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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An accessible and rigorous approach to thermodynamics and statistical mechanics
In Statistical Thermodynamics: An Information Theory Approach, distinguished physicist Dr. Christopher Aubin delivers an accessible and comprehensive treatment of the subject from a statistical mechanics perspective. The author discusses the most challenging concept, entropy, using an information theory approach, allowing readers to build a solid foundation in an oft misunderstood and critically important physics concept.
This text offers readers access to complimentary online materials, including animations, simple code, and more, that supplement the discussions of complex topics in the book. It provides calculations not usually provided in comparable textbooks that demonstrate how to perform the mathematics of thermodynamics in a systematic way.
Readers will also find authoritative explorations of relevant theory accompanied by clear examples of applications and experiments, as well as:
- A brief introduction to information theory, as well as discussions of statistical systems, phase space, and the Microcanonical Ensemble
- Comprehensive explorations of the laws and mathematics of thermodynamics, as well as free expansion, Joule-Thomson expansion, heat??engines, and refrigerators
- Practical discussions of classical and quantum statistics, quantum ideal gases, and blackbody radiation
- Fulsome treatments of novel topics, including Bose-Einstein condensation, the Fermi gas, and black hole thermodynamics
Perfect for upper-level undergraduate students studying statistical mechanics and thermodynamics, Statistical Thermodynamics: An Information Theory Approach provides an alternative and accessible approach to the subject.
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Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
Acknowledgments
About the Companion Website
1 Introduction
1.1 What is Thermodynamics?
1.2 What Is Statistical Mechanics?
1.3 Our Approach
Notes
2 Introduction to Probability Theory
2.1 Understanding Probability
2.2 Randomness, Fairness, and Probability
2.3 Mean Values
2.4 Continuous Probability Distributions
2.5 Common Probability Distributions
2.6 Summary
Problems
References
Notes
3 Introduction to Information Theory
3.1 Missing Information
3.2 Missing Information for a General Probability Distribution
3.3 Summary
Problems
References
Further Reading
Notes
4 Statistical Systems and the Microcanonical Ensemble
4.1 From Probability and Information Theory to Physics
4.2 States in Statistical Systems
4.3 Ensembles in Statistical Systems
4.4 From States to Information
4.5 Microcanonical Ensemble: Counting States
4.6 Interactions Between Systems
4.7 Quasistatic Processes
4.8 Summary
Problems
References
Notes
5 Equilibrium and Temperature
5.1 Equilibrium and the Approach to it
5.2 Temperature
5.3 Properties of Temperature
5.4 Summary
Problems
References
Notes
6 Thermodynamics: The Laws and the Mathematics
6.1 Interactions Between Systems
6.2 The First Derivatives
6.3 The Legendre Transform and Thermodynamic Potentials
6.4 Derivative Crushing
6.5 More About the Classical Ideal Gas
6.6 First Derivatives Near Absolute Zero
6.7 Empirical Determination of the Entropy and Internal Energy
6.8 Summary
Problems
References
Notes
7 Applications of Thermodynamics
7.1 Adiabatic Expansion
7.2 Cooling Gases
7.3 Heat Engines
7.4 Refrigerators
7.5 Summary
Problems
References
Further Reading
Notes
8 The Canonical Distribution
8.1 Restarting Our Study of Systems
8.2 Connecting to the Microcanonical Ensemble
8.3 Thermodynamics and the Canonical Ensemble
8.4 Classical Ideal Gas (Yet Again)
8.5 Fudged Classical Statistics
8.6 Non‐ideal Gases
8.7 Specified Mean Energy
8.8 Summary
Problems
Notes
9 Applications of the Canonical Distribution
9.1 Equipartition Theorem
9.2 Specific Heat of Solids
9.3 Paramagnetism
9.4 Introduction to Kinetic Theory
9.5 Summary
Problems
References
Notes
10 Phase Transitions and Chemical Equilibrium
10.1 Introduction to Phases
10.2 Equilibrium Conditions
10.3 Phase Equilibrium
10.4 From the Equation of State to a Phase Transition
10.5 Different Phases as Different Substances
10.6 Chemical Equilibrium
10.7 Chemical Equilibrium Between Ideal Gases
10.8 Summary
Problems
References
Notes
11 Quantum Statistics
11.1 Grand Canonical Ensemble
11.2 Classical vs. Quantum Statistics
11.3 The Occupation Number
11.4 Classical Limit
11.5 Quantum Partition Function in the Classical Limit
11.6 Vapor Pressure of a Solid
11.7 Partition Function of Ideal Polyatomic Molecules
11.8 Summary
Problems
Reference
Notes
12 Applications of Quantum Statistics
12.1 Blackbody Radiation
12.2 Bose–Einstein Condensation
12.3 Fermi Gas
12.4 Summary
Problems
References
Notes
13 Black Hole Thermodynamics
13.1 Brief Introduction to General Relativity
13.2 Black Hole Thermodynamics
13.3 Heat Capacity of a Black Hole
13.4 Summary
Problems
References
Notes
Appendix A: Important Constants and Units
References
Appendix B: Periodic Table of Elements
Appendix C: Gaussian Integrals
Note
Appendix D: Volumes in ‐Dimensions
Notes
Appendix E: Partial Derivatives in Thermodynamics
Reference
Notes
Index
End User License Agreement
List of Tables
Chapter 3
Table 3.1 Combinations of bits corresponding to each coin flip in Example 3....
Table 3.2 Combinations of bits corresponding to each die roll in Example 3.3...
Table 3.3 The number of times in a text of 106 400 characters each letter in...
Chapter 4
Table 4.1 A list of all microstates corresponding to three electrons in a ba...
Table 4.2 The possible sums and how to obtain them on two dice, which are as...
Chapter 6
Table 6.1 Specific heats for various monatomic and diatomic gases.
Table 6.2 Coefficients for the Van der Waals equation for a select set of ga...
Chapter 9
Table 9.1 The molar mass and specific heats (per mass and per mole) at const...
Chapter 11
Table 11.1 The melting points for the six naturally occurring noble (and thu...
Appendix A
Table A.1 Base and some derived SI units.
Table A.2 Important fundamental constants that we will need. Those which are...
Table A.3 Some important unit conversions, where I include not just metric u...
List of Illustrations
Chapter 1
Figure 1.1 Various subfields of physics in relation to each other, when cons...
Figure 1.2 Adding a third dimension, (the number of particles in a system)...
Chapter 2
Figure 2.1 A box divided into boxes with a marble tossed into one of them....
Figure 2.2 A box divided into boxes with a marble tossed into one of them....
Figure 2.3 A box divided into boxes along the ‐axis with a marble to be t...
Chapter 3
Figure 3.1 A system with outcomes, with a specific state shown where the b...
Figure 3.2 A system with boxes and compartments per box, with the ball s...
Figure 3.3 Representing the flipping of four coins using our ball‐in‐a‐box e...
Figure 3.4 Six out of the sixteen possible outcomes in an ensemble of system...
Chapter 4
Figure 4.1 A plot of vs. for the classical one‐dimensional simple harmon...
Figure 4.2 One of the possible microstates for a system of 10 six‐sided di...
Figure 4.3 A system constructed from an ensemble with low entropy.
Figure 4.4 A system constructed from an ensemble with high entropy.
Figure 4.5 Plots of vs. for , and 100.
Figure 4.6 The percent difference between the exact number of states for the...
Figure 4.7 Phase space for a single classical oscillator in one dimension. F...
Figure 4.8 A schematic of equally spaced energy levels for a single particle...
Figure 4.9 Two systems, and , in thermal contact with each other. The par...
Figure 4.10 A system with a lid which is allowed to move, and under the infl...
Figure 4.11 Our system above before (on the left) and after (on the right) y...
Figure 4.12 The three paths to perform the integral for Exercise 4.6.
Chapter 5
Figure 5.1 A container with oxygen gas in the left half, and either vacuum o...
Figure 5.2 The number of states as a function of the energy for our system...
Figure 5.3 The product of and as a function of the energy for our syst...
Figure 5.4 The entropy as a function of energy for a sample system that can ...
Figure 5.5 Various temperature scales that have been defined in the history ...
Chapter 6
Figure 6.1 A system (with rigid boundaries) in thermal contact with a heat...
Figure 6.2 Two systems in thermal contact with each other, where the partiti...
Figure 6.3 A graphical representation of the motion of states as we change t...
Figure 6.4 Three functions that all have the same slope (derivative) but are...
Figure 6.5 The same as Figure 6.4 but with the tangent lines extended to the...
Figure 6.6 (a) One of the curves in Figures 6.4 and 6.5, with different valu...
Chapter 7
Figure 7.1 Comparing vs. for a diatomic ideal gas undergoing isothermal ...
Figure 7.2 Free expansion of a system at three stages: (a) the gas is contai...
Figure 7.3 The molar energy vs. the temperature for three different volu...
Figure 7.4 The temperature vs. the molar volume of nitrogen for three di...
Figure 7.5 The setup for the throttling process. On the left is a gas at tem...
Figure 7.6 The setup for the throttling process, this time focusing on mol...
Figure 7.7 The inversion curve for nitrogen gas, where the solid curve denot...
Figure 7.8 A schematic of a perfect heat engine: heat flows from a reservo...
Figure 7.9 A schematic of a real heat engine. In this case, energy is tran...
Figure 7.10 (A) The – diagram for the Carnot cycle, where Stage 1 is from
Figure 7.11 A schematic of a perfect refrigerator, where energy is “transf...
Figure 7.12 A schematic of a real refrigerator, where work must be done on t...
Chapter 8
Figure 8.1 A system with rigid boundaries in thermal equilibrium with a he...
Figure 8.2 vs. for our simple two‐state system in Example 8.2.
Figure 8.3 vs. for our simple two‐state system in Example 8.2.
Figure 8.4 (a) Two identical systems, both with molecules of a classical i...
Figure 8.5 Three identical molecules; if we interchange any of them we have ...
Figure 8.6 An example of a realistic intermolecular potential as a functio...
Figure 8.7 The Lennard–Jones potential, Eq. (8.56), as a function of .
Figure 8.8 A comparison of the Sutherland potential with the Lennard–Jones...
Figure 8.9 An ensemble of systems with one system in a state with energy s...
Chapter 9
Figure 9.1 A schematic of molecules in a solid (only showing a two‐dimension...
Figure 9.2 The hexagonal structure of one layer of carbon atoms. Graphite is...
Figure 9.3 The molar specific heat as a function of temperature from the Ein...
Figure 9.4 Experimental data for the specific heat of diamond from Ref. Adap...
Figure 9.5 The molar specific heat as a function of temperature in the Debye...
Figure 9.6 The Brillouin function for three different values of .
Figure 9.7 The mean magnetization per ion for three different substances (I ...
Figure 9.8 The allowed values of and for a classical particle with speed...
Figure 9.10 Results for from a simulation of a system of molecules. In t...
Figure 9.9 The Maxwell speed distribution with the three relevant speeds sho...
Figure 9.11 The volume of gas that we are considering when determining how m...
Figure 9.12 A container divided into two sections, with a small slit between...
Chapter 10
Figure 10.1 A plot of entropy vs. energy for a simple two‐state system with ...
Figure 10.2 The entropy vs. some thermodynamic parameter for a system with a...
Figure 10.3 An example of a system in contact with a heat and work reservoir...
Figure 10.4 A simple example of a phase diagram. The line shows the values o...
Figure 10.5 A schematic of the phase diagram of water. The star labels the c...
Figure 10.6 The Van der Waals equation of state, vs. , for different valu...
Figure 10.7 The same as Figure 10.6, but for a single temperature where ther...
Figure 10.8 The same as Figure 10.7, but with the specific volumes from that...
Figure 10.9 The area to the left of the Van der Waals curve for a given temp...
Figure 10.10 The molar Gibbs free energy vs. the pressure as a result of the...
Figure 10.11 The proper dependence pressure has on volume for a given temper...
Figure 10.12 The Maxwell construction for the phase transition point: Vary t...
Figure 10.13 The proper vs. diagram for various temperatures (as opposed...
Chapter 11
Figure 11.1 A system (with rigid boundaries) in thermal contact with a hea...
Figure 11.2 Allowed states for the system in Example 11.3 if the particles o...
Figure 11.3 Allowed states for the system in Example 11.3 if the particles o...
Figure 11.4 Allowed states for the system in Example 11.3 if the particles o...
Figure 11.5 The Planck distribution function as a function of the single‐par...
Figure 11.6 The Bose–Einstein distribution function shown as a function of t...
Figure 11.7 The Fermi–Dirac distribution function as a function of the singl...
Figure 11.8 The quantum distribution functions for FD (dotted lines), BE (da...
Figure 11.9 A typical dependence of the ground state electronic energy as a ...
Figure 11.10 A diatomic molecule as a rigid rotor, which rotates with angula...
Chapter 12
Figure 12.1 The integrand in Eq. (12.15), showing the peak at .
Figure 12.2 A blackbody in a container filled with a photon gas at temperatu...
Figure 12.3 A comparison of the ground state occupation number for classical...
Figure 12.4 as a function of for rubidium, data from Ref. [9]. The dotte...
Appendix D
Figure D.1 The function (curve) shown along with the values of the factori...
Appendix E
Figure E.1 The hilly region described by Eq. (E.1). The thin lines are lines...
Figure E.2 Pressure as a function of temperature and volume for one mole of ...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
Acknowledgment
About the Editors
Begin Reading
Appendix A: Important Constants and Units
Appendix B: Periodic Table of Elements
Appendix C: Gaussian Integrals
Appendix D: Volumes in ‐Dimensions
Appendix E: Partial Derivatives in Thermodynamics
Index
End User License Agreement
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Statistical Thermodynamics
An Information Theory Approach
Christopher Aubin
Fordham University
Bronx, New York
United States
Copyright © 2024 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per‐copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750‐8400, fax (978) 750‐4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748‐6011, fax (201) 748‐6008, or online at http://www.wiley.com/go/permission.
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Library of Congress Cataloging‐in‐Publication Data is Applied forHardback ISBN: 9781394162277
Cover Design: WileyCover Image: © Hector Roqueta Rivero/Getty Images. Image of the handwritten Boltzmann principle provided by Christopher Aubin.
For my father.
Preface
The first time I tookStatistical Physics as an undergraduate physics major, I hated the class. This is a pretty strong sentiment, but while I was able to learn the basics of statistical mechanics and thermodynamics, I didn’t fully appreciate the beauty of the subject. Then I opted to take the undergraduate course again during my first semester of graduate school (at a different school, taught an entirely different way), and my feelings about the subject changed drastically. I attributed this largely to the organization of the course and how the material was developed.
Since then, I have found that this is a very strange subject in physics. As I discuss in the introduction, this is one of the four “core” fields of physics, but unlike the other three (classical mechanics, electricity and magnetism, and quantum mechanics), there doesn’t seem to be anywhere near the same sort of consensus on how to approach the subject. As I have developed this course over the past 13 years, I have worked hard to try to motivate students to find the beauty in statistical mechanics and thermodynamics. There is a plethora of fantastic textbooks out there, all of which approach the subject from different starting points, covering topics in different orders, and of course having different priorities. Many of them have been excellent resources as I developed my class throughout the past 13 years and this book, although much of my inspiration comes from the textbook by Frederick Reif, Fundamentals of Statistical and Thermal Physics. Even still, my approach has a slightly different starting point, that of information theory, that has rarely been seen at the undergraduate textbook level. Perhaps this is because it is a more theoretical approach, lacking the direct and immediate applications that a thermodynamic starting point has, but I do believe that this is precisely where the beauty of the field becomes clear.
The approach that I take here (which I first learned about from Claude Bernard, who taught the undergraduate class I took while in graduate school), I find gives a wonderful first‐principles approach to the field. While it begins very abstractly, I find it allows the physical applications of thermodynamics to fall out naturally while also setting us up to apply our methods to some very strange ideas. And, this abstract approach allows the broad nature of the subject, that it can be applied to literally any area of physics and engineering, to be made patently clear to the student. Sure the lack of direct practical applications (for example, I do not spend nearly enough time on the topic of heat engines, for reasons I explain in Chapter 7) might cause some to find this less useful. However as is often the case, having an abstract approach provides us with much more power when solving problems.
The basic approach I take is to start from probability in Chapter 2 and information theory in Chapter 3, as the title suggests. From this point, I will move to a discussion of the underlying statistical nature of systems with large numbers of particles in Chapter 4. It is only at this point where I define the concept of temperature in Chapter 5. This is difficult at first, as temperature is such an important concept that everyone has an everyday understanding of, but by waiting for a bit to define it allows a deeper understanding of its meaning, beyond how hot or cold it is outside. This lays the groundwork that, with few assumptions, allows us to formulate all of thermodynamics, which I work through in detail in Chapters 6 and 7.
After the basics of thermodynamics have been laid out, I start fresh with the canonical ensemble in Chapters 8 and 9, thus moving back to our statistical description again. From there, I will work through applications to phase transitions and chemical reactions in Chapter 10 and then quantum statistics in Chapters 11 and its applications in Chapter 12. I will end with a huge detour to discuss the fascinating area of black hole thermodynamics in Chapter 13, not because it is essential for an undergraduate education, but rather because it is a great way to demonstrate how wide‐reaching the subject can be.
This textbook is designed to be a one‐semester course in undergraduate statistical mechanics. The later topics in the text I have alternated in the times I have taught the course, often due to the interest of the students. Throughout the book are of course examples, but more importantly many of the simpler derivations I have left out. In place of those I have included Exercises that the student should do, to assist with the development of the subject. Additionally, sprinkled throughout the text are references to Jupyter notebooks (using Python) that can be accessed at the companion site to better understand some of the smaller details that we will cover. These are meant on one hand to be a quick way to see some of the results I will derive and also as a way for students to learn a little bit about coding if that is of interest. And of course at the end of each chapter are more detail problems, as any good book has, for the student to master the material and see where this exciting field can lead us.
Christopher Aubin
Acknowledgments
It doesn’t matter how many authors are listed on a book, as there are many people who help but are behind the scenes. First and foremost I must thank Claude Bernard for introducing me to this approach to the subject. He was instrumental in two ways, the first of which was showing me this approach to statistical mechanics. Additionally, he and Michael Ogilvie were my PhD advisors, so without them I wouldn’t be where I am today. And of course I must thank my students; well over 100 of them have been the unknowing guinea pigs as I developed this course and book.
In terms of this work, I am grateful for Maarten Golterman and Martin Ligare for their useful comments and discussions on the manuscript. I would like to thank Stephen Holler for many discussions during the last 10 years that helped me along the way while preparing this (even if he didn’t always realize it). Finally, Vassilios Fessatidis has been a great help in discussions throughout the years and specifically helped me with some Greek to understand the origins of many terms in this subject.
But a book of this sort is not only about the physics, as I must acknowledge the work of my family. Roger, Diane, and Carrie Aubin (my father, mother, and sister) have been my strongest supports throughout my entire life. While my father didn’t live to see this book published, he would always love to hear me talk about physics (even if he really didn’t care about the subject). And last, but certainly not least, I am so grateful for the love and support of my husband, Corbett, who has put up with me and my physics rambling for more time than anyone not in the field should have to.
Christopher Aubin
About the Companion Website
This book is accompanied by a companion website.
www.wiley.com/go/Aubin/StatisticalThermodynamics
This website includes:
• Computer code in the form of Jupyter notebooks
1Introduction
Before jumping right into everything, I want to spend a brief moment discussing statistical mechanics and thermodynamics. Why do we need these subjects and what are they? How do they differ and how do they connect with each other? Of course, you can jump to Chapter 2 and start learning the necessary material, but this will be a useful overview to frame our discussion over the next several hundred pages.
For most people, including undergraduate physics students, statistical mechanics tends to be one of the less well‐known subjects in physics. Every student knows about classical mechanics (even if not by that name when they first start studying physics), electricity and magnetism, and even quantum mechanics. Additionally, these subjects (along with more “exciting ideas” such as general relativity) are known by many non‐physics students as well. However, statistical mechanics is one of the “core subjects” that all physicists and engineers should understand in depth along with classical mechanics, electricity and magnetism, and quantum mechanics.1
Students are of course familiar with thermodynamic concepts (heat, temperature, pressure, etc.), and while many schools offer a course with this title, it is also often called Statistical Physics, Statistical Mechanics and Thermodynamics, or some variation on these. When registering for this class, many times the only idea students have regarding the course is that it “is the hardest class you’ll ever take in college, it makes no sense, and you no longer will understand what temperature is.”2 The salient question is then: What is the relationship between the (slightly) more familiar thermodynamics and this unfamiliar statistical mechanics? The quick answer, to some degree, is that they are two different approaches one can take to understand the same subject. I plan to give an overview in this chapter on how they relate and what the difference is when we consider this subject from either point of view. The goal is to understand how they are connected, why we study one over the other, and most importantly, why we’re going to start with statistical mechanics before moving on to the study of thermodynamics. After I discuss the two, as well as the approach we will take in this book, we will finally get started on the actual subject.
1.1 What is Thermodynamics?
Considering the constituent words that make it up, thermodynamics is merely the study of the motion of heat. Coming from the Greek, thermodynamiki, or, ϑερμοδυναμική, is a combination of thermótita (ϑερμότητα) which means “heat” and dynamikí (δυναμική) which means “dynamics.” In the early nineteenth century, when this field began to develop, the focus was on heat engines: using heat as a form of energy that can be converted into work (the reverse is easy to do, doing work to create heat). That the study of heat engines, and thus thermodynamics, exploded during this time is not surprising given that this overlaps with the Industrial Revolution. A steam locomotive is an early example of a widely used heat engine: You use the steam produced by burning coal to turn gears (thereby turning the wheels of the train and propelling it forward). In the process, and as the field progressed, four laws of thermodynamics were postulated and these allowed the introduction and development of various concepts such as temperature, entropy, enthalpy, and so forth.3
Many of these concepts are familiar from an everyday perspective (like temperature), or from hearing about it colloquially (like entropy). Still others, like enthalpy, are not well known by those who never took a chemistry class. The question remains though: What do these quantities describe physically? The meanings of these ideas can often be muddled (or worse, misunderstood) when first studying thermodynamics. Additionally, we will see that while starting from a thermodynamic perspective is nice from a conceptual (and historical) point of view, it can be very limiting. Finally, it is only natural to ask where thermodynamics fits in with regard to other areas of physics.
As physics developed throughout the centuries, it was necessary to begin to divide it into different subfields we can categorize very roughly by the relevant length scales and speeds in a given problem. A depiction of this is shown in Figure 1.1. If the relevant length scale of our system is denoted as , and the relevant speed is denoted as , then we can divide physics up as follows. Starting with classical mechanics (“the original physics”), and take on “everyday values”: values that are common for buildings, cars, people, etc. I’ll admit that the term “everyday values” is not a great way to describe these quantities, as classical physics is valid for a fairly wide range of sizes and speeds. The motion of dust ( m) as well as that of planetary orbits (Neptune is an average distance of about m from the sun), both can be described classically. Additionally, planets orbit the sun at high speeds (from our perspective) and they can be described classically (Mercury travels around the sun at almost m/s = 180 000 km/h). The precise length or speed scale is not important here; it just matters that there is a range such that the classical description is valid, and at some point it breaks down.
Figure 1.1 Various subfields of physics in relation to each other, when considering a range of length scales and speeds of the system under consideration.
Around the turn of the twentieth century, experiments were performed at much smaller length scales, and classical mechanics began to fail. Enter quantum mechanics, which becomes relevant as one nears molecular scales, so and smaller.4 Around the same time, the theory of special relativity was being developed, which is important to consider when our system approaches high speeds, specifically close to the speed of light . In each of these cases, we consider one of these quantities to change: In ordinary (non‐relativistic) quantum mechanics, speeds are not too high, while in special relativity, the length scales are those of classical mechanics, all shown in Figure 1.1.
Electricity and magnetism, the third of our four “core” subjects, doesn’t quite fit into any of these categories, so I added it to the middle of our plot, somewhat spilling over into various other subfields. It belongs a bit more in the special relativity section (as it was important in the development of this theory and in fact is already relativistic by nature), but often can be thought of as a classical topic. One thing to note for clarity though, while it overlaps various categories, it is specifically not quantum mechanical.
A couple of other subjects are thrown into the plot (just for some level of completeness) that aren’t always studied in an undergraduate curriculum. For large systems, general relativity (the fundamental theory of gravity) takes over for classical mechanics (in the figure, I have infinity in quotes to imply this is for very large systems without needing to specify an actual scale).5 For very small and fast systems, we combine quantum mechanics and special relativity to formulate quantum field theory (needed to study particle physics, including a quantum description of the electromagnetic field as well as the strong and weak nuclear interactions). While the use of a single length (or speed) scale is overly simplistic, systems with multiple scales that are wildly different are quite difficult to study in physics; this is one of the reasons why it is difficult to combine general relativity and quantum field theory (hence we have no quantum theory of gravity!).
All of the fields above are solvable directly in terms of laws that allow us to determine the equations of motion. For example,
Classical mechanics
: We can use Newton’s laws of motion to determine the trajectories of particles as long as we know
all
of the forces acting on them given a set of initial conditions.
Electricity and magnetism
: With Maxwell’s equations (and appropriate initial and boundary conditions), we can determine the electromagnetic field due to any charge and current distribution. Add in the Lorentz force, and we can describe (with classical mechanics) how charged particles move in electromagnetic fields.
Quantum mechanics
: We can solve the Schrödinger equation, along with appropriate initial and boundary conditions, to determine the wavefunction which can be used to calculate expectation values of physical observables.
While more complicated, the same can be said for the other advanced topics, as long as we have the relevant equations (and initial and/or boundary conditions). Keep in mind when we say we can solve these equations, this is all in principle: An exact solution is rarely possible in practice, while often approximate solutions are.
But what about thermodynamics, where do we put this in our figure? In Figure 1.1, I have implicitly assumed that the number of objects in our system is small, by which I mean one, two, or maybe three. A thermodynamic system, such as the air in the room you’re sitting in while reading this or the cup of coffee you are drinking to stay awake while doing so, involves a ridiculously large number of particles. Many systems we will focus on will have something on the order of a mole of molecules, around (the mole is a base unit in the International System of Units (SI), which, along with other units, I discuss in Appendix A).
Now imagine we had a classical system with this many molecules interacting. Knowing all of the forces between the molecules is difficult, but let us suppose we did, so that we could write down a system of equations for the trajectories of all the molecules. Solving this system of equations would be completely intractable, as even the three‐body problem cannot be solved in most cases analytically!
But let’s make an absurd assumption to describe a naïve way to think about how intractable this is. We’ll assume that we actually could solve a problem with such a large number () of molecules. For a three‐dimensional system we would have equations, and from those would need to determine each particle’s trajectory for . I’ll even assume that we are so good at solving these problems that we only need one second to figure out (and write down) each of these 3‐D trajectories. Even with these unrealistic assumptions, it would take us
to actually solve for all of these trajectories. Not only is this much longer than our average lifespan, the age of the universe is roughly yr, so even if we could solve such a problem, we wouldn’t have enough time to do so! This is a ridiculous example, as it is not really a proper measure of how or why to solve physics problems; however, it does illustrate an important limitation that can arise.
More importantly, even if we could determine all of these trajectories, it wouldn’t lead to anything useful for practical purposes. For example, consider the air molecules (really just the oxygen molecules) in your room: What about those molecules matters to you? You aren’t interested in the individual trajectories of every single molecule, nor do you care which of the many molecules are near you. You really only care that there are oxygen molecules near you so that when you inhale, you can capture them. As such, we are more interested in what we call the bulk properties, which are large‐scale properties of the entire system of oxygen molecules in the room (such as the pressure, density, and temperature, all of which we will discuss in detail when the time comes).
To add thermodynamics to Figure 1.1, we would need to add a third axis, shown in Figure 1.2. The additional dimension is the number of molecules, , ranging from one to “,” where I am using the same notation for infinity as I did for . We will generally just suggest , and while we will look at the large‐ limit, taking to infinity will often be troublesome mathematically during intermediate steps.6 I won’t give too many examples of how big needs to be yet, but we’ll see how it comes into play as we go through the subject. That large corresponds to the subject of thermodynamics actually becomes clearer when we look at it from the point of view of statistical mechanics, which we’ll do in Section 1.2.
Figure 1.2 Adding a third dimension, (the number of particles in a system), to Figure 1.1.
One last note before moving on: When we have only one or two particles, of course the problems are (often) relatively easy to solve. When we have a very large number (), the problems are also relatively easy to solve, but only because of the questions we will ask.7 That middle ground, the few‐body problem—when we have 10 or 20, or maybe a few 100—is actually way more difficult!
1.2 What Is Statistical Mechanics?
How does all of this relate to statistical mechanics? Thermodynamics as we will think of it is an empirical theory of the large systems mentioned above. This arose from centuries of experimental observations of bulk properties such as the number of molecules , the pressure , the volume , the temperature ,8 the internal energy ,9 and plenty of other quantities. The four laws of thermodynamics allow us to study these quantities: specifically how they are related, how they change when we make changes to the system, etc. In a sense, this is akin to the study of optics, where so much can be understood without knowing that light results from an electromagnetic wave solution to Maxwell’s equations. Geometric optics is a phenomenological theory about how light behaves with a particular set of assumptions, even though it is not the fundamental theory and it has limitations. It is useful but you cannot delve too deeply into the subject without more knowledge of the underlying theory.
Similarly, statistical mechanics is the fundamental theory that gives rise to thermodynamics, and in some sense we can consider it similar to electricity and magnetism. From it, we can derive all of the phenomenological results in thermodynamics (including the four laws), but with fewer, and more general, assumptions. Not only that, but a deeper understanding of the system can be surmised from statistical mechanics, and more importantly, there are physical observations that can only be understood if we start from the more fundamental theory.
This analogy with optics is far from perfect. In that case, we have a well‐defined fundamental theory that can be used to derive the results in optics, but it’s not really the same here. Statistical mechanics is more of a methodology rather than a fundamental theory that is used so that we can make predictions based on a very small set of assumptions. This methodology involves (essentially) one major assumption and, as we shall approach it, an application of information theory.
1.3 Our Approach
Given the name, it should be clear that understanding statistics will be required to study statistical mechanics. As we are not able to solve the complicated equations of motion for these large systems (and as mentioned, we don’t really want to), instead we use a probabilistic approach. We replace those quantities that we ordinarily care about (such as the trajectories in classical mechanics) with, for example, the bulk properties referred to in Section 1.1.
Some aspects of our approach are the same as in any physics problem. As always we start by precisely defining our system. From that, given that we will take a probabilistic approach, we need to define an ensemble of systems: a set of all states that are accessible to the system. This is where our method will diverge from an ordinary physics problem.
When we discuss states, we will now need to distinguish between the possible microstates of a system (for example, all of the trajectories of the particles in our example above) and the possible macrostates of our system, which are dependent on what we are interested in measuring. The allowed macrostates will be specified by the ensemble we will use (discussed in Chapters 4, 8, and 11), but initially the macrostates I’ll discuss will have a given constant energy . It’s easy to see that generally many different microstates correspond to a single macrostate. As a quick example of this, consider a classical one‐dimensional simple harmonic oscillator with mass . This is a conservative system, so the total energy is constant. There are an infinite number of microstates (one for each position and corresponding momentum ), but there is only a single macrostate defined by the total energy of the system.10
To calculate physical observables (the primary goal in any physics problem) here, we have to be able to assign a probability that our system will be in a given state, which we can do by considering everything that we know about the system. What we will need to do is figure out how to quantify this information, or more importantly, the missing information of our system. We will actually find that not knowing things isn’t necessarily a bad thing!11
Figure 1.2 makes it apparent that because statistical mechanics is relevant for , we can apply these methods to any problem in classical mechanics, electricity, and magnetism, quantum mechanics, etc. That is, it can be applied to any problem in physics, so long as the number of particles in the system is very large; as such, we will need a cursory knowledge of these other subfields.12 As such, this is really a “capstone course” in physics, as we get to review everything else you studied in other courses. This is one of the reasons that this can be so difficult, as the starting point for our problems can come from anywhere, but is also one of the reasons this field is so rich and interesting!
As mentioned, we cannot get right into statistical mechanics without covering probability and information theory first. We will cover probability in Chapter 2 at a cursory level; primarily just considering it from an everyday understanding (without much rigor or formalism). In Chapter 3, we will go over the basics of information theory, which will set the foundation for one of the most important concepts in thermodynamics: entropy. While it will not be used much beyond setting up the initial stages of our course, it will give us a wonderful foundation upon which to understand what thermodynamics (and more importantly statistical mechanics) can do for us at a deeper level.
Notes
1
One of the reasons I say this is that a standard physics PhD qualifying exam usually includes these four subjects. More importantly, I say this because the results of statistical mechanics find their way throughout all of physics and engineering, as we shall see.
2
This is a combination of several comments made by my students in the past, and I have edited out some more colorful words used.
3
I will define all of these terms precisely later.
4
We will be more concrete with the classical vs. quantum regime when the time comes.
5
Even though most of the universe can be understood with Newtonian gravity, for precise measurements, general relativity must enter the picture.
6
Taking the
thermodynamic limit
, when , is often ultimately done for many cases, for realistic systems, we will want to keep finite but very large.
7
Many students tend to disagree with this statement while taking this class. However, the fact that we even have an undergraduate version of this class shows that these problems are “easy” in some sense!
8
Which you know about from an everyday perspective, but for now forget you know anything about it.
9
Many textbooks use for this quantity to distinguish it from mechanical energy, but I tend to prefer this notation. If we used , we would have to contend with a possible confusion with potential energy anyway.
10
We will return to this example many times, more specifically in
Exercise 4.1
.
11
Ignorance is bliss, after all.
12
You won’t need to know how to solve these problems for this course, just the results of such solutions, specifically the total energy of a state of the system.
2Introduction to Probability Theory
As discussed inChapter 1, before we can truly study statistical mechanics, we first will have a brief overview of probabilities. I will not get into the rigor that is needed for a fully satisfactory treatment (there are entire courses for that). Rather I will focus on the basics of the subject that we will need here. Much of what I will discuss in this chapter will be familiar in an everyday sense; what makes it difficult is trying to quantify what we already know.
After completing this chapter, you should be able to
understand the meaning of probabilities, and calculate them for simple random systems,
understand the role of mean (average) values in statistical systems, and
have an idea of the difference between discrete and continuous probability distributions.
2.1 Understanding Probability
Even though it is so prevalent in our daily lives, probability is often misunderstood by many people. In some way, this misunderstanding arises because this mathematical concept is used so regularly in a colloquial sense. You may look at the weather forecast and see that it states there is a 65% chance of rain. If it doesn’t rain, you might think, “As usual, the weather forecast was wrong!” And if it says there is a 10% chance of rain and it does rain, you might think the same thing. It is very easy to have such a misconception of what these percentages (probabilities) are telling us.
Whenever we study probabilistic systems—those for which we don’t know what will happen definitively—then there is a specific meaning when we say there is a certain chance of something occurring. If something has a 65% chance to occur, then what we mean by this statement is as follows: If we have a very large1 number of identically prepared systems, then on average we would expect this to happen in 65% of those systems. In the weather example, we would need a very large number of identically prepared Earths (which is an odd idea to consider) and in roughly 65% of those Earths, it would rain. This very large number of identical systems this defines an ensemble.2 I have used the terms on average and roughly here, because we have to be careful as we discuss these concepts, and I’ll be more concrete in Section 2.2. What we need to keep in mind is that whenever I refer to a system, I really mean an ensemble of systems.3
2.2 Randomness, Fairness, and Probability
A common application of probability in everyday life is to games of chance, which I broadly define as any activity that involves (among other things) flipping a coin, drawing from a deck of cards, rolling dice, or other sorts of games you might find at a casino. As we discuss probabilities, I will use specific (non‐physics) examples to understand the various ideas that will arise.
What is most important to remember is that you already understand so much about probability without realizing it. You know that if you flip a coin, there is a 50‐50 chance of getting “heads,” a one out of six chance that you roll a four on an ordinary six‐sided die, and with a standard deck of playing cards, there’s a 1 in 52 chance you’ll draw the ace of spades. However, in order to apply these ideas more generally, and to ensure we don’t misunderstand these statements, we need to formalize these ideas more.
We begin by considering a system which can have some number of outcomes or events, and while perhaps in principle we could determine which of these events could occur, we will say there is no way to know exactly what will happen. In the case of the coin toss, in principle we can account for the initial force which causes the coin to begin to spin as it moves up and down under the force of gravity, and thus could determine which side will be face up upon landing. However, this can be difficult (you can consider the physics of coins, both under the assumption that the coin is very thin so there are only two outcomes in Ref. [1] as well as a thick coin, where there’s a significant chance it will land on its side in Ref. [2]). For all intents and purposes we will say that we cannot determine the outcome at all.
Once we have our system and the possible outcomes, then we wish to count how many of these outcomes there are. From this we will be able to determine the probabilities of each outcome.
Example 2.1 Let’s consider the coin, the die, and a deck of cards, all examples we will return to several times.
In the case of the coin, there are two possible outcomes, (usually referred to as heads or tails), for each event, or coin toss.
If the event under consideration is rolling a six‐sided die, then we have six possible outcomes (usually one through six), and each outcome is the side which is facing up after a roll.
An ordinary deck of playing cards has 52 cards (we are ignoring jokers), where we have 13 different faces (2–10, jack, queen, king, ace) and four suits (spades, hearts, clubs, diamonds) of each. The event in this case is drawing a card, and we have 52 outcomes (if you care only about which
specific
card you draw).
As alluded to in the third example, the outcomes we consider may change depending on what we are interested in—perhaps we only care about the value (say you are hoping for an ace specifically) of the card, not the suit. This must be taken into account as we calculate probabilities.
How then do we determine those probabilities? Unless we have some knowledge about the system (perhaps you know the die is weighted, or you were able to palm cards while shuffling the deck of cards), we will assume that the outcomes are all completely random. That is, we cannot assume any one outcome is more or less likely than any other. For our examples above, we will assume (unless proven otherwise) that the coins or dice are fair and the deck of cards is thoroughly shuffled that we can assume every outcome is equally likely. A given coin is different on either side and the “pips” on a die are such that each side of the coin or die are technically not equally weighted; however, this difference is rarely going to be significant when considering the probability of a particular outcome. The key point is that unless we have some prior knowledge about our system, there won’t be any reason to assume any individual outcome is more likely.4
Let’s now be a bit more precise about the probability of an individual outcome, which I will label as . In this definition I will be a little vague about the variables and terms, as they will become more clear with the exercises and examples below. Suppose a given outcome can occur times out of a total of possible events. The individual probability that outcome will happen is defined by
Keep in mind that is just an arbitrary label; don’t fixate on it too much, and note that the sum over is over all possible outcomes. For example, we could write for the coins (assuming a completely random outcome as discussed above):
We could use or 2, or , or even just write out heads and tails for the labels. Similarly, for a six‐sided fair die:
where refers to the roll of the die. And finally, for any of the possible card draws, we would get ; equal probabilities for any card draw. It’s important to note that it must be true that summing over all probabilities should give you one:
and given that , you can easily show this is true from Eq. (2.1). This is because if we have correctly accounted for all possible outcomes, there is a 100% chance that something will occur.
However, I must point out that Eq. (2.1) is only true in terms of what we expect to occur. That is, just because when you roll a die you expect that 1/6 of the outcomes will result in a four, this doesn’t mean that you will always get exactly that number of rolls. You might find that only 92 out of 600 rolls of a die result in a four, even though from above you would expect 100 of them. I clarify this when I discuss what I call experimental probabilities below in Eq. (2.5), but for now we will only consider this ideal probability.5
Exercise 2.1 What if you rolled two six‐sided dice, what are the possible outcomes? Is rolling a 1 on one and a 4 on the other more or less likely than rolling a 2 on one and a 6 on the other?
Usually we only care about the sum of the pips on the two dice, so in that case, what is the probability of rolling each allowed sum (so a 5 or an 8 in the aforementioned rolls)?
The previous exercise shows how you are familiar with probabilities, even if you don’t think so. In Exercise 2.1, you can work out the probabilities two different ways. The simplest (brute force) approach is to enumerate all of the possibilities and then apply Eq. (2.1). When rolling two dice, we can get the following:
a 2 if both dice resulted in a 1,
a 3 if one die is a 1 and the other is a 2,
a 4 if one result is 1 and the other is 3 or if both resulted in a 2,
etc.
In these cases, we see there is one way to obtain a 2, two ways to obtain a 3, three ways to obtain a 4, and so forth. Counting up all of the possible outcomes we find a total of 36 outcomes, and then using Eq. (2.1), we have that the probability, for example, to roll a total of 3 is .
Alternatively, we could realize that the outcomes of each die are statistically independent and then combine the individual probabilities of each outcome. Two outcomes are statistically independent if a given result does not depend on any prior results. That is, the result of the first die does not rely on the result of the second die.6 If we want to know the probability of rolling a 3, we would ask, “What is the probability that we would roll a 1 on the first die and a 2 on the second die, or a 2 on the first die and a 1 on the second die?” To determine the probability of two things occurring (so we want this and that to happen), we multiply the individual probabilities, and to determine the probability of one or another event to happen, then we add the individual probabilities. In the case of rolling a 3 on two dice,
