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Kinetic theory of gases, the: an anthology of classic papers with historical commentary
This book introduces physics students and teachers to the historical development of the kinetic theory of gases, by providing a collection of the most important contributions by Clausius, Maxwell and Boltzmann, with introductory surveys explaining their significance. In addition, extracts from the works of Boyle, Newton, Mayer, Joule, Helmholtz, Kelvin and others show the historical context of ideas about gases, energy and irreversibility. In addition to five thematic essays connecting the classical kinetic theory with 20th century topics such as indeterminism and interatomic forces, there is an extensive international bibliography of historical commentaries on kinetic theory, thermodynamics, etc. published in the past four decadesThe book will be useful to historians of science who need primary and secondary sources to be conveniently available for their own research and interpretation, along with the bibliography which makes it easier to learn what other historians have already done on this subject. .
668 pages, Paperback
First published January 1, 1965
22 people want to read
About the author
Stephen G. Brush
51 books4 followersA scholar in the history of science, Stephen George Brush earned his BS in physics at Harvard University and his D.Phil. at Oxford University. After a year as a National Science Foundation Postdoctoral Fellow at Imperial College London, Brush worked as a physicist at the Lawrence Livermore National Laboratory in California in the area of statistical mechanics from 1959 until 1965. He was a lecturer in Physics at Harvard University from 1965 until 1968, and a historian of science at the University of Maryland, College Park from 1968 until his retirement as Distinguished Professor of the History of Science in 2007.
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September 19, 2023
Those wise enough to know the value of going back to the primary sources will revel in this collection of noteworthy papers on the kinetic theory of gases, by the historian of science Stephen G. Brush. All of the major advances are represented here, conveniently translated into English when this is not the original language. The subject goes back to Robert Hooke and Isaac Newton, who, stimulated by the nascent classical mechanics first began seriously to think about the air not as a featureless fluid but as an aggregate of tiny particles interacting according to some force law. Yet, their view differs from the modern one that would achieve acceptance by a sizable portion of the scientific community during the nineteenth century – they thought of the corpuscules of air as packed together (presumably due to the force of their gravity) and resisting closer approach as a result of their elastic springiness which accounts for pressure. Newton, in his contribution to the present volume, even estimates the force law that would explain the known characteristics of ideal gases.
The inception of the modern view of a kinetic theory may be attributed to Daniel Bernoulli in the eighteenth century, in a paper that failed to receive much attention at the time. As Brush notes, central to the kinetic theory is not just the supposition that heat represents a form of motion but also the idea that atoms in a gas move freely most of the time and that air pressure is the effect of countless collisions with the walls of the container. The caloric theory, in some versions, is compatible with a theory of heat as being due to internal oscillations of the bound state between molecules of air and the caloric particles. Thus, it was by no means easy to disentangle the two theories, since as Brush adverts, there was indeed a considerable body of evidence in favor of the caloric theory! See, for instance, Laplace’s derivation of ideal gas law from caloric theory and what it explains, included in Part I.
The real leap comes with the work of the neglected John Herapath and of James Joule. For until one has a clear idea of energy and its conservation, the mechanical theory of heat may appear inconsequential. As Brush tells us:
While Joule’s experiments, strictly interpreted, show only that the various forms of energy are interconvertible but do not appear to single out any one of them as more fundamental than the others, Joule’s popular lecture ‘On Matter, Living Force and Heat’ and Helmholtz’s memoir ‘On the Conservation of Force’ indicate quite clearly that mechanical energy is regarded as the basic entity. [p. 21]
To a large degree it is to Clausius that we owe the establishment of the kinetic theory as a bridge between the atomic theory and thermodynamics; he perceived not only that the atomic theory could thus be used to explain macroscopic thermodynamics, but also that experimental data on thermal properties of matter could be translated into specific statements about he nature of atoms themselves. [pp. 23-24]
Part II follows the development of the kinetic theory at the hands, principally, of James Clerk Maxwell and Ludwig Boltzmann. The significance of their work is to go from simple considerations of the equilibrium state to a full-blown theory of irreversible processes and of how a real gas tends to equilibrium from an arbitrary initial state.
In general, Maxwell’s treatment amount to an informal heuristics of what Chapman-Enskog do for real in the second decade of the twentieth century (their memoirs are too long and technical to make the cut here). The calculation of internal friction between two gas layers makes for the most interesting part of Maxwell’s memoir. Maxwell’s approach seems as if more powerful since one gets to transport equations without the intermediary of the collision operator, but it requires one wield better physical intuition to get there. This reviewer finds it remarkable Boltzmann had his transport equation and the H-theorem already in 1872! Note, it helps to be familiar with a modern exposition to appreciate what Maxwell and Boltzmann do for the first time; it is hard to imagine what the original readers made of their work! The heuristic derivation of the Boltzmann equation is clear, after all. It is also nice how Boltzmann recovers Maxwell’s transport equations by substituting a perturbative Ansatz for the distribution function into the Boltzmann equation (the more logical procedure!). A modern exposition will, it is true, display a higher level of technique but, for all their missteps, Maxwell and Boltzmann are more the real scientists and founders of a discipline.
Finally, it should be exciting to read in the last section of Part II the original papers on the controversy between Boltzmann and his adversaries Joseph Loschmidt and Ernst Zermelo, which everyone will have seen cursorily treated in semi-popular expositions. As one ought to expect, there is more to the dispute than suggested by textbook caricatures: Loschmidt was an old friend whom Boltzmann respects for his contribution to the kinetic theory (he was the first to arrive at a realistic estimate of the size of an air molecule) while Zermelo was a mathematician and protege of Max Planck in Berlin who wants to overturn the entire atomistic theory. At the time Planck himself was an opponent of Boltzmann yet the latter succeeds in converting the former to his own point of view! Indeed, Planck learned from Boltzmann his counting method which he puts to such good use in his blackbody radiation theory.
Part III reprints several papers written by Stephen Brush himself, whose editorial work in this volume will afford a useful reference on account of occasional observations that throw light on his subject, such as on the significance of the fluctuation-dissipation theorem. The papers in Part III make for interesting reading in view of Brush’s immense erudition. The first, entitled ‘Gadflies and geniuses in the history of gas theory’ rectifies a failing common to much writing in the history of science, which understandably tends to focus on the accomplishments of the outstanding original thinkers. But Brush points out how often these geniuses were stimulated to do their best work only when prompted by incisive criticism on the part of those whom he styles as ‘gadflies’! A good review of this topic as applied to the history of the kinetic theory follows: Boyle and Linus; Bernoulli (a ‘genius without a gadfly’), Herapath (a ‘gadfly without a genius’); Clausius and Buys-Ballot; Maxwell, Loschmidt, Boltzmann and Guthrie; Boltzmann, Culverwell and Burbury; and lastly Boltzmann and Zermelo. Everyone will have heard of Boltzmann, but who – apart from Brush – remembers Linus, Buy-Ballot, Guthrie, Culverwell and Burbury? Perhaps these gadflies receive at last their due.
The second reprinted paper, ‘Interatomic forces and gas theory from Newton to Lennard-Jones’ argues for a failure of hypothetico-deductive method because one cannot reliably determine the supposed interatomic potential from transport data alone. In response, Brush lowers the goal to a kind of fitting procedure which should at least be internally consistent, if not necessarily an accurate representation of the actual molecular properties (whatever they may be!).
The third paper on irreversibility and indeterminism can be gone through quickly if one already has read the book up to here. Brush’s point is that the shift to indeterminism happened earlier than quantum mechanics and indeed is responsible for why it could so readily be accepted in quantum theory. The final reprint in this volume consists in historical notes on statistical mechanics in the context of the philosophy of science. Here, Brush has the audacity to question whether thermodynamics has really been reduced to statistical mechanics, after all:
In the present context it seems appropriate to point out that the attempt to reduce thermodynamics to mechanics led to the introduction of statistical ideas which changed the absolute character of thermodynamics. The ‘reduced theory’ thereby was modified but at the same time strengthened since it could now be applied with more confidence to a wider range of phenomena, in particular those involving fluctuation phenomena such as a Brownian movement. It was just this kind of application of a generalized thermodynamics that led to the final establishment of the existence of atoms at the beginning of the 20th century. But the ‘reducing theory’ was also affected: the introduction of statistical thermodynamics and in particular the recognition that irreversibility involves some kind of randomness at the molecular level helped to undermine the rigid determinism of Newtonian mechanics. The result was that physicists in the first part of the 20th century were prepared to accept a new mechanics based on indeterminism. In this way one can see a close connection between the three philosophical aspects of the history of statistical mechanics – reduction, statistical law and irreversibility. [p. 532]
It is always salutary for the modern reader to be reminded of how we got to where we are now! The present volume as whole must be warmly recommended to those who wish, under Stephen Brush’s adept editorial guidance, to gain an overview of the entire course of kinetic theory. The point is not merely as an academic to reconstruct some obscure and possibly forgotten history, but to learn how to think for oneself as a scientist – what is too often neglected nowadays, when rather than to educate it seems those in control of the system aim mainly just to train another generation of puzzle solvers in the reigning paradigm. Else, why has there been essentially no progress on the problem of measurement in quantum mechanics since John von Neumann in 1932?
The inception of the modern view of a kinetic theory may be attributed to Daniel Bernoulli in the eighteenth century, in a paper that failed to receive much attention at the time. As Brush notes, central to the kinetic theory is not just the supposition that heat represents a form of motion but also the idea that atoms in a gas move freely most of the time and that air pressure is the effect of countless collisions with the walls of the container. The caloric theory, in some versions, is compatible with a theory of heat as being due to internal oscillations of the bound state between molecules of air and the caloric particles. Thus, it was by no means easy to disentangle the two theories, since as Brush adverts, there was indeed a considerable body of evidence in favor of the caloric theory! See, for instance, Laplace’s derivation of ideal gas law from caloric theory and what it explains, included in Part I.
The real leap comes with the work of the neglected John Herapath and of James Joule. For until one has a clear idea of energy and its conservation, the mechanical theory of heat may appear inconsequential. As Brush tells us:
While Joule’s experiments, strictly interpreted, show only that the various forms of energy are interconvertible but do not appear to single out any one of them as more fundamental than the others, Joule’s popular lecture ‘On Matter, Living Force and Heat’ and Helmholtz’s memoir ‘On the Conservation of Force’ indicate quite clearly that mechanical energy is regarded as the basic entity. [p. 21]
To a large degree it is to Clausius that we owe the establishment of the kinetic theory as a bridge between the atomic theory and thermodynamics; he perceived not only that the atomic theory could thus be used to explain macroscopic thermodynamics, but also that experimental data on thermal properties of matter could be translated into specific statements about he nature of atoms themselves. [pp. 23-24]
Part II follows the development of the kinetic theory at the hands, principally, of James Clerk Maxwell and Ludwig Boltzmann. The significance of their work is to go from simple considerations of the equilibrium state to a full-blown theory of irreversible processes and of how a real gas tends to equilibrium from an arbitrary initial state.
In general, Maxwell’s treatment amount to an informal heuristics of what Chapman-Enskog do for real in the second decade of the twentieth century (their memoirs are too long and technical to make the cut here). The calculation of internal friction between two gas layers makes for the most interesting part of Maxwell’s memoir. Maxwell’s approach seems as if more powerful since one gets to transport equations without the intermediary of the collision operator, but it requires one wield better physical intuition to get there. This reviewer finds it remarkable Boltzmann had his transport equation and the H-theorem already in 1872! Note, it helps to be familiar with a modern exposition to appreciate what Maxwell and Boltzmann do for the first time; it is hard to imagine what the original readers made of their work! The heuristic derivation of the Boltzmann equation is clear, after all. It is also nice how Boltzmann recovers Maxwell’s transport equations by substituting a perturbative Ansatz for the distribution function into the Boltzmann equation (the more logical procedure!). A modern exposition will, it is true, display a higher level of technique but, for all their missteps, Maxwell and Boltzmann are more the real scientists and founders of a discipline.
Finally, it should be exciting to read in the last section of Part II the original papers on the controversy between Boltzmann and his adversaries Joseph Loschmidt and Ernst Zermelo, which everyone will have seen cursorily treated in semi-popular expositions. As one ought to expect, there is more to the dispute than suggested by textbook caricatures: Loschmidt was an old friend whom Boltzmann respects for his contribution to the kinetic theory (he was the first to arrive at a realistic estimate of the size of an air molecule) while Zermelo was a mathematician and protege of Max Planck in Berlin who wants to overturn the entire atomistic theory. At the time Planck himself was an opponent of Boltzmann yet the latter succeeds in converting the former to his own point of view! Indeed, Planck learned from Boltzmann his counting method which he puts to such good use in his blackbody radiation theory.
Part III reprints several papers written by Stephen Brush himself, whose editorial work in this volume will afford a useful reference on account of occasional observations that throw light on his subject, such as on the significance of the fluctuation-dissipation theorem. The papers in Part III make for interesting reading in view of Brush’s immense erudition. The first, entitled ‘Gadflies and geniuses in the history of gas theory’ rectifies a failing common to much writing in the history of science, which understandably tends to focus on the accomplishments of the outstanding original thinkers. But Brush points out how often these geniuses were stimulated to do their best work only when prompted by incisive criticism on the part of those whom he styles as ‘gadflies’! A good review of this topic as applied to the history of the kinetic theory follows: Boyle and Linus; Bernoulli (a ‘genius without a gadfly’), Herapath (a ‘gadfly without a genius’); Clausius and Buys-Ballot; Maxwell, Loschmidt, Boltzmann and Guthrie; Boltzmann, Culverwell and Burbury; and lastly Boltzmann and Zermelo. Everyone will have heard of Boltzmann, but who – apart from Brush – remembers Linus, Buy-Ballot, Guthrie, Culverwell and Burbury? Perhaps these gadflies receive at last their due.
The second reprinted paper, ‘Interatomic forces and gas theory from Newton to Lennard-Jones’ argues for a failure of hypothetico-deductive method because one cannot reliably determine the supposed interatomic potential from transport data alone. In response, Brush lowers the goal to a kind of fitting procedure which should at least be internally consistent, if not necessarily an accurate representation of the actual molecular properties (whatever they may be!).
The third paper on irreversibility and indeterminism can be gone through quickly if one already has read the book up to here. Brush’s point is that the shift to indeterminism happened earlier than quantum mechanics and indeed is responsible for why it could so readily be accepted in quantum theory. The final reprint in this volume consists in historical notes on statistical mechanics in the context of the philosophy of science. Here, Brush has the audacity to question whether thermodynamics has really been reduced to statistical mechanics, after all:
In the present context it seems appropriate to point out that the attempt to reduce thermodynamics to mechanics led to the introduction of statistical ideas which changed the absolute character of thermodynamics. The ‘reduced theory’ thereby was modified but at the same time strengthened since it could now be applied with more confidence to a wider range of phenomena, in particular those involving fluctuation phenomena such as a Brownian movement. It was just this kind of application of a generalized thermodynamics that led to the final establishment of the existence of atoms at the beginning of the 20th century. But the ‘reducing theory’ was also affected: the introduction of statistical thermodynamics and in particular the recognition that irreversibility involves some kind of randomness at the molecular level helped to undermine the rigid determinism of Newtonian mechanics. The result was that physicists in the first part of the 20th century were prepared to accept a new mechanics based on indeterminism. In this way one can see a close connection between the three philosophical aspects of the history of statistical mechanics – reduction, statistical law and irreversibility. [p. 532]
It is always salutary for the modern reader to be reminded of how we got to where we are now! The present volume as whole must be warmly recommended to those who wish, under Stephen Brush’s adept editorial guidance, to gain an overview of the entire course of kinetic theory. The point is not merely as an academic to reconstruct some obscure and possibly forgotten history, but to learn how to think for oneself as a scientist – what is too often neglected nowadays, when rather than to educate it seems those in control of the system aim mainly just to train another generation of puzzle solvers in the reigning paradigm. Else, why has there been essentially no progress on the problem of measurement in quantum mechanics since John von Neumann in 1932?
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September 15, 2024Library-of-Congress QC175 B896 2003 AMP Library
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