Boltzmann Ludwig - Ludwig Boltzmann: the man who trusted atoms

Ludwig Boltzmann

Ludwig Boltzmann at the age of 58, when he was a professor in Vienna.

The Man Who Trusted Atoms

CARLO CERCIGNANI

Dipartimento di Matematica
Politecnico di Milano

Great Clarendon Street, Oxford OX2 6DP

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Carlo Cercignani 1998

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First published 1998
First published in paperback 2006

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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, 18441906. 2. Atomic structureHistory.
3. PhysicistsAustriaBiography. I. Title.
QC16.B64C47 1998 530.092dc21 9817743

Printed in Great Britain
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ISBN 0198501544 9780198501541
ISBN 0198570643 (Pbk.) 9780198570646 (Pbk.)

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To my daughter Anna,
who helped me through Boltzmanns dense German

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. Boltzmanns ideas were central to Max Plancks 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 Boltzmanns 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 significanceand even of deep philosophical interestsome 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 Boltzmanns 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 Boltzmanns 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 universes big bang origin, and about the as yet unknown physical principles which came importantly into play at that crucial moment.