Ludwig Boltzmann Ludwig Boltzmann at the age of 58, when he was a professor in Vienna. Ludwig Boltzmann The Man Who Trusted Atoms CARLO CERCIGNANI Dipartimento di Matematica Politecnico di Milano Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Carlo Cercignani 1998 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 1998 First published in paperback 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law. or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Cercignani, Carlo. Ludwig Boltzmann: the man who trusted atoms / Carlo Cercignani. Includes bibliographic references and index. 1. Boltzmann. Ludwig, 1844–1906. 2. Atomic structure–History. 3. Physicists–Austria–Biography. I. Title. QC16.B64C47 1998 530’.092–dc21 98–17743 Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn, Norfolk ISBN 0–19–850154–4 978–0–19–850154–1 ISBN 0–19–857064–3 (Pbk.) 978–0–19–857064–6 (Pbk.) 1 3 5 7 9 10 8 6 4 2 To my daughter Anna, who helped me through Boltzmann’s dense German FOREWORD Our now standard picture of matter, as presented by the physics of today, tells us that ordinary macroscopic materials are made up of atoms. Although the essentials of this picture go back to early Greek times, its general acceptance is remarkably recent. From about the middle of the nineteenth century, a gradually increasing number of physicists were indeed coming to accept the reality of atoms, but there were still a great many who regarded this “atomic hypothesis” as merely a convenient fiction which did not reflect any genuine reality at a submicroscopic level. This “hypothesis” did, nevertheless, enable (non-obvious) macroscopic qualities of substances to be deduced. In principle, knowing the laws governing the individual atoms provides a means for deducing the overall properties of materials. Yet, there is no remotely practical procedure for calculating the behaviour of a macroscopic body from a detailed calculation of the motions of all of its constituent atoms. The number of atoms making up any ordinary macroscopic system is far too enormous. A cubic centimetre of air, for example, contains some 1019 atoms. Thus, to deduce how a macroscopic material must behave, according to our standard picture, it is necessary to employ statistical arguments. The laws governing macroscopic behaviour are obtained from the forming of appropriate statistical averages of physical parameters of individual atoms. Such procedures were beginning to be worked out by a number of physicists in the second half of the nineteenth century, but the outstanding figure among these was Ludwig Boltzmann. Boltzmann stands as a link between two other great theoretical physicists: James Clerk Maxwell in the nineteenth century and Albert Einstein in the twentieth. Maxwell, who is best known for his discovery of the laws governing electric and magnetic fields and light, first found the formula for the probability distribution of velocities of particles in a gas in equilibrium, but it was Boltzmann who derived the equation governing the dynamical evolution of the probability distribution, according to which the state of a gas, not necessarily in equilibrium, will actually change. Boltzmann’s ideas were central to Max Planck’s later analysis of black body radiation at the turn of the century, in which he introduced the quantum of action, thereby firing the opening shot of the quantum revolution. In 1905, Einstein not only picked up on this idea and developed it further (in effect showing that the “atomic hypothesis” applied even to light itself!) but was also influenced by Boltzmann’s concepts in two of his other famous papers of 1905, one in which he provided a method of determining molecular dimensions and the other in which he explained the nature of Brownian motion, whereby small particles in suspension in a fluid jiggle around owing to the impact of the molecules that constitute the fluid. Both of these papers gave enormous support to the “atomic hypothesis”, leading to the confidence that we now feel in this picture of reality. The Boltzmann equation was also mathematically important, in being the first equation describing the time-evolution of a probability. But it was also fundamental in other ways. It opened up profound issues of physical significance —and even of deep philosophical interest—some of which are only partially resolved even today. For unlike the underlying dynamical equations governing the constituent particles themselves, the Boltzmann equation does not remain unchanged when the direction of time is reversed. The time-asymmetry of the Boltzmann equation arises as an aspect of the second law of thermodynamics, according to which the entropy of a system out of equilibrium increases with time. The crude meaning of the term “entropy” is “disorder”; so the second law tells us, roughly speaking, that the order in a system is continually being reduced. But it was another of Boltzmann’s fundamental contributions to give precision to the very notion of entropy, by identifying it with a specific multiple of the logarithm of the volume in phase space defined by the macroscopic parameters specifying the state of the system. Accordingly, Boltzmann showed how the second law could become amenable to precise mathematical treatment. How can it be that a time-asymmetrical dynamical equation can arise, describing the overall behaviour of some macroscopic system, when its constituent particles all satisfy time-symmetrical laws? Boltzmann thought deeply about these issues, in the face of much contemporary criticism, and realized (correctly) that the origin of the asymmetry must be traced back to a highly special state in the remote past, and must ultimately have its roots in cosmological considerations. However, virtually nothing was known about cosmology in Boltzmann’s day, so he was in no position to move that part of the argument much further than this, except for the introduction of some intriguing but speculative ideas. These issues are very much alive today, when a good deal is now known about the overall spatial and temporal nature of the actual universe; for they tell us something very significant about the nature of the universe’s “big bang” origin, and about the as yet unknown physical principles which came importantly into play at that crucial moment. What is the present status of the “atomic hypothesis” to which Boltzmann was so strongly committed? Although this is indeed the presently accepted picture of the submicroscopic nature of ordinary matter, there is now an additional twist, which is provided by the very quantum theory for which Boltzmann’s ideas acted as an unwitting midwife. Quantum particles are not like classical ones, in several respects. The statistics that they satisfy is subtly different from that of Boltzmann, which applies to the classical situation where each particle is allowed to have its own identity. Moreover, quantum particles do not have uniquely specified locations or velocities, and collections of them must, strictly speaking, be treated as a holistic (“entangled”) whole, rather than as a collection of individuals each of which has a state on its own. Moreover, the distinction between a continuous field and a collection of individual particles is by no means as clear as it was in the classical picture (as Planck’s and Einstein’s treatment of black body radiation was beginning to reveal). In view of these features, it is a remarkable fact that Boltzmann’s “classical” atomic picture works so extraordinarily well under normal conditions. There are, it seems to me, still important unresolved issues in relation to this. Yet Boltzmann himself was no dogmatist in holding to only one picture of things at the expense of all others. It is hard to imagine that he could have been, in view of the uncertainties and restlessness in his character, as manifested particularly in his later years, and of the depth of that sensitivity which drove him to his final tragic end. Carlo Cercignani provides us, here, with a most fascinating and authoritative account of Boltzmann’s life and scientific influence, and of the effects that Boltzmann’s contemporaries had on him and he on them. This is a very valuable account of an important era in scientific history. Roger Penrose