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Statistical Mechanics Statistical Mechanics Henri J.F. Jansen Department of Physics Oregon State University October 12, 2008 II Contents 1 Foundation of statistical mechanics. 1 1.1 Introduction. 1 1.2 Program of statistical mechanics. 4 1.3 States of a system. 5 1.4 Averages. 10 1.5 Thermal equilibrium. 14 1.6 Entropy and temperature. 16 1.7 Laws of thermodynamics. 19 1.8 Problems for chapter 1 . 20 2 The canonical ensemble 23 2.1 Introduction. 23 2.2 Energy and entropy and temperature. 26 2.3 Work and pressure. 28 2.4 Helmholtz free energy. 31 2.5 Changes in variables. 32 2.6 Properties of the Helmholtz free energy. 33 2.7 Energy fluctuations. 35 2.8 A simple example. 37 2.9 Problems for chapter 2 . 39 3 Variable number of particles 43 3.1 Chemical potential. 43 3.2 Examples of the use of the chemical potential. 46 3.3 Di®erential relations and grand potential. 48 3.4 Grand partition function. 50 3.5 Overview of calculation methods. 55 3.6 A simple example. 57 3.7 Ideal gas in ¯rst approximation. 58 3.8 Problems for chapter 3 . 64 4 Statistics of independent particles. 67 4.1 Introduction. 67 4.2 Boltzmann gas again. 74 III IV CONTENTS 4.3 Gas of poly-atomic molecules. 77 4.4 Degenerate gas. 79 4.5 Fermi gas. 80 4.6 Boson gas. 85 4.7 Problems for chapter 4 . 87 5 Fermions and Bosons 89 5.1 Fermions in a box. 89 5.2 Bosons in a box. 106 5.3 Bose-Einstein condensation. 113 5.4 Problems for chapter 5 . 115 6 Density matrix formalism. 119 6.1 Density operators. 119 6.2 General ensembles. 123 6.3 Maximum entropy principle. 125 6.4 Equivalence of entropy de¯nitions for canonical ensemble. 134 6.5 Problems for chapter 6 . 136 7 Classical statistical mechanics. 139 7.1 Relation between quantum and classical mechanics. 139 7.2 Classical formulation of statistical mechanical properties. 142 7.3 Ergodic theorem. 144 7.4 What is chaos? . 146 7.5 Ideal gas in classical statistical mechanics. 149 7.6 Normal systems. 151 7.7 Quadratic variables. 151 7.8 E®ects of the potential energy. 152 7.9 Problems for chapter 7 . 154 8 Mean Field Theory: critical temperature. 157 8.1 Introduction. 157 8.2 Basic Mean Field theory. 161 8.3 Mean Field results. 164 8.4 Density-matrix approach (Bragg-Williams approximation. 168 8.5 Critical temperature in di®erent dimensions. 177 8.6 Bethe approximation. 181 8.7 Problems for chapter 8 . 188 9 General methods: critical exponents. 191 9.1 Introduction. 191 9.2 Integration over the coupling constant. 192 9.3 Critical exponents. 197 9.4 Relation between susceptibility and fluctuations. 203 9.5 Exact solution for the Ising chain. 205 9.6 Spin-correlation function for the Ising chain. 210 CONTENTS V 9.7 Renormalization group theory. 218 9.8 Problems for chapter 9 . 226 A Solutions to selected problems. 229 A.1 Solutions for chapter 1. 229 A.2 Solutions for chapter 2. 237 A.3 Solutions for chapter 3 . 242 A.4 Solutions for chapter 4. 247 A.5 Solutions for chapter 5. 251 A.6 Solutions for chapter 6. 257 VI CONTENTS List of Figures 4.1 Fermi Dirac distribution function. 71 7.1 H¶enon-Heilespotential. 148 ¤ 1 ¤ 8.1 ¯ = 2 , h = 2 . 166 8.2 ¯¤ = 2 , h¤ = 0:1 . 166 8.3 ¯ = 1 , J = 1 , q = 3 . 185 8.4 ¯ = 2 , J = 1 , q = 3 . 186 9.1 Ideal case to ¯nd critical exponent, with wrong guess of critical temperature. 201 9.2 Ideal case to ¯nd critical exponent. 202 9.3 Non-analytic case to ¯nd critical exponent. 202 9.4 Finite sample case to ¯nd critical exponent. 203 9.5 Cluster results to ¯nd critical exponent. 203 9.6 Magnetization of the one dimensional Ising chain. 210 9.7 Magnetization of the one dimensional Ising chain, as a func- tion of log(h). ............................211 VII VIII INTRODUCTION Introduction. Thermodynamics and statistical mechanics are two aspects of the study of large systems, where we cannot describe the majority of all details of that sys- tem. Thermodynamics approaches this problem from the observational side. We perform experiments on macroscopic systems and deduce collective properties of these systems, independent of the macroscopic details of the system. In ther- modynamics we start with a few state variables, and discuss relations between these variables and between the state variables and experimental observations. Many theorists ¯nd this an irrelevant exercise, but I strongly disagree. Thermo- dynamics is the mathematical language of experimental physics, and the nature of this language itself puts constraints on what kind of experimental results are valid. Nevertheless, true understanding needs the details on a microscopic level. We need to describe the behavior of all particles in the sample. That is an impossible task, and we need to resort to some kind of statistical treatment. Discussions on the foundations of statistical mechanics are very interesting, and help us understand the limitations of statistical mechanical treatments. This is analogue to the situation in quantum mechanics. Textbooks in quantum mechanics sue two approaches. One can simply start from SchrÄodinger'sequa- tion and discuss the mathematical techniques needed to solve this equation for important models. Or one can start with a discussion of the fundamentals of quantum mechanics and justify the existence of the SchrÄodingerequation. This is always done in less depth, because if one wants to go in great depth the discus- sion starts to touch on topics in philosophy. Again, I ¯nd that very interesting, but most practitioners do not need it. In statistical mechanics one can do the same. The central role is played by the partition function, and the rules needed to obtain observable data from the partition function. If these procedures are done correctly, all limitations set by thermodynamics are obeyed. In real life, of course, we make approximations, and then things can go wrong. One very important aspect of that discussion is the concept of the thermodynamic limit. We never have in¯nitely large samples, and hence we see e®ects due to the ¯nite size, both in space and in time. In such cases the statistical assumptions made to derive the concept of a partition function are not valid anymore, and we need alternative models. In many cases the modi¯cations are small, but sometimes they are large. Quanti¯cation of small and large now depends on intrinsic time and length scales of our system. The important di®erence between quantum mechanics and statistical me- chanics is the fact that for all atomic systems quantum mechanics is obeyed, but for many systems the ¯nite size of a sample is important. Therefore, in statistical mechanics it is much more important to understand what the as- sumptions are, and how they can be wrong. That is why we need to start with a discussion of the foundations. Again, we can go in greater depth that we INTRODUCTION IX will do here, which leads to important discussions on the arrow of time, for example. We will skip that here, but give enough information to understand the basic ideas. In order to introduce the partition function one needs to de¯ne the entropy. That can be done in many di®erent ways. It is possible to start from chaos and Lyaponov exponents. It is possible to start with information theory and derive a fairness function. Although the latter approach is fundamentally more sounds, it does not illuminate the ideas one needs to consider when discussing validity of the theory. Therefore, I follow the old fashioned approach based on probability of states. Working with statistical approaches is easier when the states of a system are labelled by an integer. Continuous variables need some extra prescriptions, which obscures the mathematics. Therefore we start with quantum mechanical systems. Counting is easiest. We follow the approach in the textbook by Kittel and Kroemer, an excellent textbook for further reference. It is aimed at the undergraduate level. Here we take their ideas, but expand on them. The toy model is a two state model, because counting states is simplest in that case. Almost every introduction to quantum statistical mechanics uses some form of this model, with states labelled up and down, plus and minus, black and white, occupied and empty, and so on. This basic model is simple enough to show mathematically how we use the law of large numbers (ergo, the thermodynamic limit) to arrive at Gaussian distributions. All conclusions follow from there. The ¯rst few chapters discuss the de¯nition of the entropy along these lines (chapter one), the partition function (chapter two), and the grand partition function, where the number of particles varies (chapter three). Remember that in thermodynamics we use N for the number of mols in a material, and use the constant R, the molar gas constant. In statistical mechanics we use N as the number of particles, and use kB, the Boltzmann constant. The ¯rst three chapters describe quantum states for the whole system. A very important application is when the energies of the particles are independent, in which case we can simplify the calculations enormously. This is done in chapter four. In chapter ¯ve we apply this formalism to the important examples of non-interacting fermions and bosons. The mathematical background needed is also explained in the textbook by Huang. This chapter is a must read for anybody who is going to work in solid state physics.
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