Foundations of Physics, Vol. 18, No. 4, 1988
Book Review
The Shaky Game: Einstein, Realism, and the Quantum Theory. By Arthur Fine. University of Chicago Press. Chicago and London, 1986, xi + 186 pp., $29.95 (hardback).
Arthur Fine is a philosopher of science who for some fifteen years now has been engaged, by his own description in this book, in "the technical defense of... quantum realism" (p. 3). What this means is that he has been attacking statements of the form "Quantum theory has shown us that ... is false," where in place of the ellipses one might put "the notion that noncommuting observables have simultaneous exact values," or "the idea that the propositional structure of the microworld is that of classical logic," or even "the ideal of an objective, determinate, locally causal reality for the microworld." The arguments for such statements are generally in the form of "no-go" results, purporting to show that implementation of some such realist assumption leads to quantitative inconsistencies with the predictions of quantum theory. Fine's defense of realism has been "technical" in showing that such arguments depend in turn on other assumptions, usually disguised in formal relations, which are themselves outside anything in quantum theory itself. His most recent work, concerned with the most powerful of the no-go results surrounding Bell's theorem, has appeared in journals as disparate as the Journal of Philosophy and Physical Review Letters. Along the quixotic way of this long defense, he "gradually began to realize," as he tells us in his first chapter, "that in some cases Einstein had pioneered similar ideas and had been severely criticized for so doing" (p. 3). It is the results of Fine's investigations into Einstein's attitudes toward the quantum theory that are gathered here. Everyone seems to know of Einstein's skepticism about quantum theory. Generally, what they know about is his unwillingness to accept an irreducible role for probability in fundamental physics. His terse "God does not play dice" has slipped into the index of standard quotations. But Fine presents a second focus of Einstein's concern over the quantum theory. 471 0015-9018/88/040o-0471506.0o/0 © 1988 Plenum Publishing Corporation
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Speaking of quantum theorists generally in a letter of 1950 to Erwin Schr6dinger, Einstein brooded, "Most of them simply do not see what sort of a risky game they are playing with reality" (p. 2). It is the image of a risky, or, as he calls it, a shaky game that Fine takes as his central theme. Fine takes the quote as encapsulating Einstein's concern (from the mid 1920's on) with the direction in which physics was to grow. For, Einstein appreciated that physics involves elements of "free construction" and a kind of intellectual play. In this way it is a game, a game played with reality as its subject. But the game is "shaky," as Fine says, "because without firm foundations or superstructures, [itsl outcome is uncertain. Indeed not even the rules of play are fixed. It follows that at every step we have to be guided by judgment calls" (p. 2). I think [Einstein] understood that the dispute over the quantum theory was important precisely because past scientific practice, which he saw as developing a program for causal and realist theories, did not have built-in rules that would fix the character of future science. What he saw, I think, was that just because science is a shaky game, the realist program was at risk. (p. 2) There are large themes here. And the book clearly has two major pieces of work to do. First, Fine has to make a case for his construal of this second focus of Einstein's concern over the quantum theory. And second, he must make a case for the extrapolation of this image of a "shaky game" beyond quantum theory in the 1920's and 30's to a characterization of science in general, or, as he puts it, of "all the constructive work of science and of the philosophical or historical programs that seek to place and understand it" (p. 2). Considering that t h e n i n e essays (seven previously published) that make up the book were written over a period of ten years, they make a remarkably coherent assault on these large tasks. Chapters 2-6 take up the first. Chapter 2 reassesses the established insinuation that Einstein's failure to embrace quantum theory was due to a conceptual conservatism born of a kind of scientific senility. Fine argues against this view by illustrating Einstein's positive contributions to quantum theory right through 1926 and his actual anticipation of some of its central features. But the force of the chapter is a reexamination of the nature of Einstein's criticisms of the views of Niels Bohr, in particular the doctrine of complementarity, with its restriction to classical concepts. In Fine's analysis, Einstein comes out the more radical of the two men, the more critically purist in his rejection of "conservative" conceptual structures. This is a decidedly heretical conclusion. And the four chapters which follow it are equally heretical. In Chapter 3 we learn that the famous Einstein-Podolsky-Rosen paper of 1935 was actually written by Boris
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Podolsky alone, "after much discussion" (as Einstein put it) with Einstein and Nathan Rosen, and that the central point (so far as Einstein was concerned) was "smothered in the formalism" (p. 35) in Podolsky's text. In Chapter 5 we learn that discussions of this central point with Erwin Schr6dinger in the summer of 1935 were instrumental in Schr6dinger's formulation of his famous "cat paradox." In Chapter 4 we learn that what is generally referred to as "Einstein's statistical interpretation" has little support in Einstein's own writings. What Fine finds instead is a very vague set of notions, presented only as a rhetorical setting for calling attention to the incompleteness of the quantum theory. And in Chapter 6 we learn that Einstein's vaunted realism was not really a cognitive doctrine at all, not a set of specific beliefs about nature. "Rather," summarizes Fine, "his realism functions as a motivational stance toward one's scientific life, an attitude that makes science seem worth the effort" (p. 7). Where all this heretical conceptual biography comes from is largely Einstein's letters, melded and synthesized by Fine. He began the project just at the time when Einstein's letters were becoming more accessible than they had been since Einstein's death. Some of these letters had been generally available, for instance in volumes edited by Przibam, by Born, and by Speziali. But Fine has found several important ones that have not previously been considered by Einstein biographers. (As the complete publication of all Einstein's writings by the Princeton University Press has begun, such detective work will soon be much simpler.) Fine's analyses and syntheses here are all more or less heretical. But he is self-conscious and good-natured about his heresy, and his treatments are well documented. They have to be taken seriously. For, the larger issues growing out of these Einstein essays are very important, particularly to philosophers and historians of science--those, as Fine says, "who seek to place and understand" science. If Einstein feared for the future of his motivational realism in the shaky game of science, Fine exults (in the seventh essay) in the failure of cognitive realism. In the eighth essay he attacks other "antirealisms" that recently have been put forward to take its place (constructive empiricism, for instance). In the ninth, to turn the collection round full-circle to Einstein's concerns, he drives home the point in detail that realism, while not contradicted by anything in quantum theory (the Bell's theorem work particularly included), is "a program lying outside the life of physics itself" (p. 11). This last phrase is the key to the positive proposal within the critical thrust of these last three chapters. Fine bids us give up all these global strategies for interpreting and providing a setting for science as a whole and adopt what he calls the "natural ontological attitude (NOA)." In place of global realism (or any other ism) we are simply to accept the shakiness
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of the game of science and abide by the "in-house series of judgment calls"~both theoretical and experimental- that go into the acceptance as "true" of some existence claim within the community of scientists at some particular time. We are not to seek some further "meaning" for "true." Fine's goal thereby is to make the philosophy of science connect as sclosety as possible with on-going science, a goal with which most scientists can surely empathize. Roger Jones
Philosophy Department University of Tennessee Knoxville, Tennessee 37996-0480
1This review was written during a period of support provided by the National Science Foundation.
Foundations of Physics, Vol. 18, No. 4, 1988
Book Review
Maximum Entropy and Bayesian Methods in Applied Statistics. Edited by James H. Justice. Cambridge University Press, Cambridge, England 1986, 2 + 319 pp., $44.50 (hardcover).
A small band of determined Bayesians has been meeting annually for the last 10years. This book covers the meeting on August 5-8, 1984 in Calgary, Alberta, Canada. The participants have in common a belief that the maximum entropy principle and Bayesian methods provide a superior approach to problems of inference. Their meetings are characterized by a broad variety of application examples and philosophical discussions. The topics of previous meetings have ranged over the reconstruction of images, the interpretation of molecular beam interactions, the species of butterflies to be found in forests, the analysis of ocean waves to find signals, the interpretation of seismic records, the compositions of reactants in competing chemical reactions and the estimation of the density distribution in the earth. The intellectual leader of the band is Edwin T. Jaynes, a physicist at Washington University in St. Louis, upon whose work most of the participants base their own. The principal ideas are: 1. The central problem of statistical inference is to assign numbers to probabilities which are attached to symbols which represent unambiguous statement. The numbers range from 0 to 1 representing "false" or "true." These numbers represent "a unique encoding of incomplete information" and as such represent a state of knowledge, not of things. The numbers assigned as probabilities are not "degrees of belief." They are not surrogates for frequencies. This group is solidly united in rejecting the frequency definition of probability and if, during the meetings, one of them strays into such an interpretation he or she will be quickly challenged by the others. 2.
When all that is given is a probability distribution, the unique 475 0o15-9018/88/0400-0475506.00/0 © 1988 Plenum Publishing Corporation
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measure of what is not known is Shannon's information measure, - k Y" Pi In Pi. 3. To remove the "degree of belief' connotation from the process of assigning probabilities, and to guard against accidentally introducing information not available, the maximum entropy principle (MAXENT) is invoked. According to this principle, that probability distribution is selected which maximizes the entropy, subject to constraints provided by the available information. This process provides the "minimally prejudiced assignment subject to what is known." (See The Maximum Entropy Formalism, Levine and Tribus, MIT Press, Cambridge, Massachusetts, 1979.) The book contains 17 articles. Although the topics vary widely, the articles are connected by a common methodology. The first two articles are by Ed Jaynes. "Bayesian Methods: General Background" discusses what is different about the way Bayesians (should) reason and how classical statisticians (normally) approach problems. Since many people write articles which are supposed to represent Bayesian methods but do not and many classically trained statisticians sometimes use some Bayesian methods, the categories into which Jaynes separates people are not so sharply defined. Nevertheless, there are many strong opinions on the use or non-use of Bayesian methods, and in this article Jaynes examines their basis. This reviewer sides with Jaynes in believing that all the existing textbooks of statistical do a very poor job of introducing Bayesian methods. Jaynes begins with a history of the development of statistical inference, from Herodotus, Bernoulli, Bayes, Laplace, Jeffreys, Cox, and Shannon. He describes the different kinds of discourse which keep Bayesians and classically trained statisticians from understanding one another. Example: If a classically trained person is asked to assign a probability to the value of the gravitational constant, he believes that the value of g has a distribution, otherwise the problem makes no sense. To a Bayesian it is a probability that is distributed; g is an unknown constant. A Bayesian has no problem with assignment of probabilities to hypotheses. The classically trained person objects that an hypothesis is not a random variable. The classically traited person believes that "p" represents an "approximation" to something; the Bayesian interprets it as a representation of incomplete information. The second article, "Monkeys, Kangaroos and N," discusses the role of maximum entropy in image reconstruction. The problem is related to the completely random activity of monkeys and typewriters, on the one hand, and the much more restricted problem of the kangaroos, on the other hand ("If 3/4 of the Kangaroos are left-handed and 3/4 drink Foster's, what
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fraction of them are left-handed and drink Foster's?"). The issue is how to portray more precisely what we know before we start the image reconstruction. As the number of pixels, N, grows larger, the differences between these two approaches becomes greater. Jaynes discusses the more accurate elucidation of prior information. In "The Theory and Practice of the Maximum Entropy Formalism" R.D. Levine discusses the justification for applying MAXENT to systems in which N is small is presented, with examples from statistical physics. Stephen F. Gull and John Fielden (whose company name, "Maximum Entropy Consultants," gives a clue to his beliefs) discuss the application of Bayes Theorem and the MAXENT principle to nonparametric interpretation of statistical information. J. Aczel and B. Forte discuss alternative measures beyond Shannon's measure in "Generalized Entropies and the Maximum Entropy Principle." In "The Probability of a Probability" John Cyranski discusses the relation between the "degree of belief' interpretations and the use of MAXENT. N. C. Dalkey, in "Prior Probabilities Revisited," presents an analysis with which this reviewer disagrees strongly. The key sentence is "The present paper presents an approach to the estimation of prior probabilities when these are unknown." Prior probabilities are assigned They are not "unknown." The person assigning them may be in doubt as how best to portray his or her prior knowledge and may not have carried out the examination of the prior knowledge required to make an honest assignment. The objective of the paper seems to be to figure out what prior knowledge will give the best agreement with the posteriori observations. This strikes me as something akin to painting the bulls-eye around the arrow after it has been shot. "Band Extensions, Maximum Entropy and the Permanence Principle" by Ellis, Gohberg, and Lay discusses the application of MAXENT to signal processing. In an extremely important paper, "Theory of Maximum Entropy Image Reconstruction," John Skitling describes how to use MAXENT in the reconstruction of one- and two-dimensional images. (The reconstructed photograph of a blurred license plate on the cover of the book is due to the work of Skilling and colleagues). In this paper Skilling connects the "Monkeys and the Kangaroos" to the problem of image reconstruction. In "The Cambridge Maximum Entropy Algorithm" John Skilling describes the computational methods underlying the application of MAXENT to image reconstruction. C. G. Gray applies MAXENT to spectroscopic data in "Maximum Entropy and the Moments Problem: Spectroscopic Applications." When
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only the first few moments of a distribution are given, what should be said about the remaining moments? MAXENT gives answers which agree surprisingly well with experimental data. Paul Fougere, in "Maximum-Entropy Spectrum from a Nonextendable Autocorrelation Function," discusses how to apply MAXENT to estimate the n + 1 sample from a simple of n, given the autocorrelation function based on the sample of n. P. A. Tyraskis discusses different methods to analyze data from multichannels in "Multichannel Maximum Entropy Spectral Analysis using Least Squares Modelling." A different approach to the same problem is given by Bruce R. Musicus and Rodney Johnson in "Multichannel Relative-Entropy Spectrum Analysis." E. Rietsch examines data on the density of the earth as obtained from different sources and uses these as constraints on a MAXENT estimate of the variation in the density of the earth with depth in the paper "Maximum Entropy and the earth's Density." James Justice, in "Entropy and some Inverse Problems in Exploration Seismology," applies MAXENT to the interpretation of seismic signals at the surface of the earth in the construction of estimates of density variations and structure in the interior. Ramarao Inguva and James Baker-Jarvis apply MAXENT to quantum mechanical inverse scattering, the electromagnetic inverse problem, and the problem of inverse scattering in their paper "Principle of Maximum Entropy and Inverse Scattering Problems." This book demonstrates that, in the 30 years since the presentation of MAXENT, enough new and original problems have been solved to justify treating it as a fundamental principle of physics and engineering. It is a pity it is still treated as an obscure topic in existing textbooks and still not a central topic in the teaching of statistics. Myron Tribus 350 Britto Terrace Fremont, California 94539
Foundations of Physics, Vol. 18, No. 4, 1988
Book Review
The Anthropic Cosmological Principle. By John D. Barrow and Frank J. Tipler. Clarendon Press, Oxford, and Oxford University Press, New York, 1986, xx + 706 pp., $29.95 (hardback). Barrow and Tipler have had a fruitful collaboration in writing many papers that are novel and provocative, often controversial and occasionally irritating, but always interesting and stimulating. This book follows in that tradiction. The authors have an amazing breadth of academic interests, which they use to tackle a controversial subject from a wide variety of angles, some of which will entertain, some of which will provoke, and some of which will inspire the reader. The anthropic principle is basically the selection effect that our own existence has on the properties of the universe we observe. Its controversial aspect is its value as an explanatory principle. It does seem to help explain certain features of the world we inhabit because of the requirement that this world be habitable, but it is not clear why such a habitable world should exist at all. Barrow and Tipler define several versions of the anthropic principle in the Introduction of their book, each having varying degrees of acceptance. The least controversial is the weak anthropic principle (WAP), which says roughly that what we observe about the universe is restricted by the requirements of our existence as observers. A stronger but more controversial version is the strong antropic principle (SAP), which says that the universe must have the properties necessary for life to develop at some time within it. The authors put forward an even more speculative extension they call the final anthropic principle (FAP): Intelligence must develop within the universe and then never die out. The book proceeds from the Introduction with nearly two hundred pages of discussion of the nontechnical historical concepts, from the Old Testament to Teilhard de Chardin, that have provided the philosophical background for the anthropic principle. Then Barrow and Tipler recount 479 0015-9018/88/0400-0479506.00/0 © 1988 Plenum Publishing Corporation
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the modern technical development of the anthropic principle in physics and astronomy, which began primarily with Robert Dicke's 1961 WAP explanation of the Eddington-Dirac large number coincidence to/(e2/m~c3) ~e2/(GmNme). A large number of further implications of the WAP in physics, astrophysics, classical cosmology, and biochemistry are covered in three further chapters. One chapter discusses the curious implications of quantum mechanics for the anthropic principle and how the various versions of the principle would be formulated in a quantum mechanical description. Another chapter summarizes Tipler's widely published space-travel argument against the existence of extraterrestrial intelligent life (if it existed, at least within the Galaxy, it would most probably have already come here). The final chapter, which is the most speculative and perhaps the most interesting, discusses the future of the universe and proposes that it may end in an "Omega Point" with life achieving omnipotence, omnipresence, and omniscience. In these chapters Barrow and Tipler amass an enormous collection of material with at least some tangential relevance to the anthropic principle. For example, Chapter 5 shows how a large number of empirical quantities in physics, astrophysics, chemistry, and biology are determined, at least to within an order of magnitude or so, by the fundamental constants of nature. There are also a number of amusing sidelights, such as how to catch fleas (p. 119) and why the government should not interfere with the economic system (p. 173). In an attempt to include as many arguments and calculations as possible, the authors have shown little caution in resorting to half-baked ideas and often have not been careful to eliminate fairly obvious errors. As an example of the former, on p. 105 they argue that "many modern cosmologists" believe the Hartle-Hawking wave function is the key to a "single logically possible universe" and that the necessity of quantum mechanics would imply that "only one unique Universe--the one we live in--is logically possible." However, Hawking has emphasized that his wave function cannot be derived purely from quantum mechanics and is only a proposal for the state of the universe (albeit the most natural one in his opinion). Barrow and Tipler go on to admit that their own "discussion sounds a bit woolly," but they fail to point out here that a different wave function is logically possible and could give a different universe with different observational properties having nearly unit conditional probabilities. Even in the many-worlds interpretation, the significance of the wave function is not only the "domain of possibilities" that it contains but also the measure (traditionally interpreted as probabilities) it gives to these possibilities. Essentially this point is admitted on p. 496 with the statement that "different boundary conditions imply different physics."
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Also on the subject of the quantum state of the universe, the authors should have pointed out that their quantization of the Friedmann universe on pp. 490ff, with the resulting wave function depending on "conformal" time as well as the radius of the universe, is nonstandard. Many other workers in quantum cosmology would question what it would mean for the wave function to depend upon some unobservable parameter such as conformal time. There are also several minor errors that ! readily noted. For example, the claim on the bottom of p. 316 that L 3 v 0(2 L 3 implies v oc L -1 leads me to question whether the authors have ever tested their "strategy for escaping from grizzly bears." On p. 328 it is claimed that the thermal pressure of electrons is millions of times less that of nucleons. On p. 413, by erroneously taking A~m/Znt o > h rather than the correct limit (Aminc2/G)(cto)3to>h, it is even argued that observations limit the cosmological constant to be nearly 65 orders of magnitude smaller than the smallest measurable value allowed by the Heisenberg uncertainty principle within the age of the universe, to N 101° years! These sorts of errors have little direct consequence for the general argument of the book, but they do serve as a warning to the reader not to accept any calculation before checking it himself. Barrow has informed me that many of the errors (such as the one about grizzly bears) are typographical, arising from the fact that this was the first book Oxford University Press ever typeset directly from computer disk. He hopes all of these will be corrected in the second printing. Despite the misinterpretations and errors of the book, I was impressed by how many restrictions the weak anthropic principle apparently places on our observations. It does seem to be a strong selection effect on where we can live in the world or even on which world we can inhabit in the many-worlds interpretation of quantum mechanics. I am tempted to go beyond the book and speculate that if the universe is described by one of the new superstring theories with no arbitrary fundamental constants, then the weak anthropic principle (plus other observed facts about our world) may be necessary to explain the effective physical constants we observe in our four-dimensional space-time, if they vary from world to world. On the other hand, I am skeptical that either the strong anthropic principle or the final anthropic principle are true in the form they are defined in the book, namely that intelligent life exists or persists respectively in all possible worlds given by the quantum state of the universe. Surely there is some nonzero amplitude for a world or component of the wave function to exist without life, unless the wave function is more finely tuned to life that we have any right to believe. I suspect it may even be impossible to fine-tune any wave function, consistent with some fairly sim-
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pie dynamical "theory of everything" (e.g., a superstring theory), so that in all component worlds, life does not die out. Thus I feel that the exceedingly optimistic view of the future, given in the last few pages of the book, which more or less deifies the universe, is science fiction rather than reasonable speculation or likely extrapolation using the known laws of physics. However, I recommend that anyone with a strong background in physics and cosmology, and an interest in the anthropic principle, read the book for himself and make his own critical evaluation. Don N. Page
Department of Physics The Pennsylvania State University University Park, Pennsylvania 16802
Foundations of Physics, Vol. 18, No. 4, 1988
Book Review
The Nature of Irreversibility. By Henry B. Hollinger and Michael J. Zenzen. D. Reidel Publishing Co., Dordrecht, The Netherlands, 1985, xi + 340 pp., (hardcover). In The Nature of h'reversibility Henry Hollinger and Michael Zenzen undertake the arduous task of delving into the origins of irreversibility and making it comprehensible to the scientific layman with some background in physics. The book begins with the grandiose promise to chip away the encrustment of mathematics that has hidden the origins of irreversibility but, regrettably, winds up scavenging the technical mathematical details for the clues without ever catching the culprit. Be that as it may, it would not signal the defeat of such a difficult task of uncovering the origins of irreversibility, were it not for a fundamental error in their logic. The central theme of the book is that "All of the equations are invariant under time reversal, as indeed they must be. Irreversibility in the sense of violating invariance under time reversal is logically untenable within dynamical schema. Even the Boltzmann equation, upon which the paradoxes about irreversibility were initially erected, and which is still said to be irreversible, is actually invariant under time reversal..." They fault Onsager's derivation of the reciprocal relations based on temporal reversibility because "there is no such thing as temporal reversibility." They claim that "macroscopic reversibility is the origin of the reciprocal relations, and the problem is to explain how that comes about out of microscopic
irreversibility." The authors are taking on the giants--Boltzmann, Ehrenfest, Onsager, and just about everyone else who holds the conventional notion of irreversibility in macroscopic physics. "We are tempted to conclude that the diffusion equations are really 'irreversible' in the sense of being not invariant under time reversal and we are tempted to identify that irreversibility with the commonly expressed irreversibility of diffusion, with the idea associated with 'decay,' 'dissipation,' 'anistropy in time,' etc." The test of time-reversal, 483 0o15-9018/88/040o-0483506.0o/0 (~) 1988 Plenum Publishing Corporation
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according to them, is not only "changing time t into - t and reversing the signs of the motion variables" (which include energy and material fluxes!) but also in changing the diffusion coefficient, D, to - D since the diffusion coefficient "is proportional to time"! By the time that the authors reach their conclusion that, "we do not believe in 'temporal irreversibitity' under any circumstances...," it is rather obvious that there is something definitely amiss. Their error was to employ a local equilibrium MaxwelI-Boltzmann distribution in an expression where the reversible Liouville equation has been used. For small times, the distribution function f ( x , v, t) can be expanded as f ( x , v, O) + tO/Otf(x, v, 0). But the authors use the expression f ( x , v, t) = f ( x , v, O) - tv O/~?xf ( x , v, 0) which implies that f satisfy the reversible Liouville equation 3/~?tf+ v#/Oxf= O. Then, according to them, the flux of mass is given by Jm -~ ~ d V V m f ( x , v, t) = ~ d V V m ( f - tv 8/3xf). Previous reasoning has suggested to the authors that the correct f to use is the local equilibrium one given by the Maxwell-Bottzmann formula fo = n(m/2z~kT) 1/2 e x p [ - m ( v - v o ) 2 / 2 k T ] where they are prepared to allow the particle number, n, and absolute temperature, T, to vary (slowly) with x and t. Then evaluating the expression for the flux of mass and comparing it with Fick's first law, J~ = - D 8/Ox p, they come out with the expression D = m ~ d V VVt(Jo/n), showing that the diffusion coefficient is, indeed, pl"oportional to time! However, the f they have chosen, namely local equilibrium distribution, is incompatible with the fact that the distribution satisfy the reversible Liouville equation and it is precisely the expression f ( x , v, t)= f ( x , v, O ) - tv O/Oxf(x, v, 0) which is responsible for the diffusion coefficient having acquired an explicit dependence upon time. In fact, the authors are aware of this incompatibility as their preceding discussion of the entropy balance equation and, in particular, the entropy production, a = - k S dv In f ( 8 / S t f + v O/ax f ) , shows. Any f satisfying Liouvitle's equation will automatically lead to the conclusion that "entropy cannot be created or destroyed." The authors then go on to argue that the correct f to use in the expression for the entropy, s(x, t) = - k ~ dvfln f is the "most probable" f which, admittedly, "has very little similarity to the actual evolving f..." This, they claim, among other things, gives the correct expression for the entropy production since the most probable f does not satisfy the Liouville equation and consequently the entropy production does not vanish. According to them, this resolves a lot of the errors in discussions about entropy. But, by the same token, it is incompatible for them to have used the "most probable" f in the first-order expansion of the distribution function in time in which the Liouville equation has been used.
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The authors have, unwittingly, found their own mistake and we are put at ease to learn that the transport coefficients do not depend explicitly on time and that macroscopic equations are, indeed, irreversible as they should be. Their arguments are based on deceptively simple examples and their reasoning is often circuitous and contradictory. In Chapter 7, a time te is introduced as the "time at equilibrium" or "period of equilibrium" only to learn in Chapter 8 that "equilibrium is 'stagnant' and constant in time in every way." By definition, equilibrium is a time-independent state, so what sense does it make to talk about "young and old equilibria"? We are told that the law of mass action is a stochastic assumption albeit the kinetic equations can be solved deterministically, given the initial concentrations; that equilibrium equations of state "reek with irreversibility" and "irreversibility exhibits its full nature at equilibrium" although there is no motion at equilibrium; that macroscopic reversibility is due to microscopic irreversibility and not the other way around; that the dissipative nature of fluids is due to the fact "that they are always leaving equilibrium, never approaching..."; and that entropy "creation" (production) "can be both positive or negative" which is in flagrant contradiction with the Second Law. In fact, they are "hesitant" about the Second Law (or better the second part of that law) "because of the common feeling that we might be leaving Newtonian dynamics and hence all possibility of "understanding," maybe even waving in the air the kinds of statistical, quantum mystical, cosmologically overwhelming arguments that have already clone so much to obscure our objective." According to the authors, the entropy production does not contain any element of irreversibility; rather, the origin of irreversibility is to be found in the (linear) phenomenological relations relating thermodynamic forces to the fluxes in the immediate neighborhood of equilibrium. Are we then to conclude that irreversibility has the same domain of validity as the linear phenomenological relations? In the simple case of heat conduction, the positive semidefiniteness of the entropy production attests to the fact that heat will not spontaneously flow in the direction of increasing gradient of temperature, independently of whether Fourier's law is valid or not. The authors are skeptical about "formalism and theorems" and claim that "more mathematical sophistication will not lead one out of the labyrinth of irreversibility." Yet their presentation rests heavily on the kinetic theory of gases and the Boltzmann equation as seen from the point of view of a chemist. Frequently, technical details are emphasized at the expense of given the reader a balanced presentation. Often symbols are not defined and cited references are not to be found in the bibliography. Early
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in the book, the authors manipulate terms like entropy and entropy production as if they belong to the comon English language before defining exactly what they mean. An entirely new dictionary must be created; for, otherwise, the unsuspecting reader will be thrown into a state of utter confusion. No distinction is made between isolated and closed systems or microcanonicat and canonical ensembles; "the general formula" of entropy S-= k In f2 "is sometimes expressed in a more familiar way, S = - k Z P~ln Pi'! Their "statistical entropy S = k ln(Ax Av) has, apart from Boltzmann's constant k, the peculiar dimensions of the logarithm of an action. "Stochastic" has nothing to do with randomness. "Deterministic" means "not determined" and "dynamic" does not refer to the causal aspects of the motion, but rather to a set of closed equations. "Complex" equations refer to a linear combination and "recent" equilibrium really means a state of "local" equilibrium. The term "fluid" includes everything from a gas to a solid. In a final chapter, the authors compare their approach to Prigogine's which they refer to as some type of messianic prophesy that the "origin of irreversibility will be uncovered by some new and mysterious physics that is still waiting to be discovered." The fact of the matter is that they provide the answer themselves--neither the phenomena nor the mathematics are new. "What is new is a sudden jump in our awareness..." As an illustration of the necessity for a "wider view toward applications of the mathematical theory," the authors cite a chemical scheme that has been reviewed by Prigogine. This example turns out to be the Lotka mechanism which was devised--over six decades ago--as a simple model for undamped oscillations in an ecological system. It is only the unawareness of the audience which makes it a novelty. Moreover, the authors' interpretation is incorrect and contradicts an earlier statement that the reciprocal relations are "the basic 'law' of nonequilibrium thermodynamics." In Chapter 8, the authors go to great length to distinguish steady states from equilibrium states only to contradict themselves in the Lotka example of Chapter 11 by confusing a far from equilibrium steady state, where the rates of the reverse reactions are negligible, with an equilibrium state. Furthermore, if they had cast the linearized rate equations as a set of linear force-flux relations, they would have realized that their "basic law" of nonequilibrium thermodynamics had, in fact, been violated. The Nature of Irreversibility is on a level which is accessible to the scientific layman, whether he be a philosopher of science and technology, physicist, chemist, or anyone who has tried, at some time or another, to grapple with the apparent irreversibility of Nature. And this is all the more reason for giving an as accurate account as possible of the nature of
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irreversibility. Granted that there is a need for a simple and accurate presentation of the origins of irreversibitity which would be accessible to the scientific layman, The Nature of Irreversibility has only made the need more urgent. Bernard H. Lavenda Universit(t degli Studi di Camerino Dipartimento di Science Chimiche 62032 Camerino (MC), Italy
