Microstates, Entropy and Quanta Don Koks Microstates, Entropy and Quanta An Introduction to Statistical Mechanics 123 ISBN 978-3-030-02428-4 ISBN 978-3-030-02429-1 (eBook) https://doi.org/10.1007/978-3-030-02429-1 Library of Congress Control Number: 2018960736 © Springer Nature Switzerland AG 2018, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover art: Central to statistical mechanics is the idea of counting the states accessible to a system. When these states exist within some continuous space, they cannot be counted. Instead, we “tile” the space into cells, with each cell defining a state, and then we count those cells. The ball on the front cover is a schematic of this tiling of the velocity space of a free particle that moves in three spatial dimensions. For real particles, the cells are so much smaller than the size of the ball that, to all intents and purposes, the ball is a smooth sphere. The number of cells can then easily be found from the sphere’s volume. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For my ancestors, and all those who have gone before. Preface Another book on introductory statistical mechanics? You might think that a century-old subject would have nothing left unsaid; but that is perhaps not the case. Unlike most other fields of physics, one can compare a dozen books on statistical mechanics and find a dozen different approaches to the discipline. At one extreme are authors who revel in arcane abstraction, but whose books go mostly unread. At the other extreme are very readable books that lack the mathematics to carry the reader very far beyond a set of physical assumptions. Most readers are looking for something in between; but that space is vast and subjective, with plenty of room for another book to aim for the \Goldilocks Zone" of being just right. That's why I wrote this book: I think that the field of introductory statistical mechanics still has plenty of scope for an author to try a different mix of mathematical exposition and physical reasoning. The physics part of this mix is a build of statistical mechanics from the ground up, anchored to a bedrock of physical concepts. With this approach, I hope to have revealed the necessity and importance of the subject's core ideas, such as entropy and temperature. The mathematics part of the mix has been an emphasis on a strong logical reasoning that has a clean outline, yet avoids the notational clutter and obscure discussions that are so often associated with statistical mechanics, and which can make it so hard to learn. Thus, beside the calculations of representative physical quantities, you will find here various mathematical analyses that I believe are important to physicists. Much of this mathematical foundation is given in the first chapter, such as details of integrating the gaussian function, and the correct use of infinitesimals, partial derivatives, and units of measurement. By the time you reach that chapter's end, you might be wondering whether you are really reading a book on statistical mechanics after all. And yet, you will encounter those topics time and again as you work through the rest of the book. The choice of how and where to begin describing a subject is always highly author dependent. The concepts that I introduce methodically, as needed, vii viii Preface are sometimes merely postulated with a breezy stroke of the pen in books that announce themselves as introductions. Postulatory approaches to other subjects can certainly work well; for instance, I admire Feynman's approach to electromagnetism in his Lectures on Physics, since, although he postulates Maxwell's equations at the very start, we never lose sight of the physics in his discussions. In contrast, I struggle to see any physics at all in some postulatory approaches to statistical mechanics, which can so easily ignore the difficult questions that interest physicists. I commence the subject of statistical mechanics with an archetypal obser- vation: why does a drop of ink placed in a bathtub disperse? Once dispersed, might it ever re-assemble into a drop? This question showcases the impor- tance of counting the number of ways in which a system's constituents can be arranged, and leads to statistical mechanics proper via its fundamental postulate. That discussion demands knowledge of the concept of energy, a concept that was useful and intriguing to early astronomers studying plane- tary orbits, but whose wider application was not well understood in the early days of thermodynamics, 150 years ago. With a more modern understanding of energy (or perhaps \acceptance" is a better word, since we still don't know what it is|if, indeed, asking what it is has any meaning), we are in a good position to write down the laws of thermodynamics. Then, we can explore heat engines, chemical processes and equilibria, and heat flow. The flow of heat is a stepping stone to appreciating diverse related areas, such as parti- cle diffusion and, in fact, the signal processing performed in a modern radar receiver. But no system is ever truly isolated; and the question of how to analyse a system in contact with the wider world brings us to the Boltzmann dis- tribution, with examples in paramagnetism, atomic energy levels, molecular and crystal heat capacities, and data-transmission theory. The Boltzmann distribution also sheds light on the motion of gas particles. I use that theory to explore an atmosphere, as well as the molecular details of viscosity and thermal conductivity. Quantum ideas then emerge, via Einstein's and Debye's theories of heat capacity. The notion of fermions and bosons forms a springboard to the study of electronic heat capacity, electrical conduction, thermal noise in electric circuits, the spectra of light produced by hot bodies, some cosmology, the greenhouse effect, and the modern technologies of light-emitting diodes and the laser. I have sprinkled the text with occasional short digressions, discussing top- ics such as the factorial function in number theory, the energy{momentum tensor in relativity, a little bit of signal processing, and decrying the short- comings of modern analytical astronomy. Hopefully, these asides will only enrich your interest without being a distraction. Unlike some books on statistical mechanics, I have chosen to discuss a lot of material before introducing the Boltzmann distribution. Thus, in those pre- Boltzmann chapters, I invoke the equipartition theorem to approximate the Preface ix particles in an ideal gas of temperature T as each having, say, translational energy 3=2 kT . Later, when studying the Boltzmann distribution, we learn that only their average translational energy is 3=2 kT . Some authors will avoid this initial simplification by introducing the Boltzmann distribution very early on. But I think that using the simple approximation initially, and leaving Boltzmann for later, is useful pedagogically. A subject as old as statistical mechanics is bound to carry baggage picked up along the way, created as a normal part of its development, when physicists and chemists were searching for the best path through the new forest they had discovered. The choice of what might best be discarded is a little subjective. I have tended to minimise the use of phrases and topics that appear to be generally confusing, unattractive, or not useful. For example, I cannot imagine that the conventional vernacular that de- scribes various flavours of ensemble, along with free energies, partition func- tions, and Maxwell relations, does anything to attract new adherents to sta- tistical mechanics. Pseudo wisdom that you can find in books on the subject, such as \The trick to solving this problem is to use the grand canonical en- semble", is apt to give the impression that statistical mechanics is all about finding the right trick using the right ensemble to get the right answer. The language of ensembles is not especially deep, and after explaining what it means and how it's used, I tend to avoid it, because the \correct ensemble to use" should be clear from the context being studied; it is not some arbitrary choice that we make. Free energies have a range of uses in thermodynamics (and I certainly use them in this book), but they are probably more relevant to the history of the subject, when early physicists and chemists worked hard to ascertain the nature of what was then a cutting-edge new quantity called energy. Nowadays, we view free energies as useful combinations of more fun- damental parameters such as energy, temperature, and entropy.