Ludwig Boltzmann: The Man Who Trusted Atoms

455 Pages·2006·4.33 MB·English

by  Carlo Cercignani

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
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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, 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