PHILOSOPHICAL ISSUES IN THE FOUNDATIONS OF 5 J\.TISTICAL MECHANICS
Lawrence Sklar
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PHILOSOPHICAL ISSUES IN THE FOUNDATIONS OF 5 J\.TISTICAL MECHANICS
Lawrence Sklar
Philosophical issues in the foundations of statistica mec anics
LAWRENCE SKLAR
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Lawrence. Physics and chance : philosophical issues in the foundations of statistIcal mecnamcs / Lawrence ~K1ar. p. cm. Includes bibliographical references and index. ISBN 0-521-44055-6 1. Statistical mechanics. 2. Phvsics - Philosoohv. 1. Title. QC174.8.S55 92-46215 1993 CIP 530.1'3 - dc20 ~K1ar,
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Preface
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Philosophy and the foundations of physics The structure ot this book l. Probability 2. Statistical explanation 2 0/'
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measure zero pro em
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the non-equilibrium theory
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References Index
421 429
.
The aim of this work is to continue the ex loration into the foundational questions on the physical theory that underpins our general theory of siderations into the fundamentals of our physical description of the world. a p YSlca eory lS s a lS lca mec nlCS. The history of this foundational quest is a long one. It begins with an intense examination of the remises of the theo at the hands of ames Clerk Maxwell, Ludwig Boltzmann, and their brilliant critics. It continues to the philosophical community. And the quest has persisted as a set o cu concep a a enges, ln a s y rna e ever ric er y e development of ever more sophisticated technical resources with which to treat the roblems. I ho e that this book will encoura e others in the philosophical community to join with those in physics who continue to
has often guided me to the crucial questions to be addressed. James Jo ce and Robert Batterman, as students and as collea ues, have been enormously helpful to me in my thinking about these issues. I have Frank Artzenius, and Abner Shimony. manuscript of this book. I am also grateful to Terence Moore of Cambrid e Universi Press, to the two referees for the Press for their help in bringing the book to publication, and to Ronald Cohen for his editorial The research contained in this book has been supported by a number fully acknowledged here.
1
I. Phlloso h
and the foundations of hies
There are four fundamental theories that constitute, at present, the foundational illars of our h sical theo of the world: eneral relativi , quantum mechanics, the theory of elementary particles, and statistical of these fundamental theories presents its own budget of scientific and would examine the so-called foundational issues in these areas vary in a marked and interesting way from t eory to t eory. General relativity - at present the most plausible theory of the structaken to include the theory of gravitation - is in many ways the most remain: Should we accept general relativity, or some alternative to it like the Dicke-Brans scalar-tensor t eory? Are t ere genera izations 0 t e theory that might encompass other forms of interaction over and above ?
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tions of causal "niceness," for example? But we are, at least, clear about of the theory is its totally classical nature, and all expect that some day we wi ve a new quant1ze t eory to ta e its pace. ut at east we know what theory we are talking about when we begin to explore the Philosophically, too, the situation is particularly clean in the case of
logy, and so on have been discussed long enough and hard enough by
2
Physics and chance
the answers, ar least there is a general consensus about the right questions to ask. uantum mechanics is a ain a theo whose scienti c claims are at least fairly clear. The basic structure of the fundamental theory is avail-
what the theory should be. Just as general relativity runs into the fundamental problem that it is not a quantized theory, leaving us to speculate about 'ust what our ultimate uantized s ace-time theo will be so quantum mechanics runs into difficulties when one tries to extend it to
fields, whether in the vein of Wightmannian axiomatic theories, theories of scattering of t e LSZ type, S-matrix t eories, or somet . g else. Philoso hicall uantum mechanics like eneral relativi resents us with a now well-defined set of foundational questions to be answered.
in the orthodox theory. This understanding will require the resolution of suc su si iary pro ems as wave-partic e ua ity, e non-causa interde endence of se arated s stems the nature of uncertain relations the existence or non-existence of hidden variables, and so on.
what the questions are that need to be answered. In the latter case we at east ow qUlte e nlt1ve y w at some POSSl e answers are, even we aren't sure which are ri ht and which are wron . In the uantum case I think many will agree that most (perhaps all) of the answers themselves
understand much more clearly what they are saying. e eory 0 e ementary partlc es agam as ltS own speCla avor as an area for foundational research. Here we are concerned with the a eold question of the fundamental constitution of matter. This theory presents
Introduction
3
particles by, perhaps, gluon forces, But just why there are the structures and forces there are, and just what fundamental laws govern the comosition of macro-ob'ects out of their micro-constituents remains something of a mystery to us, despite the recent progress of theories such as
theory also remain less well characterized and far less well explored than those concerne with space-time theories an quantum mechanics, Althou h some attention has been aid to such uestions as wh ex lanation so frequently consists in reducing macro-objects to their microscopic
theories devoted solely to accounting for the thermal behavior of macroSCOPIC matter, t ese ranc es 0 p YSICS now stan eSl e e 0 er three as fundamental com onents of our scientific world-view, Independent of the other theories, they are an indispensable supplement
to move from the fundamental theory governing the behavior of the mlcroscoplC constituents 0 matter an t e t eory 0 e constitutlon 0 macroscopic ob'ects out of their microscopic parts to a full explanation of the macroscopic behavior in terms of the microscopic constitution
distribution of radiation and the laws of particle scattering are among the e wor w ose escnp lon an exp ana lon requITes vanous aspects 0 the application of thermodynamics and statistical mechanics, Originally formulated to provide the general laws governing the thermal
4
Physics and chance
world ranging from the distribution of elementary particles upon interaction to the distribution of stars interacting by gravity in massive clusters can all be sub'ected to the thermod namic view oint. Thermodynamics has a peculiar place when viewed in the context of
all physical phenomena are to be described. This framework is modified and generalized by general relativity in the curious revolutionary shift due to Einstein which makes the s atio-tem oral arena of henomena a dYnamic and causal participant in events and assimilates the gravitational
quantum terms. In addition to these two fundamental theories (theories not yet proper y reconCI e to eac 0 er we nee t e e -t eoretic rinci les that describe the s ecific nature of material existents in their most primitive and elementary form - that is, the field theory of the
discipline such as thermodYnamics. If we consider the dYnamic evolue c aractenze y t e t eones note , tlon 0 t e wor as It wou there seems little lace for additional fundamental lawlike structure. We can describe any system at a time by its quantum state, and project its
by the general quantum theory, and the specific rules of dYnamical intere e ementary partlc es. at e se action prOVI e y t e e t eory 0 is there to do? That there is another, fundamental, level of descri tion that is vital and fruitful - a level involving such concepts as equilibrium,
apparently radically different fundamental natures, is the surprising truth o ermo ynamlcs. Many no longer think of thermodYnamics as an autonomous science, however. Since the middle of the nineteenth century there has been a
Introduction
5
of internal motion of microscopic parts of matter, progressing through the development of the kinetic theory of gases in the nineteenth century, and evolvin into the com lex and multi-faceted disci line of statistical mechanics, there has been a continuous program designed to show us
Part of the motivation of at least some of the discoverers and developers of statistica mechanics has een the hope that the eeper un erstandin of thermod namic henomena rovided b this new a roach would eliminate the need to invoke thermodynamic-like principles of the
are reducible to, and intelligible in terms of, the features posited by the ot er n amenta t eories 0 t e wor an e aw i e nature 0 t ose features as these other fundamental theories describe them. This ho e for the elimination of thermodynamic principles as autonomous elements
theses that remain underivable from the fundamental laws of kinematics or ynamlCS. ese are now ta en to e 0 a pro a 1IStlC or statlstlca nature. Just what these seemingly autonomous principles of statistical mech-
themselves be derived from the fundamental laws of general kinematical an ynamlca t eones IS arge y e su Ject matter 0 IS 00. Unlike the relatively closed disciplines of general relativity and uantum mechanics, statistical mechanics presents us not with a single, well-
aspects of the theory are well understood and universally accepted, such crucla areas as e correc approac 0 e In ro uc 10n In 0 e eory of irreversibility and the approach to equilibrium and the proper statistical mechanical definition of entropy are the subject of intense and
we find the same fundamental issues at the foundational level arousing con roversy among con emporary eoris s as arouse con oversy m e early years of the theory. Not that no progress has been made nor that the issues are as unclear now as they were then. But it is still true that h r r n h
6
Physics and chance
concerned with an application of the conceptually clarifying techniques
sophers than either our theory of space-time or quantum mechanics. Not that foundational issues in statistical mechanics are ignored, of course, for there exists alar e rou of h sicists who devote substantial effort to the task of clarifying this theory at its most fundamental level. But
attack on the so-called problem of the direction of time, philosophers have generally kept their distance from the conceptual perplexities of statistical mechanics. I believe that some of this reticence can be accounted for by the very
But in the case of
task. 1S context es to a ope to brin to ether in one lace a sufficient! e1ementa ,com rehensive, and organized survey of the fundamental physical and philosophical
would like to understand what the crucial debates ope a 1S wor W1 e su C1en y compre ciently accessible, that it will encourage some to detailed, technical works of the physicists that
are all about. Further, enS1ve, an yet su proceed to the more are more difficult of
Introduction
7
upon it. I doubt if I shall here resolve the major conceptual questions to everyone's satisfaction. On some uestions I will offer m own a raisal of the answers and my own conclusions about the right direction in which to
other. I will consider my ambitions amply fulfilled if this work can stimulate the same application of energy an ta ent in attac . g the ndamenta roblems I will surve and ex ound as is resent! devoted to the fundamental philosophical questions in space-time theories and quantum
think of and that might bear fruit under a kind of extensive investigation that is not possi e or t em ere. D. The structure of this book
which we will be concerned. As is not uncommon in philosophy, these orm a rat er ense networ 0 Issues, so t at reso vlng one questIon seems to resu ose fior resolution of all the others. To some extent then, the sorting out of issues and the order in which they will be treated
exploration of the substantive issues will follow a preliminary outline of e IStOry 0 ermo ynamlcs an statlstlca mec anlcs. IS survey 0 the hi h oints in the develo ment of these fields is essential to introduce to the reader the main problems and their overall context. 1. Probability
8
Physics and chance
a mea ure 0 su jec ive gr e 0 e ie. om e concep a issues concerning the role of probability in statistical mechanics, we shall see, hinge on a prior understanding of probability from a general philosophical h r wh in the particular theories we are examining will vary with our conception
role of probability in the particular context of statistical mechanics that are rather more specific to the very special function played by probabilities in this articular realm of h sics. In other words even havin ado ted some general philosophical stance, there will be additional conceptual
I do not expect to resolve to general philosophical satisfaction the issue of the correct interpretation of probabilistic assertions. I will lay out briefl what some of the ma'or alternatives are and what some difficulties with them are taken to be. I will opt for one interpretation as most
sophical questions I certainly don't have available. Attention will be paid, owever, to t e pro ems more spec' c to t e ro e 0 pro a i ity in statistical mechanics because these have been rather ne lected in the literature in comparison with the broader, and admittedly more funda-
Scientific theories are su osed to ex lain what ha ens in the world. The features of the world that statistical mechanics sets out to explain
existence of a small set of macroscopic parameters sufficient to charactenze equll num an a somew at arger set su Clent to c aractenze some a roaches to e uilibrium, the lawlike inter-relationshi s amon these macroscopic parameters - these are the sorts of things to be ac-
statistical and probabilistic assertions postulated by it. What is the nature o an exp anatlon a res s upon a s a lS ca or pro a llS lC assump 10n. Again, we have available to us a rather wide variety of accounts in methodological philosophy of science as to just what statistical explana-
Introduction
9
once again it will be seen that statistical explanation as it functions in statistical mechanics has its own idiosyncratic features that deserve special attention in their own ri ht. We shall see that man of the erennial debates in the foundations of statistical mechanics rest upon assumptions
debates even if not resolving them.
calculate certain quantities. Associating these quantities with macroscopica y measure parameters we 0 er an exp anation 0 the equili rium ro erties of the s stem. But what justified our choice of fundamental statistical assumptions?
the appropriate one to choose to associate with the macroscopic quantity? ere 1S an attempt to 0 er at east partia answers to some 0 ese uestions that relies heavil on certain features characteristic of the laws governing the micro-constituents of the macroscopic system. This is so-
laws governing its micro-constituents and other features that are the result, ra er,o e vast num er 0 m1cro-constituents at rna e up a macros stem. I shall t ,b disentan lin a number of distinct uestions that tend to get muddled together in the literature, to make clearer just how
line will be that there are a number of quite distinct questions one can as an at some resu ts are answers to some questions ut not to others. Putting this simply, the response is likely to be, "of course that is true," but, as we shall see, failure to make the fine distinctions here has
The study of non-equilibrium in statistical mechanics presents quite a different set of scientific problems than those that arise in the equilibrium
10
Physics and chance
cally, in this case from the equilibrium case. In particular, the relationship of the generalizations to the underlying dynamical laws governing the micro-constituents is uite different in the two cases. The roles of initial conditions and laws in the explanations, the modes
5. Cosmology and statistical mechanics Since Boltzmann, it has been frequently alleged that a full understanding
Recent work on observational and theoretical cosmology has made the interre ationship 0 cosmo ogica an statistica mechanica features of the world a vital area of scientific ex loration. The aim of this section will be to concisely present the present state
philosophically. Once again, what I hope to show is that there are a num er 0 quite istinct questions to e answere ,questions at can easil be confused with one another. When these are disentan led from one another, the earlier exposition of statistical explanation will allow us
concerning the origin of irreversibility.
for in the theory of thermodYnamics. The alleged relationship of thermoYnam1cs to statlstlca mec an1CS, ten, 1S t at 0 one t eory t at as been reduced to another. We will examine a number of standard hilosophical accounts of the reductive relationship and seek to place the
Introduction
11
outstanding general methodological issues. Indeed, in some cases there will have to be a simple dogmatic adoption of one plausible position amon others. Rather I want to focus on the roblems that are s ecial to this particular reduction, maintaining, as I will, that once again the
has very special features vis-a.-vis other physical theories. Its descriptive concepts have peculiarities not share in general y theoretical concepts of h sics. And certainl this will be true of the conce ts of statistical mechanics as well. Naturally, then, the interrelation of these two theories
7. The direction of time
Most of the attention that has been paid by philosophers to the founda-
The issue has been explored rather as a means of providing resources use m lscussmg t e so-ca e pro em 0 t e rrectlon 0 tIme. ere, the resources of statistical mechanics have been invoked in order to offer a reductionistic account of the very notion of past-future asymmetry it-
dYnamics and, allegedly, explained by statistical mechanics, that grounds our very notIon 0 e lstmctlon etween past an ture. e I ea ere is present in Boltzmann's work, and it has received its most detailed exegesis by Reichenbach.
ible to the entropic aSYffiffietry of the world in time. Failure to clearly un ers n JUs w a tn 0 re uc Ion one oug 0 ave In mm ere nas led, I believe, to some confusion in the literature. I will then explore, to some degree, the plausibility of the reductionist program. It is to be
difficult problem than has previously been gained.
12
Physics and chance
ence to, and exposition of, a substantial amount of physics, the history of science, and philosophy. In order to keep the work within reasonable bounds man fascinatin conce tual roblems in the foundations of statistical mechanics will simply have to be ignored altogether or, at best,
of phase changes. The scientifically knowledgeable rea er will be especially struck by the overall focus in this book on classical statistical mechanics and the apparent neglect of quantum statistical mechanics. Because we know
what is the point in directing our philosophical attention to a version of the theory we now to e incorrect an outmo e ? The reason is clear. It is that the articular conce tual roblems on which we focus - the origin and rationale of probability distribution assumptions over initial
the theory is essential. For example, quantum ensembles of finite systerns cannot escape recurrence resu ts as can c aSSlca ensem es 0 nlte s stems. And, some aIle e, onl b reference to uantum limitations can the limits on initial ensembles be correctly delineated. Where one
varieties of classical statistical mechanics as the objects for conceptual exp oration. is way 0 olng t figs is not 1 losYncratic, ut common in the physics literature devoted to foundational issues. Whether things can be carried off profitably in this way can only be determined by
in the problems treated but who lack an extensive back-
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works that I have found most useful in laying out the essential philosophical and physical issues in a clear way. The bibliographies are annotated in the hope that they will provide a road map into the literature for those who would like to pursue the material covered in this work . 'c• ,1 ..J...L:> 1-..... 11 ...... ............. .. , "".... , ~
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ing out in greater depth the exposition of philosophical positions expounded and criticized here. Given the lack of coherence in the physics, noted earlier. this exPlicit 2uide to the literature is. I believe. a particularly essential part of this work. ....
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physical theory is marked with an asterisk (*), Material demanding comprehension ot more advanced mathematics and phySICS IS marked with two asterisks (**).
2
selective historical survey of the development of thermodynamics, kinetic prehensive. Nor is the material presented in a manner that would suit historians of science. Neither issues of chronolo and attribution nor the far more important questions of placing the specific scientific results in goal is merely to present some important developments in the history of will facilitate our later conceptual exploration. Many of the scient' c resu ts note in this chapter wi 0 y e mentioned here and not referred to again. They are being noted simply to pects of the historical development - those dealing with the conceptual detail, especially in Chapters 5 through 7. The reader ought not to be iscourage ,t en, . certain conceptua an oun ationa issues seem to be too sketchily treated in this chapter to be grasped with full clarity, nor xt
abeyance. These conceptual and foundational questions and issues will
sions into novel areas. I then take up the development of the kinetic approac to matter, t e eve opment 0 lnetlc t eory at t e an s 0 Maxwell and Boltzmann toward a genuinely statistical or probabilistic theo the develo ment of formal statistical mechanics and some as ects of the evolution of foundational questions through the probing critiques developments are noted. Once again these developments are treated
15
Historical sketch foundational issues they will be taken up in detail in Chapter 7. cs
erties and laws The discovery of the right concepts with which to describe nature and the discovery of the lawlike relations among these features are processes that cannot be disentan led from one another. The a ro riate conce ts are the ones that allow the formulation of lawlike regularities among the
of development of a science, a progression that requires great philosop ica su t ety in 011 er to e y characterize , ut whose supe cia nature is clear. In the be innin we characterize the world in terms of concepts directly connected to our sensory awareness of things. Gradually
qualitative and comparative to fully metric concepts. Finally, as the concepts are uti 1ze to construct aws escn 1ng t e regu arities we iscover in nature we move to a sta e where the ve meanin of the relevant conceptual terms is more closely associated with the role of the terms in
One point of origin of the full thermodynamic theory is the discovery 0 erst so-ca e equatton 0 state or gases. e re evant aw is formulated usin conce ts whose scientific role is alread wellestablished in kinematics (the description of motion) and dynamics (the
the pressure imposed upon it, and that it exerts on the walls of its conta1ner, 1S constant. But this law neglects the additional relevant parameter of temperature, and it is in the introduction of the concept of temperature, and the
with felt hotness and coldness. The introduction of thermometers detaches
16
Physics and chance
equa 10n 0 s a e k is a constant. Even the consideration of simple experiments on the resulting temmixed suggests the necessity of an extensive quantity - heat - to parallel s me temperature but twice as much heat content. Moving to the results obtained on the mixing of different substances, there develops a slow awareness of the need for such notions as heat ca ad - how much heat needs to be added to a given substance to raise its temperature to
deal with static properties of matter, it is soon realized that a theory of the laws of change of the features introduced is needed as well. Temerature difference is invoked as the drivin force for heat ow and invoking the notion of the distribution of temperature over matter, soon
formulated governing the combined flow of matter and heat, although the erivation 0 these equations requires the concomitant progress in the develo ment of the full abstract thermod amic theo . Here ro ress is obtained by imposing stringent assumptions upon the flow of matter
to move to more complex equations for which those first derived prove o y approxlmations. Eu er's equatIons, or examp e, escri e t emotion of an incom ressible non-viscous fluid. Retainin incom ressibili but allowing viscosity (momentum transfer by diffusion rather than by transfer
ity, and so on leads to the ability to characterize ever more complex sltuatl0ns In w lC matter an eat are 0 In a c anglng state nve y their interactive forces.
2. Conseroation an irreversi ility Conservation. Concurrent with the gradual development of ever more comp ex an gene constltutlve equatlons - e vanous equatl0ns re tmg the lawlike interrelations amon the descriptive arameters of particular kinds of matter such as ideal gases in their macroscopically static states,
Historical sketch
17
general principles that are taken to hold for all particular material systerns, and that together with the more particularized constitutive equations rovide the resources needed for the full theo . There are two parallel paths in this progress, and insight into both strands develops
.
the overarching principle of the conservation of energy. The other is the growing realization of the fundamental role p ayed by irreversibility in the theo assimilated after the conce tual innovations of absolute temperature and entropy into the famous Second Law of Thermodynamics.
frictional motion to raise the temperature of an object suggested the specu ation that heat, the gain 0 which accounte or a rise in temperature mi ht be nothin more than some kind of motion of internal constituents of matter. This internal motion was not directly evident because,
scopically determined is
metric experiments on mixing can be accounted for by the idea that heat is some t tng t at can in a it matter an e trans erre rom one piece of it to another. Somethin like the densi of this so-called caloric substance in the matter is taken to account for temperature, with subtle
18
Physics and chance
as a quasi-substance that can carty energy across matterless space. Next, iii engines 0 vi a y impo n 0 the Industrial Revolution. Most important of all, though, is the growing ability to carefully quantify amounts of heat and amounts of mechanical motion and the abili to show that the disa earance of a iven uanti of overt motion always results in a corresponding increase in a specific
observable motion. When this is combined with the growing prominence of conservation principles in the foundation of physics, the suggestion is unavoida Ie that there is one kind of uali of thin s - ener - that is never created or destroyed. The apparent disappearance of quantity of
the cost of a corresponding disappearance from the world of heat, is then rea y nothing more than the conversion of motion from the realm of the overtl observable to that of the invisible microsco ic and vice versa. By the midpoint of the nineteenth century, Clausius is writing "
Irreversibility. The facts about the interconvertibility of mechanical energy an eat ou ine in east section e , y means 0 t e princip es of the conservation of ener and the identification of heat as non-overt internal energy, to a degree of assimilation of the physics of thermal
Historical sketch
19
motion of the working fluid, and so on. Far more important, though, is the realization that even an ideally efficient en ine is limited in its abili to convert in ut heat into out ut usable mechanical motion. The crucial observation is that in order for
the working fluid is discharged at the end of the mechanical cycle. For examp e, input steam rom the oi er must e at a higher temperature than the tern erature of the condenser that turns the out ut steam into water to eliminate back pressure in the steam engine. Heat can only do "
can be extracted from the heat than is allowed by the temperature gap etween input an output reservoirs. eru e y, eat at ig temperature is valuable because work can be extracted from it b usin an extraction reservoir, at the end of the cycle, of lower temperature. Heat available at
work.
ity to get useful mechanical work out of heat energy is limited not simply Y t e act t at eat is energy an t at t e conservation aw lscusse earlier prohibits our extracting mechanical energy greater than the heat energy consumed. It is limited also by the necessity to operate our engines
The results are soon generalized into a principle of the irreversibility o processes in e wor . nergy mus e aval a e in a 19 qua 1 state to do work. Performing the work degrades the energy into "low quality." If the conversion of heat into work in the process is ideal, only
the processes that reduce them from ideality are all those that allow the qua 1 0 energy 0 egra e Wi ou pro uClng wor in e process. e net result is that there is a steady degradation of the quality of energy throughout the world. Very soon, speculation arises about the ultimate 11
20
Physics and chance
becomes a cliche of popular science.
3. Formal thermostatics are converted into a formal theory of surpassing elegance, usually called thermodynamics. Throughout, the trick is to borrow as much conceptualization as one can from the familiar mathematical resources of d namics adding such intrinsically thermal concepts as are necessary, and modifying
the new context must be taken into account. One egins with the notion 0 a system. The concepts 0 inematics and d namics are a ro riated as allowin for descri tions of s stems and of their changes over time. Thus we can talk about the volume of
of the world (adiabatic systems) and those that can exchange energy eit er wit an In etermlnate environment or wi one anot er. e tota ener flow into or out of a s stem is divided into those arts of the flow that are the result of overtly observable mechanical work, and the
overtly unchanging state called the equilibrium state. Two systems, each In IVl ua y In equl 1 num w en energe lca y IS0 ate ,mayor may no leave their respective equilibrium states when they are allowed to exchange energy with one another. When the condition of the systems is
Historical sketch
21
equivalence class of systems in mutual equilibrium. At this point, the work of Camot on heat engines is applied. The tern erature of the reservoir into which heat is dischar ed in an ideal engine able to convert all of its input heat into overt work is taken as the
heat in differing ideal engines. The net result is the definition of the full absolute temperature sca e. One then examines the routes b which a s stem in one e uilibrium state can be converted into a system in a, distinct equilibrium state. One
entropy value to each equilibrium state of a system. Entropy is so defined at a system w ose energy 1S ess egra e an so more ava1 a e or transformation into mechanical work) has a lower entro than its more degraded counterpart. The crucial fact needed to justify the introduction
ing in a cycle, will produce no effect other than the transference of heat rom a coo er 0 a warmer 0 y. n e e V1n- anc vers10n, 1 1S impossible to construct a device which, operating in a cycle, will produce no effect other than the extraction of heat from a reservoir and the "
undergoing a transition is never less than its initial state." e ne an e egan vers10ns 0 e orma eory 0 ermo ynaffilcs are available. Perhaps the high point is the 1909 formulation of the Second Law by C. Caratheodory. His basic postulate, "In the neighborhood of
The extension of thermodynamics concepts beyond their original range
chemical potentials in a manner parallel to pressure. Another important
that which is transmuted into internal energy of microscopic constituents
Statistical equilibrium thermodynamics. In the orthodox thermo-
unchanging values of the macroscopic parameters. An isolated system se es to an equi I flum state 0 e pressure, vo ume, an so on. system in perfect energetic contact with an "infinite" reservoir at a given tern erature settles to an e uilibrium state in which its tern rature and energy content are fixed. As we shall see in Sections 11,6; III; and IV this \
micro-mechanics are also introduced.
single macroscopic state but by a probability distribution over a class of
probabilistic theory, we would not be surprised to find ourselves under orIn
terms, and on a postulational basis, a novel thermodynamics is derived
24
Physics and chance
cally connected component systems. Instead of the static uniformity throughout all these pieces that one would expect in the older account, in this new account a distribution of numbers of com onents in a diversity of macroscopic states is now predicted by the theory.
and the application of the Zeroth Law of Thermodynamics, suffice to generate the specific form of the fluctuational nature of equilibrium identical to the forms familiar from the robabilistic theo built on the underlying micro-structure and micro-mechanics. •
In the orthodox presentation of thermodynamics, we attribute thermoynamic features temperature, entropy on y to systems that are in e uilibrium. In the formal version of the theo the ve meanin lness of such attributions is restricted to equilibrium states by the way the
Of course this does not mean that the theory tells us nothing about change. e very pro ems t e t eory was origina y esigne to so ve are t ose that arise when we ask for exam Ie what states can be obtained from others by processes that leave the system energetically isolated from its
and ends up with it in another. But we have no resources within this t eory to a equate y escn e systems not 1n equ11 num, nor to ea thermod micall with such details of the transition from one e uilibrium to another as the rates of flow of quantities.
equilibrium thermodynamics, one dealt with temperature distributions over a system an W1 t e ows 0 eat nven y tempera re erences. Other such laws, connecting fluxes or flows with their driving forces, exist as well, and _provide the starting point from which a fuller
Historical sketch
25
introduce such notions as the density of mass at a point, a field of velocities of mass components, a field of energy density, and so on. The trick is to extend this notion of a field of uantities to such urel thermodynamic quantities as temperature and entropy. But how can it be
entropy are only defined, within orthodox thermostatics, for systems in equilibrium? The solution is to invoke the notion of local equilibrium. If thermod namic features of thin s chan e slowl enou h from oint to point and from moment to moment, we can hope to think of a non-
with appropriate temperatures and entropies. The full understanding of just what constitutes "s ow an sma enough c ange" wou ta e us into an anal sis that oes be ond the henomenolo ical level to which we are now restricted, but the theory can be developed on a postulational
Once the appropriate field-like concepts have been introduced, one invo es a ance equations, agaIn amllar rom continuum mec anics. These are the local versions of the various rinci les of the conservation of mass, the conservation of energy and momentum, and so on. The
is, of course, not generally conserved. In a small region it can even ecrease y oWIng out t roug e oun anes 0 e regIon. at IS crucial is that within a small re ion the entro enerated is never smaller than the entropy that "flows out," so that we have a local version of the
Clausius-Duhem inequality). n non-equll num t eory e notlon 0 a s ea y-s ate process ta es a role analogous to that taken by the equilibrium state in thermostatics. Even in a system that is prevented from reaching equilibrium by inter-
26
Physics and chance
intricately interact with one another. In the case of a system that is globally ",,,,-g r n rui' i po u a e a recipro i relation among flows, so that the effect of the driving force of one flow on the flow of another quantity is matched by a reciprocal effect. The ve meanin fulness of the formal statement of the rinci Ie first of all requires that the forces and flows be characterizable in a linear way, '-'.L ..
mechanically when the system is near its "neutral" position. The reciprocity relations can then be derived from certain very general postulates of invariance. For exam Ie a rinci Ie of "detailed balance" amon states leads to these reciprocity relations.
it is possible to show that the equilibrium state is stable against small perturbations. In this exten e theory it is possible to show t at the stead -state rocesses also ossess a stabili ro e . An small deviation of the system from its steady-state flow will die out, restoring the
maximize the entropy of the system in question, in near equilibrium nonequi i rium eory one can s ow assuming e inearity an reciprocity conditions) that the stable stead -state flows will be 'ust those rocesses compatible with the restraints that minimize the production of entropy.
dum. When the system with which we deal is in a state that is far from equll num, many 0 e tracta e eatures 0 systems near-equi 1 num no Ion er a I. We cannot ho e that in eneric situations, forces and fluxes will be related in a simple linear way; that there will be simple
steady-state systems will hold. Nor can we expect an elegant principle of mintma entropy pro uctl0n to 0 genera y. If a s stem is deviant enou h from e uilibrium, so that even locall we cannot view it as at least momentarily in something like an equilibrium
Historical sketch
27
subregions, then we can, once again, apply the field-like generalizations of the thermodynamic notion. As before we introduce field-like mechanical features such as veloci and momentum fields and field-like thermodynamic features such as a field of temperatures and of entropy density.
momentum, and energy. Again, as before, a field-like surrogate for the Second Law is intro uce . Although entropy in a region may ecrease due to flow of entro across the re ion's boundaries the net result of entropy flow and entropy production in the region is always positive.
boundary conditions, it may maintain an unchanging state of flow. Sometimes these stea y-state situations wi e sta e an easi y repro uci e, so that a far-from-e uilibrium s stem no matter how often re ared will always fall into the same steady-state situation. In other cases, though,
will apparently spontaneously change into different steady-state flows. One situatIon 0 partlcu ar Interest IS were we manlpu ate a ar- rome uilibrium stem b slowl va in an extemall controllable arameter. In some cases this results in the generation of a unique, always repro-
of available options for steady-state flow. Under other appropriate circumstances or particu ar va ues 0 parameters, t e system WI s ow a fascinatin oscillatory behavior, switchin back and forth with clock-like regularity between a number of distinguishable steady-state flow modes.
stable steady-state behavior of near equilibrium systems, but instead ecoffilng amp e y vanous In s 0 pOSItIve ee ac mec anlsms. As a parameter changes, the system may, as a matter of small fluctuation, pick a particular steady-state mode, but once having moved into that
Some elegant experimental instances of steady-state behavior of sysems ar rom equll num ave een cons ruc e. one ea s a Ul between plates of differing temperature, one gets steady, stable diffusion of heat from the hotter to the cooler plate when the temperature dif-
28
Physics and chance
varying chemical concentrations, temperatures, and so on, one can generate cases of steady-state flow, of oscillatory behavior with repetitive order both in s ace and in time or of bifurcation in which the s stem jumps into one or another of a variety of possible self-sustaining flows,
"self-organizing" phenomena as those described may play an important role in biological phenomena (biological clocks as generated by oscillato flows s atial or anization of an initiall s atiall homo eneous mass by random change into a then self-stabilizing spatially inhomogeneous
caloric theory dominated the scientific consensus, so throughout the caloric perio t ere appeare numerous specu ations a out just w at in 0 internal motion constituted that ener that took the fonn of heat. Here, the particular theory of heat offered was plainly dependent upon one's
might think of heat as a kind of oscillation or vibration of the matter. ven an a vocate 0 lscreteness - 0 t e constltution 0 matter out 0 discrete atoms - would have a wide varie of choices, es eciall because the defenders of atomism were frequently enamored of complex
As early as 1738, D. Bernoulli, in his Hydrodynamics, proposed the mo e 0 a gas as cons 1 e 0 mlcroscoplC pa lC es ln rapl mo lon. Assuming their unifonn velocity, he was able to derive the inverse relationship of pressure and volume at constant temperature. Furthennore,
Historical sketch
29
common identical velocity, would be proportional to temperature. Yet the caloric theory remained dominant throughout the eighteenth century. The unfortunate indifference of the scientific communi to Bernoulli's work was compounded by the dismaying tragi-comedy of Herepath and
model of independently moving particles of a fixed velocity. He identie heat wi interna motion, but apparent y too temperature as proortional to article veloci instead of article ener . He was able to offer qualitative accounts of numerous familiar phenomena by means of
though it appeared elsewhere, it had little influence. (J. Joule later read Herepa 's wor an in act pu is e a piece exp aining an e en ing it in 1848 a iece that did succeed to a de ree in stimulatin interest in Herepath's work.)
the Royal Society in 1845. The paper was judged "nothing but nonsense" y one re eree, ut It was rea to t e oClety In a t oug not y Waterston, who was a civil servant in India), and an abstract was ublished in that year. Waterston gets the proportionality of temperature
the same, and even (although with mistakes) calculates on the model e ratlo 0 spec c eat at constant pressure to t at at constant vo ume. The work was once again ignored b the scientific communi . Finally, in 1856, A. Kr6nig's paper stimulated interest in the kinetic
Kr6nig's paper may have been the stimulus for the important papers of auslus In an aUSlUS genera lze rom onlg, w 0 a idealized the motion of particles as all being along the axes of a box, by allowing any direction of motion for a particle. He also allowed, as Kr6nig
of translational motion.
Physics and chance
30
in manreepa the average distance a molecule could be expected to travel between one collision and another. The r win rece tiveness f the scientific c mmu . theory was founded in large part, of course, on both the convincing
body of evidence for the atomic constitution of matter coming from other areas of science (chemistry, electro-chemistry, and so on). 2. Maxwell
In this paper we find the first language of a sort that could be interpreted in a probabilistic or statistical vein. Here for the first time the nature of ossible collisions between molecules is studied and the notion of the probabilities of outcomes treated. (What such reference to probabilities
uniformity with regard to the velocities of molecules, Maxwell for the rst time ta es up t e question of just w at in of istribution of ve ocities of the molecules we ou ht to ex ect at e uilibrium and answers it by invoking assumptions of probabilistic nature.
claim that the components in the yand z directions are "probabilistically In epen ent 0 e component In e x !fechon. rom t ese assumptions he is able to show that "after a reat number of collisions amon a great number of identical particles," the "average number of particles
v = 2
Av ex
Historical sketch
31
aware that his second assumption, needed to derive the law, is, as he puts it in an 18 7 paper, "precarious," and that a more convincing derivation of the e uilibrium veloci distribution would be welcome. But the derivation of the equilibrium velocity distribution law is not
from place to place, we will have transport of mass. But even if density stays constant, we can have transfer of energy from place to place by molecular collision which is heat conduction or transfer of momentum from place to place, which is viscosity. Making a number of "randomness"
, An improved theory of transport was presented by Maxwell in an 1866 paper. Here he offere a genera eory of transport, a theory that once a ain relied u on "randomness" assum tions re ardin the initial conditions of the interaction of molecules on one another. And he provided
molecular interaction, and on the relative velocities of the molecules, w ic , given non-equi i rium, ave an un nown istri ution. But or a articular choice of that otential - the so-called Maxwell otential which is of inverse fifth power in the molecular separation - the relative velocities
- that is, unchanging with time, and that this is so independently of the eta sot e orce aw among t e mo ecu es. symmetry postu ate on des of transfers of molecules from one veloci ran e to another allows him to argue that this distribution is the unique such stationary distri-
The paper then applies the fundamental results just obtained to a vanety 0 transport pro ems: eat con uctl0n, VISCOSIty, slon 0 one as into another, and so on. The new theory allows one to calculate from basic micro-quantities the values of the "transport coefficients," numbers
comparison of the results of the new theory with observational data, a oug e 1 cu les encoun ere In ca cu a Ing exac va ues In t e theory, both mathematical and due to the need to make dubious assumptions about micro-features, and the difficulties in exact experimen-
32
Physics and chance
3. Boltzmann In 1868, 1. Boltzmann published the first of his seminal contributions to 'n i h en 1'z for velocity found by Maxwell to the case where the gas is subjected to p p
,
In the second section of this paper he presents an alternative derivation of the equilibrium distribution, which, ignoring collisions and kinetics resorts to a method reminiscent of Maxwell's first derivation. B assuming that the "probability" that a molecule is to be found in a region
In a crucially important paper of 1872, Boltzmann takes up the problem of non-equilibrium, the approach to equilibrium, and the "explanation" of the irreversible behavior described b the thermod namic Second Law. The core of Boltzmann's approach lies in the notion of the distri-
some specified value of the energy x and x + dx. He seeks a differential equation that wi speci ,given the structure of this function at any time, its rate of chan e at that time. The distribution function will change because the molecules collide
this, some assumptions are made that essentially restrict the equation to a particu ar constitution 0 t e gas an situations 0 it. For examp e, the ori inal e uation deals with a as that is initiall s atiall homo eneous. One can generalize out of this situation by letting f be a function
by a "streaming" term that takes account of the fact that even without co isions t e gas wi ave its lStri ution in space c ange y emotion of the molecules unim eded aside from reflection at the container walls, The original Boltzmann equation also assumes that the gas is sufficiently
not be taken into account. In Section 111,6,1 I will note attempts at genera lzlng eyon t is constraint. In order to know how the ener distribution will chan e with time, we need to know how many molecules of one velocity will meet how
Historical sketch
33
of Collisions. One assumes the absence of any "correlation" among molecules of given velocities, or, in other words, that collisions will be "totall random." At an time then the number of collisions of molecules of velocity VI and V 2 that meet will depend only on the proportion of
This - along with an additional postulate that any collision is matched by a time-reverse collision in which the output molecules of the first collision would if their directions were reversed meet and enerate as output molecules that have the same speed and reverse direction of the
his famous kinetic equation:
of energy, as it was expressed in Boltzmann's paper. What this equation escri es is t e raction 0 mo ecu es wit ve ocity VI' I changing over time. A molecule of veloci V mi ht meet a molecule of veloci V and be knocked into some new velocity. On the other hand, molecules of
,
another).
,
Physics and chance
34
1S
justifying the claim that the
.
.
equation finally
rmo yn i : i qu qui" s a wi be ceaselessly and monotonically approached from any non-equilibrium state. It is to justifying the claim that the Maxwell-Boltzmann distribution is the uni ue 'stationa solution of the kinetic e uation that Boltzmann turns. The definition of H is arrived at by writing lex, t) as a function of velocity: H=
4. Ob 'ections to kinetic theory The atomistic-mechanistic account of thermal phenomena posited by the 'netic t eory receive a osti e reception rom a segment o. e scientific communi whose two most rominent members were E, Mach and P, Duhem, Their objection to the theory was the result of two program-
One theme was a general phenomenalistic-instrumentalistic approach to SC1ence. rom t 1S p01nt 0 V1ew, e purpose 0 SC1ence 1S e production of sim Ie, com act, lawlike eneralizations that summarize the fundamental regularities among items of observable experience, This view "
Historical sketch
35
phenomenological laws of thermodynamics. The other theme was a rejection of the demand, common especially amon En lish Newtonians that all henomena ultimatel receive their explanation within the framework of the mechanical picture of the world.
scientific treatment for only a portion of the world's phenomena. From this point of view, inetic eory was a misgui e attempt to assimilate the distinctive theo of heat to a universal mechanical model. There was certainly confusion in the view that a phenomenalistic-
irreversibility of thermal phenomena, seems to have been initially noted y axwe lmse ln correspon ence an y omson m pu lcation in 1874. The roblem came to Boltzmann's attention throu h a oint made by J. Loschmidt in 1876-77 both in publication and in discussion
guarantee to us that a gas whose micro-state consists of one just like the equll flum gas - excep a e lrec lon 0 mo lon 0 eac cons 1 en molecule is reversed - will trace a path through micro-states that are each the "reverse" of those traced by the first gas in its motion toward
36
Physics and chance
S(b') > S(a')
Figure 2·1. Loschmidt's reversibility argument. Let a system be started in
.
.
.
invariance of the underlying dynamical laws that govern the evolution of the th r must be a micro-state b' that evolves to a micro-state a' and such that the entropy of b', 5(b'), equals that of b and the entropy of a' equals that of a 5 a' as Boltzmann defines statistical entro ). 50 for each "thermodynamic" evolution in which entropy increases, there must be a corresponding "anti-thermod amic" evolution ossible in which entro decreases.
means that the second gas will evolve, monotonically, away from its equi i rium state. ere ore, Bo tzmann's H- eorem is incompati e wi the laws of the under! in micro-mechanics. (See Pi ure 2-1.) A second fundamental objection to Boltzmann's alleged demonstration
Reversibility Objection. In 1889, H. Poincare proved a fundamental t eorem on testa 1 lty 0 motIon at IS governe y e aws 0 Newtonian mechanics. The theorem onl a lied to a s stem whose energy is constant and the motion of whose constituents is spatially
at a given time in a particular mechanical state.
In 1896, E. Zermelo applied the theorem to generate the te eli e retnwan ,or ecurrence Jec lon, 0 0 zmann s mec anlcally derived H-Theorem. The H-Theorem seems to say that a system started in non-equilibrium state must monotonically approach equilib-
Historical sketch
37
Figure 2-2. poincare recurrence. We work in phase-space where a sin e oint re resents the exact microsco ic state of a system at a given time - say the position and velocity of \ I eve molecule in a as. Poincare shows for certain s stems, such as a gas confined in a box and energetically isolated from the outside world, that if the system starts in a certain microscopic state 0, then, except for a "vanishingly small" number of such initial states, when the system's evolution is followed out along a curve p, the system will be found, for any small region E of micro-states around the original microus, "a most a " suc state 0 to return to a mIcro-state In t at sma regIon E. systems started in a given state will eventually return to a microscopic state "very c ose to at Imtla state.
one likes. But such a state would have a value of H as close to the initial value as one li es as we . Hence Bo tzmann's emonstration 0 necess monotonic a roach to e uilibrium is incom atible with the fundamental mechanical laws of molecular motion. 5. The probabilistic interpretation of the theory
tion has any definitive answer. Suffice it to say that the discovery of the everS1 11ty an ecurrence Jectlons prompte t e 1scoverers 0 t e theo to resent their results in an enli htenin wa that revealed more clearly what was going on than did the original presentation of the theory.
meant. The language here becomes fraught with ambiguity and concepe tua 0 scunty. ut 1t 1S no my purpose ere e1 er 0 ay ou a possible things they might have meant, or to decide just which of the many understandings of their words we ought to attribute to them. Again,
38
Physics and chance
Maxwell's probabilism. In a train of thought beginning around 1867, vo 0 y e on flow of heat from hot to cold is only the mixing of molecules faster on the average with those slower on the average. Consider a Demon capable of seein molecules individuall a roachin a hole in a artition and capable of opening and closing the hole with a door, his choice
left, thereby sorting a gas originally at a common temperature on both sides into a compartment of hot gas and a compartment of cold gas. And doin this would not re uire overt mechanical work or at least the amount of this demanded by the usual Second Law considerations. From
Whether a Maxwell Demon could really exist, even in principle, iscussion. 1. Bri ouin an ecame in ater years a su ject 0 muc 1. Szilard offered ar uments desi ned to show that the Demon would generate more entropy in identifying the correct particles to pass through
Later, arguments were offered to show that Demon-like constructions cou aVOl at m 0 entroplc mcrease as e resu t o e emon s rocess of knowled e accrual. More recently, another attack had been launched on the very possibility
that the Demon, in order to carry out its sorting act, must first register in a memory t e act t at It IS one sort 0 partlc e or e 0 er Wl w lC it is dealin . After dealin with this article, the Demon must "erase" its memory in order to have a blank memory space available to record the
by the Demon and fed into its environment. It is this entropy generation, t ey argue, a more an compensa es or e en ropy re uc lon accomplished by the single act of sorting. In his later work, Maxwell frequently claims that the irreversibility
Historical sketch o
0
39
0
only due to limitations on our knowledge of the exact trajectories of the "in principle" perfectly deterministic, molecular motions. Later popular writin s however do s eak if va uel in terms of some kind of underlying "objective" indeterminism. •
with Loschmidt, Boltzmann began a process of rethinking of his and Maxwell's results on the nature of equilibrium and of his views on the nature of the rocess that drives s stems to the e uilibrium state. Various probabilistic and statistical notions were introduced without it being always
world") toward equilibrium emerged in Boltzmann's writings. One paper 0 1877 rep ied speci ca y to Losc idt's version 0 the Reversibilit Ob'ection. How can the H- Theorem be understood in light of the clear truth of the time reversibility of the underlying micro-
by taking the statistical viewpoint. It is certainly true that every individual mlcro-state as e same pro a llty. ut ere are vast y more mlcrostates corres ondin to the macrosco ic conditions of the s stem bein in (or very near) equilibrium than there are numbers of micro-states
will be many more of the randomly chosen initial states that lead to a un orm, equll flum, mlcro-state at t eater hme t an ere Wl e initial states that lead to a non-equilibrium state at the later time. It is worth noting that arguments in a similar vein had already appeared in a
40
Physics and chance
ways in which molecules can be placed in the momentum boxes, always onsi er a a e e ne y i ri ion, a pec a'on 0 e num er of molecules in each momentum box. For a large number of particles and boxes, one such distribution will be obtained by a vastly larger umber of assi nments of molecules to boxes than will an other such distribution. Call the probability of a distribution the number of ways it
of boxes go to infinity and the size of the boxes go to zero and one discovers that the energy distribution among the molecules correspondin to this overwhelmin I robable distribution is the familiar MaxwellBoltzmann equilibrium distribution. (See Figure 2-3.)
away from the approach that takes equilibrium to be specified as the unique stationary solution of the kinetic equation. As such it shares " recariousness" with Maxwell's ori inal ar ument. But more has been learned by this time. It is clear to Boltzmann, for example, that one must
awareness of this stems from considerations of collisions and dynamics at te us t at it is on y t e ormer met od t at wi ead to stationary distributions and not the latter. And as we shall see in the next section Boltzmann is also aware of other considerations that associate probabil-
Combining the definition of H introduced in the paper on the kinetic equation, t e ca cu ate monotonic ecrease 0 H imp ie y t at e uation the role of entro S in thermod namics (su estin that S in some sense is to be associated with -H), and the new notion of prob-
state is determined simply by the number of ways in which the macroy arrangements 0 e constituent mo ecu es 0 state can e 0 talne the s stem. As it stands, much needs to be done, however, to make this "definition" of entropy fully coherent.
41
Historical sketch A
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