(1938); the classic text is (Morse, 1966). Suffice 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 to

amenable 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 configuration 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 theory

The 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 beyond

For the study of modeling strategies, the Bohr atom offers an instructive succession ofphases.The experimental data collected in atomic spectroscopy were already quite extensive by

the beginning of the 20th century. By choosing judiciously the simplest case, Balmer (1885)had noted that the spectral lines of hydrogen obey the rule nmn�ðn

�2 �m�2Þ withn;m ¼ 1; 2; 3; . . .; this is a purely numerical model, with no built-in explanatoryjustification; in fact, hardly a model at all.Bohr (1913) proposed a model for the hydrogen atom in which he pictures one electron

circling around a nucleus on discrete orbits of energy En�n�2; n ¼ 1; 2; 3; . . .; electro-magnetic transitions between these discrete energy levels then absorb or emit lightaccording to Balmer’s formula: a potent idea but not yet a full explanation. At this stage,the Bohr model is an H-model.


In the same way as the ideal gas serves as an L-model for Clausius’ thermodynamics—see Section 2.3 above—the 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 ðxÞ�jxj�1, the eigenvalues of H ¼ �

P3i¼1q

2xiþ V ðxÞ are the Bohr

energy levels En.In turn, the Schrodinger wave equation, together with Heisenberg’s 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 difficulty, 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).

4.2. Planck’s modeling for black-body radiation

The experimental evidence available to Planck was the spectral density rT ðnÞ 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 identified by Wien (1894), Stefan (1879) and Boltzmann (1884). Inaddition, specific 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 first, Planck (1900a, 1900b) proposes in a ‘‘lucky guess’’—his own words (Jammer,1966, p. 19)—an ad hoc formula:

rT ðnÞ ¼ Ahn

ehn=kT � 1with A ¼

8pn2

c3. (4.2.1)

(i) This expression satisfies the Wien and Stephan–Boltzmann qualitative expressions;(ii) it interpolates analytically between the Wien and Rayleigh–Jeans formulas; (iii) it fitsexperimental results so well that it allows the experimental determination of the value of c,the speed of light and k, the thermodynamical Boltzmann constant. Planck’s 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 desperation’’—again his own words (Jammer, 1966, p. 22)—the construction of a heuristic model in which


the radiation exchanges energy in discrete quantas with putative resonators in the walls inthermodynamical equilibrium.Asides from his noticing that Planck’s account did not conform to Boltzmann’s

statistical counting as closely as Planck made it appear, Einstein (1906) objected thatPlanck’s treatment of his model involved a theoretical inconsistency between: (a) the use ofthe Maxwell theory of electromagnetism to compute the average energy of a resonator in aradiation field; and (b) the assumption that the energy of a resonator must changediscontinuously. Together with other empirical problems, among which the photoelectriceffect (see Section 4.3), these difficulties led Einstein to advance an alternate ontologicalpostulate, namely that the quantization is to be looked for in the radiation field 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; specifically:

Scholium 4.2.1. If the average energy huT ðnÞi of quantum oscillators of frequency n in

thermal equilibrium at temperature T is computed from Planck’s distribution (4.2.1), then for

all values of hn=kT the energy fluctuation hðDuÞ2i ¼ kT2qT huT ðnÞi is the sum of two terms

hðDuÞ2i ¼ hðDuÞ2ip þ hðDuÞ2iw with

hðDuÞ2ip ¼ huT ðnÞihn;

hðDuÞ2iw ¼ huT ðnÞi2c3

8pn2

8><>: . (4.2.2)

Hence, the particle-like contribution hðDuÞ2ip dominates when hn=kTb1, and the wave-likecontribution hðDuÞ2ip dominates when hn=kT51. In this interpretation, the particle–waveduality is thus a matter of degree, not of essence: quantum behavior occurs mostly at lowtemperatures, whereas classical behavior emerges at high temperatures. This observation—initially made on an H-model—was extended later to more general circumstances, so muchso that it is now seen as part of the core of quantum mechanics.

4.3. Einstein’s modeling for the photoelectric effect

The 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-model

in Einstein (1905a, Section 8), a paper the title of which, the reader will note, refersexplicitly to a heuristic point of view. Let me specify first 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 identified 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.Accordingly, monochromatic light was viewed as a wave Cðx; tÞ ¼ Aeiðot�k�xÞ where


n ¼ o=2p is the frequency of the light ray, l ¼ 2p=jkj is the corresponding wave length,and I ¼ jCj2 ¼ A2 is the light intensity.

3.
The photoelectric effect itself was discovered by Hertz (1887), and soon confirmed byseveral physicists, among them Hallwachs (1888) after whom the effect was oftennamed; the experiments of Lenard (1902) and Ladenburg (1903) established firmly 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 numberof electrons ejected increases with the intensity of the light; (iii) unless the frequency oflight 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 Einstein’s explanation.

Scholium 4.3.1. If light can only be absorbed in discrete light quanta of energy hn, then the

conservation of energy reads

E ¼ hn� Eo, (4.3.1)

where E ¼ 12

mv2 is the kinetic energy of the electrons emitted in the photoelectric effect; Eo is

the energy to extract an electron from the metal, and thus depends on the constitution of the

latter.

The experimental data (i)–(iv) listed above are then immediate consequences of (4.3.1).Property (iv) is even made more specific: 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 Planck’squantum 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 out—from the wealth of Einstein’s contributions—the explanation of thephotoelectric effect, which at first sight could have been regarded as a relatively minorepisode; on the contrary, I see here a clear-cut case where an exploratory H-model

drastically contributes to the evolution of a theory.

4.4. Debye’s modeling for the specific heat of solids

For the purpose of the present paper, one of the main interests of Debye’s model is thatit involves interactions between several nodes of the ambient web: quantum theory,thermodynamics, statistical mechanics, crystallography and Rontgen rays diffraction.

Dulong and Petit (1819) had proposed an argument to the effect that the specific heat—measured in calories per mole per degree—ought to be the same for all solids: 3R where R

is the universal gas constant. However, it later became apparent that the specific heat coulddecrease dramatically with temperature. By the end of the 19th century, the experimental


data became sufficiently systematic to lead to the conjecture that the specific heat of solidswould become vanishingly small as the temperature approaches 0K.On the wider network, Bravais’ speculations (1850) on crystalline solids having the

microscopic 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 confirmed in the laboratory (Friedrich, Knipping, & von Laue, 1912, 1913).While von Laue’s first 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 providesfirmer quantitative parameters for the crystalline structure on which Debye’s H-modelingis predicated.As no classical explanation of the observed temperature-dependence of the specific heat

seemed forthcoming, Debye (1912) offered the following heuristic model, improving on(Einstein, 1907, 1911).The Planck formula for black-body radiation (4.2.1) is first given a new interpretation in

terms of the vibrational energy of a solid at temperature T

UðTÞ ¼

ZdngðnÞUðn;TÞ with Uðn;TÞ ¼

hnehn=kT � 1

and

Z 10

dngðnÞ ¼ 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 fixed 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:

gðnÞ ¼ G1 if 0pnpno

0 if n4no

( )with G ¼

12pn2

s3V . (4.4.2)

G takes into account that vibrations are now sound waves in a volume V rather thanelectromagnetic waves; compare with A ¼ 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 ¼ 2 � 4p by 12p ¼ð2þ 1Þ � 4p reflects 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 satisfies

CV ’

3R for TbY;

12

5p4R

T

Y

� �3

for T5Y:

8><>: (4.4.3)

Hence, Debye’s H-model describes two extreme regimes: at high temperatures it recoversthe Dulong–Petit value; and, at low temperatures, it predicts that as the temperatureapproaches 0K, the specific heat vanishes according to CV�T3. 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 CV

decreases monotonically and continuously over the whole range of temperatures T 2 Rþ.


Note 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 back

In 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 field. I thensketch a few of its subsequent avatars.

5.1. Immediate experimental background

When the BCS model was advanced in 1957 as a ‘‘Theory of superconductivity’’ the fieldhad 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 find their explanation in the quantum theoryof solids, and more specifically as a manifestation of an electron– electron interaction

mediated 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 c

below 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 c

ffiffiffiffiffiffiMp¼ C, where C was observed to be nearly a constant across the four isotopes of

mercury. 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 model

In 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 finite cubic box L of edge 2L ¼ V 1=3

HL ¼Xp;s

�ðpÞasðpÞ�asðpÞ þ

Xp;q

bðpÞ� ~vðp; qÞbðqÞ. (5.2.1)

Here p is the momentum of an electron; �ðpÞ is its free-energy; asðpÞ� is the creation

operator for an electron of momentum p and spin s up ð"Þ or down ð#Þ; and asðpÞ

is the corresponding annihilation operator. Similarly, bðpÞ� � a"ðpÞ�a#ð�pÞ� is the

creation operator for a pair of two electrons of opposite momentum and spin, a so-called Cooper pair. And bðpÞ� ~vðp; qÞbðqÞ describes the energy transfer during a collisionbetween Cooper pairs which models the electron–phonon interaction. In the molecular field

20Compare to footnote 13 above.


approximation,—cf. Section 2.5.1—the Hamiltonian becomes

HL ¼Xp;s

EðpÞgsðpÞ�gsðpÞ, (5.2.2)

where the quasi-particles created and annihilated by gsðpÞ� and gsðpÞ are related to the original

electron creation and annihilation operators by the Bogoliubov–Valatin transformation

g"ðpÞ�¼ uðpÞ�a"ðpÞ

�� vðpÞ�a#ð�pÞ

g#ð�pÞ ¼ vðpÞa"ðpÞ�þ uðpÞa#ð�pÞ

). (5.2.3)

The one-particle energy of these quasi-particles is

EðpÞ ¼ ½�ðpÞ2 þ DðpÞ�DðpÞ�1=2, (5.2.4)

where the energy-gap DðpÞ satisfies the self-consistency equation

DðpÞ ¼ �X

q

~vðp; qÞDðqÞ2EðqÞ

tanh1

2bEðqÞ

� �. (5.2.5)

There exists then a critical temperature T c such that: (i) for all T4Tc;DðpÞ ¼ 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 DðpÞ on the temperature b ¼ 1=kT compares very well with empirical data;see Schrieffer (1964, Figs. 1–3). 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 coefficients of the Bogoliubov–Valatin transformation (5.2.3)satisfy

u�ðpÞuðpÞ ¼ 12½1þ �ðpÞ=EðpÞ�

v�ðpÞvðpÞ ¼ 12½1� �ðpÞ=EðpÞ�

u�ðpÞvðpÞ ¼ � 12½DðpÞ=EðpÞ�

9>=>;. (5.2.6)

In particular, this entails u�ðpÞuðpÞ þ v�ðpÞvðpÞ ¼ 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 model

There are two conceptual difficulties with the BCS model. First, the Hamiltonian (5.2.1)is invariant under the gauge transformations defined by

asðpÞ ! eiyasðpÞ; a�s ðpÞ ! e�iya�s ðpÞ (5.3.1)

whereas the Hamiltonian (5.2.2) is not. Second, the energy spectrum EðpÞ depends ontemperature whereas no temperature dependence had been put in the originalHamiltonian.The way out of these difficulties was found by Haag (1962), and polished in a series of

papers; for references later than Emch and Guenin (1966), see Emch and Liu (2002,Section 14.1). This solution came from the treasure chest of the mathematical theory of


C�-algebras and their representations.21 For presentations of this theory, tailored to theneeds of the quantum physics of systems with an infinite number of degrees of freedom, seeEmch (1972) or Haag (1996).

The argument goes essentially as follows. The BCS modeling involves a passage to thethermodynamical 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 prðAÞ corresponding, at natural temperature b ¼ 1=kT , to the thermo-dynamical limit of the equilibrium state r, has a non-trivial center Zr � prðAÞ

00\ prðAÞ

0,i.e. ZraCI , where prðAÞ

00 is the weak-operator closure of prðAÞ, and prðAÞ0 is the set of

all operators that commute with all observables in prðAÞ or equivalently in prðAÞ00.

In this limit, the energy gap is still an element of the center Zr, but it is not a‘‘c-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 DðpÞ ! ei2y DðpÞ, u�ðpÞ ! ei2y u�ðpÞ, vðpÞ ! vðpÞ andgsðpÞ ! e�iygsðpÞ. The gauge symmetry of the theory is thus preserved. In addition,prðAÞ

00 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 BCS–Haag 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 identification of purethermodynamical phases in quantum statistical mechanics as extremal KMS states, thoseequilibrium states that generate primary representations. This identification was formalized

from 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-transfer

This 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 ‘‘high’’temperature superconductivity in special ceramics, the perovskites. Along this line of

21The usage of the word ‘‘representation’’ here coincides with its use in model theory: the Hilbert space

representations are realizations of the abstract objects called C�-algebras. In fact, these objects were first called

B�-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 faithfully

realized as a C�-algebra, and the notational distinction faded out.


laboratory 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 distortion—known in molecular chemistry as the Jahn–Tellereffect—and thus they conjectured that these materials could also exhibit an essentialingredient of the BCS model, namely strong electron bindings.Yet, another high Tc superconducting material—MgB2—was discovered recently, the

behaviour of which could be accounted for directly by the BCS model. The immediatesignificance of this discovery is that, in technological applications, metals are much easierto handle than ceramics; this evidently stimulated much empirical activity (e.g. Canfield &Bud’ko, 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 phenomena—here, spontaneous symmetry breaking—that emerge in the repair ofan initial formal shortcut.

6. Conclusions

Two 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 specific case studies.Here, H-modeling is explicitly demonstrated in more examples than are presented to

illustrate L-modeling. This is not accidental, but it reflects the very nature of thesestrategies. Indeed H-modeling appeals more immediately to what physicists commonly call‘‘intuition’’, a feature difficult 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 landscape—theoretical, experimental, or both—

that guided the stategy to be developed. I also described the contributions the specificmodeling 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 clarify

the issues that modeling strategies are designed to address; and that an awareness of these


issues is essential to the resolution of controversies on what a given model doesaccomplish.

Acknowledgments

I 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.

Appendix

As I surely omitted some of the reader’s 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 Bohr–EinsteinGedanken experiments; early explorations of the quantum measurement process. In

nuclear physics: the tunnel effect and the emission of a-particles; the Heisenberg–Wignerisospin; the compound nucleus; the shell model. In condensed matter physics: the imperfectBose gas; the ‘‘jellium’’ model; the scaling hierarchy (Kadanoff–Wilson–Fisher); theGross–Pitaevski 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; the‘‘standard’’ 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. In condensed matter physics: the stability of matter (Dyson–Lenard andLieb–Thirring); the residual entropy of ice (Pauling and Lieb); the X–Y model; the Dysonhierarchical model; free-induction relaxation and return to equilibrium; Kondo effect andvan Hove diagramatics. In non-relativistic quantum mechanics: yet some others of theBohr–Einstein Gedanken experiments; the existence of superselection rules; the Bellinequalities; the quantum measurement process. In relativistic quantum field theory: PðFÞ2and several others. In general relativity: the perihelion of Mercury; the Schwarzschildsolution; the Penrose–Hawking singularity theorems. In probability theory: the von Misesproposal for his ‘‘collectives’’; the de Finetti modeling of degrees of belief.

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