Contents 1 From Microscopic to Macroscopic Behavior 1 1.2 Some qualitative observations .4 Quality of energy .5 Some simple simulations .6 Work, heating, and the first law of thermodynamics .7 Measuring the pressure and temperature .8 *The fundamental need for a statistical approach .9 *Time and ensemble averages .10 *Models of matter .1 The ideal gas .11 Importance of simulations. 24 Suggestions for Further Reading .5 Pressure Equation of State .6 Some Thermodynamic Processes .com CONTENTS ii 2.8 The First Law of Thermodynamics .9 Energy Equation of State .10 Heat Capacities and Enthalpy .12 The Second Law of Thermodynamics .13 The Thermodynamic Temperature .14 The Second Law and Heat Engines .16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures .17 The Thermodynamic Pressure .18 The Fundamental Thermodynamic Relation .19 The Entropy of an Ideal Gas .20 The Third Law of Thermodynamics. 65 Appendix 2B: Mathematics of Thermodynamics. 73 Suggestions for Further Reading.
80 3 Concepts of Probability 82 3.1 Probability in everyday life .2 The rules of probability .4 The meaning of probability .1 Information and uncertainty .5 Bernoulli processes and the binomial distribution .6 Continuous probability distributions .7 The Gaussian distribution as a limit of the binomial distribution .8 The central limit theorem or why is thermodynamics possible? .9 The Poisson distribution and should you fly in airplanes? .10 *Traffic flow and the exponential distribution .11 *Are all probability distributions Gaussian?. 128 Suggestions for Further Reading .com CONTENTS iii 4 Statistical Mechanics 138 4.2 A simple example of a thermal interaction .2 *One-dimensional Ising model .3 A particle in a one-dimensional box .4 One-dimensional harmonic oscillator .5 One particle in a two-dimensional box .6 One particle in a three-dimensional box .7 Two noninteracting identical particles and the semiclassical limit .4 The number of states of N noninteracting particles: Semiclassical limit .5 The microcanonical ensemble (fixed E, V, and N) .6 Systems in contact with a heat bath: The canonical ensemble (fixed T, V, and N) 165 4.7 Connection between statistical mechanics and thermodynamics .8 Simple applications of the canonical ensemble .10 Simulations of the microcanonical ensemble .11 Simulations of the canonical ensemble .12 Grand canonical ensemble (fixed T, V, and µ) .13 Entropy and disorder. 181 Appendix 4A: The Volume of a Hypersphere. 183 Appendix 4B: Fluctuations in the Canonical Ensemble.
185 Suggestions for Further Reading .2 Thermodynamics of magnetism .3 The Ising model .4 The Ising Chain .2 ∗ Spin-spin correlation function .3 Simulations of the Ising chain .5 Absence of a phase transition in one dimension .5 The Two-Dimensional Ising Model .com CONTENTS iv 5.2 Computer simulation of the two-dimensional Ising model .6 Mean-Field Theory .7 *Infinite-range interactions. 224 Suggestions for Further Reading. 228 6 Noninteracting Particle Systems 230 6.2 The Classical Ideal Gas .3 Classical Systems and the Equipartition Theorem .4 Maxwell Velocity Distribution .5 Occupation Numbers and Bose and Fermi Statistics .6 Distribution Functions of Ideal Bose and Fermi Gases .7 Single Particle Density of States .8 The Equation of State for a Noninteracting Classical Gas .9 Black Body Radiation .10 Noninteracting Fermi Gas .1 Ground-state properties .2 Low temperature thermodynamic properties .12 The Heat Capacity of a Crystalline Solid .1 The Einstein model. 273 Appendix 6A: Low Temperature Expansion.
275 Suggestions for Further Reading. 286 7 Thermodynamic Relations and Processes 288 7.3 Applications of the Maxwell Relations .1 Internal energy of an ideal gas .2 Relation between the specific heats .4 Applications to Irreversible Processes .1 The Joule or free expansion process .2 Joule-Thomson process .5 Equilibrium Between Phases .2 Clausius-Clapeyron equation .3 Simple phase diagrams .4 Pressure dependence of the melting point .5 Pressure dependence of the boiling point .6 The vapor pressure curve. 303 Suggestions for Further Reading. 305 8 Classical Gases and Liquids 306 8.2 The Free Energy of an Interacting System .3 Second Virial Coefficient .5 High Temperature Expansion .7 Radial Distribution Function .1 Relation of thermodynamic functions to g(r) .3 Variable number of particles .4 Density expansion of g(r) .8 Computer Simulation of Liquids .9 Perturbation Theory of Liquids .1 The van der Waals Equation .2 Chandler-Weeks-Andersen theory .10 *The Ornstein-Zernicke Equation .11 *Integral Equations for g(r) .1 Debye-Hückel Theory .2 Linearized Debye-Hückel approximation .3 Diagrammatic Expansion for Charged Particles.
343 Appendix 8A: The third virial coefficient for hard spheres .com CONTENTS vi 9 Critical Phenomena 350 9.1 A Geometrical Phase Transition .2 Renormalization Group for Percolation .3 The Liquid-Gas Transition .5 Landau Theory of Phase Transitions .6 Other Models of Magnetism .7 Universality and Scaling Relations .8 The Renormalization Group and the 1D Ising Model .9 The Renormalization Group and the Two-Dimensional Ising Model. 382 Suggestions for Further Reading. 385 10 Introduction to Many-Body Perturbation Theory 387 10.2 Occupation Number Representation .3 Operators in the Second Quantization Formalism .4 Weakly Interacting Bose Gas .2 SI derived units .6 Euler-Maclaurin formula .com Chapter 1 From Microscopic to Macroscopic Behavior c 2006 by Harvey Gould and Jan Tobochnik 28 August 2006 The goal of this introductory chapter is to explore the fundamental differences between micro- scopic and macroscopic systems and the connections between classical mechanics and statistical mechanics. We note that bouncing balls come to rest and hot objects cool, and discuss how the behavior of macroscopic objects is related to the behavior of their microscopic constituents.
Com- puter simulations will be introduced to demonstrate the relation of microscopic and macroscopic behavior.1 Introduction Our goal is to understand the properties of macroscopic systems, that is, systems of many elec- trons, 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 copper coin, and a rubber band (examples of a gas, liquid, solid, and polymer, respectively). Less familiar macroscopic systems are superconductors, cell membranes, the brain, and the galaxies. We will find that the type of questions we ask about macroscopic systems differ in important ways from the questions we ask about microscopic systems.
An example of a question about a microscopic system is “What is the shape of the trajectory of the Earth in the solar system?” In contrast, have you ever wondered about the trajectory of a particular molecule in the air of your room? Why not? Is it relevant that these molecules are not visible to the eye? Examples of questions that we might 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? What is its maximum efficiency? 1 www. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 2 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 are the molecules arranged in a liquid? 6. How and why does water freeze into a particular crystalline structure? 7.
Why does iron lose its magnetism above a certain temperature? 8. Why does helium condense into a superfluid phase at very low temperatures? Why do some materials exhibit zero resistance to electrical current at sufficiently low temperatures? 9. How fast does a river current have to be before its flow changes from laminar to turbulent? 10. What will the weather be tomorrow? The above questions can be roughly classified into three groups.
Questions 1–3 are concerned with macroscopic properties such as pressure, volume, and temperature and questions 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 thermal engines, for example, the internal combustion engine and the steam turbine. Questions 4–8 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 the bridge between the microscopic and macroscopic worlds. Thermodynamics and statistical mechanics assume that the macroscopic properties of the system do not change with time on the average. Thermodynamics describes the change of a macroscopic system from one equilibrium state to another.
Questions 9 and 10 concern macro- scopic phenomena that change with time. Related areas are nonequilibrium thermodynamics and fluid mechanics from the macroscopic point of view and nonequilibrium statistical mechanics from the microscopic point of view. Although there has been progress in our understanding of nonequi- librium phenomena such as turbulent flow and hurricanes, our understanding of nonequilibrium phenomena is much less advanced than our understanding of equilibrium systems. Because un- derstanding the properties of macroscopic systems that are independent of time is easier, we will focus our attention on equilibrium systems and consider questions such as those in Questions 1–8.
FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 3 1.2 Some qualitative observations We begin our discussion of macroscopic systems by considering a glass of water. We know that if we place a glass of hot water into a cool 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 sensation of hot and cold and is important in our everyday experience.
We will find that temperature is a subtle concept. The direction or arrow of time is an even more subtle concept. Have you ever observed a glass of water at room temperature spontaneously become hotter? Why not? What other phenomena exhibit a direction of time? Time has a direction as 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 a direction of time for a single particle? Newton’s second law for a single particle, F = dp/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 behavior of macroscopic bodies and single particles is very different.
Unlike generations 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. Moreover, if we prepare two glasses of water under similar conditions, we would find that the observable properties of the water in each glass are indistinguishable, even though the motion of the individual particles in the two glasses would be very different.