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Statistical and Thermal Physics: With Computer Applications available in Hardcover, NOOK Book

- ISBN-10:
- 0691137447
- ISBN-13:
- 9780691137445
- Pub. Date:
- 07/01/2010
- Publisher:
- Princeton University Press

# Statistical and Thermal Physics: With Computer Applications

###### ADVERTISEMENT

## Overview

This textbook carefully develops the main ideas and techniques of statistical and thermal physics and is intended for upper-level undergraduate courses. The authors each have more than thirty years' experience in teaching, curriculum development, and research in statistical and computational physics.

*Statistical and Thermal Physics* begins with a qualitative discussion of the relation between the macroscopic and microscopic worlds and incorporates computer simulations throughout the book to provide concrete examples of important conceptual ideas. Unlike many contemporary texts on thermal physics, this book presents thermodynamic reasoning as an independent way of thinking about macroscopic systems. Probability concepts and techniques are introduced, including topics that are useful for understanding how probability and statistics are used. Magnetism and the Ising model are considered in greater depth than in most undergraduate texts, and ideal quantum gases are treated within a uniform framework. Advanced chapters on fluids and critical phenomena are appropriate for motivated undergraduates and beginning graduate students.

- Integrates Monte Carlo and molecular dynamics simulations as well as other numerical techniques throughout the text
- Provides self-contained introductions to thermodynamics and statistical mechanics
- Discusses probability concepts and methods in detail
- Contains ideas and methods from contemporary research
- Includes advanced chapters that provide a natural bridge to graduate study
- Features more than 400 problems
- Programs are open source and available in an executable cross-platform format
- Solutions manual (available only to teachers)

## Product Details

ISBN-13: | 9780691137445 |
---|---|

Publisher: | Princeton University Press |

Publication date: | 07/01/2010 |

Edition description: | New Edition |

Pages: | 532 |

Sales rank: | 940,718 |

Product dimensions: | 7.30(w) x 10.00(h) x 1.30(d) |

## About the Author

**Harvey Gould** is Professor of Physics at Clark University and Associate Editor of the *American Journal of Physics*. **Jan Tobochnik** is the Dow Distinguished Professor of Natural Science at Kalamazoo College and Editor of the *American Journal of Physics*. They are the coauthors, with Wolfgang Christian, of *An Introduction to Computer Simulation Methods: Applications to Physical Systems*.

## Read an Excerpt

#### Statistical and Thermal Physics

**With Computer Applications**

**By Harvey Gould Jan Tobochnik**

** PRINCETON UNIVERSITY PRESS **

**Copyright © 2010**

**Princeton University Press**

All right reserved.

All right reserved.

**ISBN: 978-0-691-13744-5**

#### Chapter One

**From Microscopic to Macroscopic**

We explore the fundamental differences between microscopic and macroscopic systems, note that bouncing balls come to rest and hot objects cool, and discuss how the behavior of macroscopic systems is related to the behavior of their microscopic constituents. Computer simulations are introduced to demonstrate the general qualitative behavior of macroscopic systems.

**1.1 Introduction**

Our goal is to understand the properties of *macroscopic* systems, that is, systems of many electrons, atoms, molecules, photons, or other constituents. Examples of familiar macroscopic objects include systems such as the air in your room, a glass of water, a coin, and a rubber band-examples of a gas, liquid, solid, and polymer, respectively. Less familiar macroscopic systems include superconductors, cell membranes, the brain, the stock market, and neutron stars.

We will find that the type of questions we ask about macroscopic systems differ in important ways from the questions we ask about systems that we treat microscopically. For example, consider the air in your room. Have you ever wondered about the trajectory of a particular molecule in the air? Would knowing that trajectory be helpful in understanding the properties of air? Instead of questions such as these, examples of questions that we do ask about macroscopic systems include the following:

1. How does the pressure of a gas depend on the temperature and the volume of its container? 2. How does a refrigerator work? How can we make it more efficient? 3. How much energy do we need to add to a kettle of water to change it to steam? 4. Why are the properties of water different from those of steam, even though water and steam consist of the same type of molecules? 5. How and why does a liquid freeze into a particular crystalline structure? 6. Why does helium have a superfluid phase at very low temperatures? Why do some materials exhibit zero resistance to electrical current at sufficiently low temperatures? 7. In general, how do the properties of a system emerge from its constituents? 8. How fast does the current in a river have to be before its flow changes from laminar to turbulent? 9. What will the weather be tomorrow? These questions can be roughly classified into three groups. Questions 1-3 are concerned with macroscopic properties such as pressure, volume, and temperature and processes related to heating and work. These questions are relevant to *thermodynamics*, which provides a framework for relating the macroscopic properties of a system to one another. Thermodynamics is concerned only with macroscopic quantities and ignores the microscopic variables that characterize individual molecules. For example, we will find that understanding the maximum efficiency of a refrigerator does not require a knowledge of the particular liquid used as the coolant. Many of the applications of thermodynamics are to engines, for example, the internal combustion engine and the steam turbine.

Questions 4-7 relate to understanding the behavior of macroscopic systems starting from the atomic nature of matter. For example, we know that water consists of molecules of hydrogen and oxygen. We also know that the laws of classical and quantum mechanics determine the behavior of molecules at the microscopic level. The goal of *statistical mechanics* is to begin with the microscopic laws of physics that govern the behavior of the constituents of the system and deduce the properties of the system as a whole. Statistical mechanics is a bridge between the microscopic and macroscopic worlds.

Question 8 also relates to a macroscopic system, but temperature is not relevant in this case. Moreover, turbulent flow continually changes in time. Question 9 concerns macroscopic phenomena that change with time. Although there has been progress in our understanding of time-dependent phenomena such as turbulent flow and hurricanes, our understanding of such phenomena is much less advanced than our understanding of time-independent systems. For this reason we will focus our attention on systems whose macroscopic properties are independent of time and consider questions such as those in Questions 1-7.

**1.2 Some Qualitative Observations**

We begin our discussion of macroscopic systems by considering a glass of hot water. We know that, if we place a glass of hot water into a large cold room, the hot water cools until its temperature equals that of the room. This simple observation illustrates two important properties associated with macroscopic systems-the importance of *temperature* and the *"arrow" of time*. Temperature is familiar because it is associated with the physiological sensations of hot and cold and is important in our everyday experience.

The direction or arrow of time raises many questions. Have you ever observed a glass of water at room temperature spontaneously become hotter? Why not? What other phenomena exhibit a direction of time? The direction of time is expressed by the nursery rhyme:

*Humpty Dumpty sat on a wall Humpty Dumpty had a great fall All the king's horses and all the king's men Couldn't put Humpty Dumpty back together again.*

Is there a direction of time for a single particle? Newton's second law for a single particle, **F** = *d***p**/*dt*, implies that the motion of particles is *time-reversal invariant*; that is, Newton's second law looks the same if the time *t* is replaced by *-t* and the momentum **p** by **-p**. There is no direction of time at the microscopic level. Yet if we drop a basketball onto a floor, we know that it will bounce and eventually come to rest. Nobody has observed a ball at rest spontaneously begin to bounce, and then bounce higher and higher. So based on simple everyday observations we can conclude that the behaviors of macroscopic bodies and single particles are very different.

Unlike scientists of about a century or so ago, we know that macroscopic systems such as a glass of water and a basketball consist of many molecules. Although the intermolecular forces in water produce a complicated trajectory for each molecule, the observable properties of water are easy to describe. If we prepare two glasses of water under similar conditions, we know that the observable properties of the water in each glass are indistinguishable, even though the motion of the individual particles in the two glasses is very different.

If we take into account that the bouncing ball and the floor consist of molecules, then we know that the total energy of the ball and the floor is conserved as the ball bounces and eventually comes to rest. Why does the ball eventually come to rest? You might be tempted to say the cause is "friction," but friction is just a name for an effective or phenomenological force. At the microscopic level we know that the fundamental forces associated with mass, charge, and the nucleus conserve total energy. Hence, if we include the energy of the molecules of the ball and the floor, the total energy is conserved. Conservation of energy does not explain why the inverse process, where the ball rises higher and higher with each bounce, does not occur. Such a process also would conserve the total energy. So a more fundamental explanation is that the ball comes to rest consistent with conservation of the total energy and with some other principle of physics. We will learn that this principle is associated with an increase in the *entropy* of the system. For now, entropy is just a name, and it is important only to understand that energy conservation is not sufficient to understand the behavior of macroscopic systems.

By thinking about the constituent molecules, we can gain some insight into the nature of entropy. Let us consider the ball bouncing on the floor again. Initially, the energy of the ball is associated with the motion of its center of mass, and we say that the energy is associated with one degree of freedom. After some time the energy becomes associated with the individual molecules near the surface of the ball and the floor, and we say that the energy is now distributed over many degrees of freedom. If we were to bounce the ball on the floor many times, the ball and the floor would each feel warm to our hands. So we can hypothesize that energy has been transferred from one degree of freedom to many degrees of freedom while the total energy has been conserved. Hence, we conclude that the entropy is a measure of how the energy is distributed.

What other quantities are associated with macroscopic systems besides temperature, energy, and entropy? We are already familiar with some of these quantities. For example, we can measure the air *pressure* in a basketball and its *volume*. More complicated quantities are the *thermal conductivity* of a solid and the *viscosity* of oil. How are these macroscopic quantities related to each other and to the motion of the individual constituent molecules? The answers to questions such as these and the meaning of temperature and entropy will take us through many chapters.

**1.3 Doing Work and the Quality of Energy**

We already have observed that hot objects cool, and cool objects do not spontaneously become hot; bouncing balls come to rest, and a stationary ball does not spontaneously begin to bounce. And although the total energy is conserved in these processes, the *distribution* of energy changes in an irreversible manner. We also have concluded that a new concept, the entropy, needs to be introduced to explain the direction of change of the distribution of energy.

Now let us take a purely macroscopic viewpoint and discuss how we can arrive at a similar qualitative conclusion about the asymmetry of nature. This viewpoint was especially important historically because of the lack of a microscopic theory of matter in the nineteenth century when the laws of thermodynamics were being developed.

Consider the conversion of stored energy into heating a house or a glass of water. The stored energy could be in the form of wood, coal, or animal and vegetable oils for example. We know that this conversion is easy to do using simple methods, for example, an open flame. We also know that if we rub our hands together, they will become warmer. There is no theoretical limit to the efficiency at which we can convert stored energy to energy used for heating an object.

What about the process of converting stored energy into work? Work, like many of the other concepts that we have mentioned, is difficult to define. For now let us say that doing work is equivalent to the raising of a weight. To be useful, we need to do this conversion in a controlled manner and indefinitely. A single conversion of stored energy into work such as the explosion of dynamite might demolish an unwanted building, but this process cannot be done repeatedly with the same materials. It is much more difficult to convert stored energy into work, and the discovery of ways to do this conversion led to the industrial revolution. In contrast to the primitiveness of an open flame, we have to build an *engine* to do this conversion.

Can we convert stored energy into useful work with 100% efficiency? To answer this question we have to appeal to observation. We know that some forms of stored energy are more useful than others. For example, why do we burn coal and oil in power plants even though the atmosphere and the oceans are vast reservoirs of energy? Can we mitigate global climate change by extracting energy from the atmosphere to run a power plant? From the work of Kelvin, Clausius, Carnot, and others, we know that we cannot convert stored energy into work with 100% efficiency, and we must necessarily "waste" some of the energy. At this point, it is easier to understand the reason for this necessary inefficiency by microscopic arguments. For example, the energy in the gasoline of the fuel tank of an automobile is associated with many molecules. The job of the automobile engine is to transform this (potential) energy so that it is associated with only a few degrees of freedom, that is, the rolling tires and gears. It is plausible that it is inefficient to transfer energy from many degrees of freedom to only a few. In contrast, the transfer of energy from a few degrees of freedom (the firewood) to many degrees of freedom (the air in your room) is relatively easy.

The importance of entropy, the direction of time, and the inefficiency of converting stored energy into work are summarized in the various statements of the *second law of thermodynamics*. It is interesting that the second law of thermodynamics was conceived before the first law of thermodynamics. As we will learn, the first law is a statement of conservation of energy.

Suppose that we take some firewood and use it to "heat" a sealed room. Because of energy conservation, the energy in the room plus the firewood is the same before and after the firewood has been converted to ash. Which form of the energy is more capable of doing work? You probably realize that the firewood is a more useful form of energy than the "hot air" and ash that exists after the firewood is burned. Originally the energy was stored in the form of chemical (potential) energy. Afterward the energy is mostly associated with the motion of the molecules in the air. What has changed is not the total energy, but its ability to do work. We will learn that an increase in entropy is associated with a loss of ability to do work. We have an entropy problem, not an energy problem.

**1.4 Some Simple Simulations**

So far we have discussed the behavior of macroscopic systems by appealing to everyday experience and simple observations. We now discuss some simple ways of *simulating* the behavior of macroscopic systems. Although we cannot simulate a macroscopic system of 1023 particles on a computer, we will find that even small systems of the order of 100 particles are sufficient to illustrate the qualitative behavior of macroscopic systems.

We first discuss how we can simulate a simple model of a gas consisting of molecules whose internal structure can be ignored. In particular, imagine a system of *N* particles in a closed container of volume *V* and suppose that the container is far from the influence of external forces such as gravity. We will usually consider two-dimensional systems so that we can easily visualize the motion of the particles.

*(Continues...)*

Excerpted fromStatistical and Thermal PhysicsbyHarvey Gould Jan TobochnikCopyright © 2010 by Princeton University Press. Excerpted by permission.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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## Table of Contents

Preface xi

Chapter 1: From Microscopic to Macroscopic Behavior 1

1.1 Introduction 1

1.2 Some Qualitative Observations 2

1.3 Doing Work and the Quality of Energy 4

1.4 Some Simple Simulations 5

1.5 Measuring the Pressure and Temperature 15

1.6 Work, Heating, and the First Law of Thermodynamics 19

1.7 *The Fundamental Need for a Statistical Approach 20

1.8 *Time and Ensemble Averages 22

1.9 Models of Matter 22

1.9.1 The ideal gas 23

1.9.2 Interparticle potentials 23

1.9.3 Lattice models 23

1.10 Importance of Simulations 24

1.11 Dimensionless Quantities 24

1.12 Summary 25

1.13 Supplementary Notes 27

1.13.1 Approach to equilibrium 27

1.13.2 Mathematics refresher 28

Vocabulary 28

Additional Problems 29

Suggestions for Further Reading 30

Chapter 2: Thermodynamic Concepts and Processes 32

2.1 Introduction 32

2.2 The System 33

2.3 Thermodynamic Equilibrium 34

2.4 Temperature 35

2.5 Pressure Equation of State 38

2.6 Some Thermodynamic Processes 39

2.7 Work 40

2.8 The First Law of Thermodynamics 44

2.9 Energy Equation of State 47

2.10 Heat Capacities and Enthalpy 48

2.11 Quasistatic Adiabatic Processes 51

2.12 The Second Law of Thermodynamics 55

2.13 The Thermodynamic Temperature 58

2.14 The Second Law and Heat Engines 60

2.15 Entropy Changes 67

2.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures 74

2.17 The Thermodynamic Pressure 75

2.18 The Fundamental Thermodynamic Relation 76

2.19 The Entropy of an Ideal Classical Gas 77

2.20 The Third Law of Thermodynamics 78

2.21 Free Energies 79

2.22 Thermodynamic Derivatives 84

2.23 *Applications to Irreversible Processes 90

2.23.1 Joule or free expansion process 90

2.23.2 Joule-Thomson process 91

2.24 Supplementary Notes 94

2.24.1 The mathematics of thermodynamics 94

2.24.2 Thermodynamic potentials and Legendre transforms 97

Vocabulary 99

Additional Problems 100

Suggestions for Further Reading 108

Chapter 3: Concepts of Probability 111

3.1 Probability in Everyday Life 111

3.2 The Rules of Probability 114

3.3 Mean Values 119

3.4 The Meaning of Probability 121

3.4.1 Information and uncertainty 124

3.4.2 *Bayesian inference 128

3.5 Bernoulli Processes and the Binomial Distribution 134

3.6 Continuous Probability Distributions 147

3.7 The Central Limit Theorem (or Why Thermodynamics

Is Possible) 151

3.8 *The Poisson Distribution or Should You Fly? 155

3.9 *Traffic Flow and the Exponential Distribution 156

3.10 *Are All Probability Distributions Gaussian? 159

3.11 Supplementary Notes 161

3.11.1 Method of undetermined multipliers 161

3.11.2 Derivation of the central limit theorem 163

Vocabulary 167

Additional Problems 168

Suggestions for Further Reading 177

Chapter 4: The Methodology of Statistical Mechanics 180

4.1 Introduction 180

4.2 A Simple Example of a Thermal Interaction 182

4.3 Counting Microstates 192

4.3.1 Noninteracting spins 192

4.3.2 A particle in a one-dimensional box 193

4.3.3 One-dimensional harmonic oscillator 196

4.3.4 One particle in a two-dimensional box 197

4.3.5 One particle in a three-dimensional box 198

4.3.6 Two noninteracting identical particles and the

semiclassical limit 199

4.4 The Number of States of Many Noninteracting Particles:

Semiclassical Limit 201

4.5 The Microcanonical Ensemble (Fixed E, V, and N) 203

4.6 The Canonical Ensemble (Fixed T, V, and N) 209

4.7 Connection between Thermodynamics and Statistical Mechanics

in the Canonical Ensemble 216

4.8 Simple Applications of the Canonical Ensemble 218

4.9 An Ideal Thermometer 222

4.10 Simulation of the Microcanonical Ensemble 225

4.11 Simulation of the Canonical Ensemble 226

4.12 Grand Canonical Ensemble (Fixed T, V, and ?) 227

4.13 *Entropy Is Not a Measure of Disorder 229

4.14 Supplementary Notes 231

4.14.1 The volume of a hypersphere 231

4.14.2 Fluctuations in the canonical ensemble 232

Vocabulary 233

Additional Problems 234

Suggestions for Further Reading 239

Chapter 5: Magnetic Systems 241

5.1 Paramagnetism 241

5.2 Noninteracting Magnetic Moments 242

5.3 Thermodynamics of Magnetism 246

5.4 The Ising Model 248

5.5 The Ising Chain 249

5.5.1 Exact enumeration 250

5.5.2 Spin-spin correlation function 253

5.5.3 Simulations of the Ising chain 256

5.5.4 *Transfer matrix 257

5.5.5 Absence of a phase transition in one dimension 260

5.6 The Two-Dimensional Ising Model 261

5.6.1 Onsager solution 262

5.6.2 Computer simulation of the two-dimensional Ising model 267

5.7 Mean-Field Theory 270

5.7.1 *Phase diagram of the Ising model 276

5.8 *Simulation of the Density of States 279

5.9 *Lattice Gas 282

5.10 Supplementary Notes 286

5.10.1 The Heisenberg model of magnetism 286

5.10.2 Low temperature expansion 288

5.10.3 High temperature expansion 290

5.10.4 Bethe approximation 292

5.10.5 Fully connected Ising model 295

5.10.6 Metastability and nucleation 297

Vocabulary 300

Additional Problems 300

Suggestions for Further Reading 306

Chapter 6: Many-Particle Systems 308

6.1 The Ideal Gas in the Semiclassical Limit 308

6.2 Classical Statistical Mechanics 318

6.2.1 The equipartition theorem 318

6.2.2 The Maxwell velocity distribution 321

6.2.3 The Maxwell speed distribution 323

6.3 Occupation Numbers and Bose and Fermi Statistics 325

6.4 Distribution Functions of Ideal Bose and Fermi Gases 327

6.5 Single Particle Density of States 329

6.5.1 Photons 331

6.5.2 Nonrelativistic particles 332

6.6 The Equation of State of an Ideal Classical Gas: Application

of the Grand Canonical Ensemble 334

6.7 Blackbody Radiation 337

6.8 The Ideal Fermi Gas 341

6.8.1 Ground state properties 342

6.8.2 Low temperature properties 345

6.9 The Heat Capacity of a Crystalline Solid 351

6.9.1 The Einstein model 351

6.9.2 Debye theory 352

6.10 The Ideal Bose Gas and Bose Condensation 354

6.11 Supplementary Notes 360

6.11.1 Fluctuations in the number of particles 360

6.11.2 Low temperature expansion of an ideal Fermi gas 363

Vocabulary 365

Additional Problems 366

Suggestions for Further Reading 374

Chapter 7: The Chemical Potential and Phase Equilibria 376

7.1 Meaning of the Chemical Potential 376

7.2 Measuring the Chemical Potential in Simulations 380

7.2.1 The Widom insertion method 380

7.2.2 The chemical demon algorithm 382

7.3 Phase Equilibria 385

7.3.1 Equilibrium conditions 386

7.3.2 Simple phase diagrams 387

7.3.3 Clausius-Clapeyron equation 389

7.4 The van der Waals Equation of State 393

7.4.1 Maxwell construction 393

7.4.2 *The van der Waals critical point 400

7.5 *Chemical Reactions 403

Vocabulary 407

Additional Problems 407

Suggestions for Further Reading 408

Chapter 8: Classical Gases and Liquids 410

8.1 Introduction 410

8.2 Density Expansion 410

8.3 The Second Virial Coefficient 414

8.4 *Diagrammatic Expansions 419

8.4.1 Cumulants 420

8.4.2 High temperature expansion 421

8.4.3 Density expansion 426

8.4.4 Higher order virial coefficients for hard spheres 428

8.5 The Radial Distribution Function 430

8.6 Perturbation Theory of Liquids 437

8.6.1 The van der Waals equation 439

8.7 *The Ornstein-Zernike Equation and Integral Equations for g(r ) 441

8.8 *One-Component Plasma 445

8.9 Supplementary Notes 449

8.9.1 The third virial coefficient for hard spheres 449

8.9.2 Definition of g(r ) in terms of the local particle density 450

8.9.3 X-ray scattering and the static structure function 451

Vocabulary 455

Additional Problems 456

Suggestions for Further Reading 458

Chapter 9: Critical Phenomena: Landau Theory and the Renormalization Group Method 459

9.1 Landau Theory of Phase Transitions 459

9.2 Universality and Scaling Relations 467

9.3 A Geometrical Phase Transition 469

9.4 Renormalization Group Method for Percolation 475

9.5 The Renormalization Group Method and the One-Dimensional Ising Model 479

9.6 The Renormalization Group Method and the Two-Dimensional Ising Model 484

Vocabulary 490

Additional Problems 491

Suggestions for Further Reading 492

Appendix: Physical Constants and Mathematical Relations 495

A.1 Physical Constants and Conversion Factors 495

A.2 Hyperbolic Functions 496

A.3 Approximations 496

A.4 Euler-Maclaurin Formula 497

A.5 Gaussian Integrals 497

A.6 Stirling's Approximation 498

A.7 Bernoulli Numbers 500

A.8 Probability Distributions 500

A.9 Fourier Transforms 500

A.10 The Delta Function 501

A.11 Convolution Integrals 502

A.12 Fermi and Bose Integrals 503

Index 505

## What People are Saying About This

**William Klein**

This is an excellent book. It is better than any other textbook I have encountered for an undergraduate course in statistical thermodynamics. The authors' use of simulations to build a student's intuition is novel, and the problems and examples are very useful. They bring out the important issues and are a real asset in getting students to think about the subject.

— *William Klein, Boston University*

**Lebowitz**

This is an ambitious book written by two experienced researchers and teachers. Starting from the microscopic dynamics of atoms and molecules, it uses statistical mechanical ideas to explain the thermodynamic behavior of macroscopic systems, and amply illustrates these ideas using hands-on computer simulations. Both teachers and students will find this book stimulating and rewarding.

— *Joel L. Lebowitz, Rutgers University*

**Jon Machta**

In addition to being a clear, comprehensive introduction to the field, this book includes a unique and welcome feature: an emphasis on computer simulations. These are integral to the exposition and provide key insights into fundamental concepts that so often confuse newcomers to the field. Simulations also give students a tool to investigate interesting topics that are normally considered too advanced for undergraduates. I highly recommend this book to anyone planning to teach undergraduate statistical and thermal physics.

— *Jon Machta, University of Massachusetts Amherst*

**Einstein**

Gould and Tobochnik are respected researchers in the field and have a good sense of what is significant. Statistical and Thermal Physics includes many problems, exercises, and enlightening commentaries. The textbook places unique emphasis on numerical simulation techniques and what one can learn from them, and closely integrates them into the presentation. This is a welcome innovation.

— *Theodore L. Einstein, University of Maryland*