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Graduate Statistical Mechanics Graduate Statistical Mechanics Leon Hostetler December 16, 2018 Version 0.5 Contents Preface v 1 The Statistical Basis of Thermodynamics1 1.1 Counting States................................. 1 1.2 Connecting to Thermodynamics........................ 4 1.3 Intensive and Extensive Parameters ..................... 7 1.4 Averages..................................... 8 1.5 Summary: The Statistical Basis of Thermodynamics............ 9 2 Ensembles and Classical Phase Space 13 2.1 Classical Phase Space ............................. 13 2.2 Liouville's Theorem .............................. 14 2.3 Microcanonical Ensemble ........................... 16 2.4 Canonical Ensemble .............................. 20 2.5 Grand Canonical Ensemble .......................... 29 2.6 Summary: Ensembles and Classical Phase Space .............. 32 3 Thermodynamic Relations 37 3.1 Thermodynamic Potentials .......................... 37 3.2 Fluctuations................................... 44 3.3 Thermodynamic Response Functions..................... 48 3.4 Minimizing and Maximizing Thermodynamic Potentials.......... 49 3.5 Summary: Thermodynamic Relations .................... 51 4 Quantum Statistics 55 4.1 The Density Matrix .............................. 55 4.2 Indistinguishable Particles........................... 57 4.3 Thermodynamic Properties .......................... 59 4.4 Chemical Potential............................... 63 4.5 Summary: Quantum Statistics ........................ 65 5 Boltzmann Gases 67 5.1 Kinetic Theory................................. 68 5.2 Equipartition and Virial Theorems...................... 70 5.3 Internal Degrees of Freedom.......................... 71 5.4 Chemical Equilibrium ............................. 74 5.5 Summary: Boltzmann Gases ......................... 76 6 Ideal Bose Gases 79 6.1 General Bose Gases............................... 79 6.2 Bose-Einstein Condensation.......................... 80 6.3 Photons and Black-body Radiation...................... 84 6.4 Phonons..................................... 85 Contents iii 6.5 Summary: Ideal Bose Gases.......................... 88 7 Ideal Fermi Gases 91 7.1 General Fermi Gases.............................. 91 7.2 Pauli Spin Susceptibility............................ 93 7.3 Sommerfeld Expansion............................. 95 7.4 Summary: Ideal Fermi Gases ......................... 98 8 Interacting Gases 101 8.1 Correlation Functions ............................. 101 8.2 Virial Expansion ................................ 104 8.3 Summary: Interacting Gases ......................... 111 9 Phase Transitions 113 9.1 Liquid-Gas Phase Transition ......................... 116 9.2 Ising Model................................... 119 9.3 Critical Exponents............................... 124 9.4 Summary: Phase Transitions ......................... 128 10 Landau-Ginzburg Field Theory 133 10.1 Introduction................................... 133 10.2 Phase Transitions................................ 135 10.3 Fluctuations and Correlations......................... 137 10.4 Symmetry Breaking .............................. 139 10.5 The Ginzburg Criterion ............................ 143 10.6 Scaling Hypothesis............................... 145 10.7 Summary: Landau-Ginzburg Field Theory.................. 149 11 Non-Equilibrium Statistical Mechanics 153 11.1 BBGKY..................................... 153 11.2 Hydrodynamics................................. 156 11.3 Summary: Non-Equilibrium Statistical Mechanics . 159 12 Review 161 12.1 Entropy..................................... 161 12.2 Ensembles.................................... 161 12.3 Classical Limit ................................. 162 12.4 Thermodynamic Relations........................... 164 12.5 Quantum Statistics............................... 165 12.6 Interacting Gases................................ 167 12.7 Phase Transitions................................ 167 12.8 Landau-Ginzburg Field Theory........................ 167 Preface About These Notes These are my class notes from the graduate statistical mechanics class I took at Michi- gan State University with Professor Luke Roberts. The primary textbook we used was Statistical Mechanics by Pathria and Beale. My class notes can be found at www.leonhostetler.com/classnotes Please bear in mind that these notes will contain errors. Any errors are certainly my own. If you find one, please email me at [email protected] with the name of the class notes, the page on which the error is found, and the nature of the error. If you include your name, I will probably list your name on the thank you page if I decide to compile and sell my notes. This work is currently licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License. That means you are free to copy and distribute this document in whole for noncommercial use, but you are not allowed to distribute derivatives of this document or to copy and distribute it for commercial reasons. Updates Last Updated: December 16, 2018 Version 0.5: (Dec. 16, 2018) First upload. Chapter 1 The Statistical Basis of Thermodynamics 1.1 Counting States Tip In general, we often want to know how to go from the Hamiltonian H of a system to the physical, macroscopic properties of the system. These properties typically include Often we will work in units where kB = 1. • N: The number of particles the system is confined to. This is typically a large number ∼ 1023 • V : The volume of the system • E: The energy of the system 46 The classical Hamiltonian H(fqi; pig) will have a huge number of variables, e.g. a 10 - dimensional space. Generally, we don't know all the coordinates qi and momenta pi of the particles in our system, but we may know the overall volume V of the system and its energy E. There are typically many microstates of a system each with the same overall energy E. In fact, the quantity Ω(N; V; E); denotes the number of microstates that correspond to the single macrostate with particle number N, volume V , and energy E. The number Ω tells us something about our uncertainty of the microscopic state of the system. For example, if Ω = 1, i.e. there is only a single microstate with N; V; E, then there is no uncertainty because we know exactly which microstate the system must be in. Typically, however, Ω ∼ 1023, which means we are very uncertain about exactly which microstate the system is in. Another useful quantity is the Boltzmann entropy S = kB ln Ω(N; V; E): The entropy is also a measure of the uncertainty of the system. Example 1.1.1: Paramagnet Suppose we have N spin-1/2 particles locked on a lattice so there are no translational degrees of freedom. Assume there are no interactions between the different particles. We will assume a magnetic field of the form B~ = Bz^: 2 The Statistical Basis of Thermodynamics The energy of a single particle is "i = −2µBSz;iB; where µB is the Bohr magneton, and Sz;i = ±1=2 is the spin of the particle in the z-direction. Our notation can be made more compact by defining the constant b = gµbB=2, and then the total energy of the system is N N X X E = "i = −b σz;i; i=1 i=1 where σz;i = ±1. For our system, there is a spectrum of energies E = {−Nb; −(N − 2)b; : : : ; (N − 2)b; Nbg ; where, −Nb corresponds to all spins being down and Nb corresponds to all spins being up. There are a total of 2N states. However, there are only N + 1 different energy levels. The question then becomes, how many states are there per energy level? For example, there is only a single state with energy −Nb, and that is the state with all spins being down ### · · · . Likewise, there is a single state with energy Nb, and that is the state with all spins being up """ · · · . Since 2N is generally much larger than N + 1, clearly, the states are not distributed uniformly among the energy levels. For example, the energy level −(N − 2)b is the state with all spins but one being down. There are N ways to arrange N spins if only one spin is up. For a given energy E, how many microstates are there with this energy? We know that E = −b (N" − N#) ; where, for example, N" is the number of spin-up particles. How can we enumerate the number of microstates with this energy? That is, how many states are there with N" up spins and N# down spins? In general, if you have N = N1 + N2 + ··· total objects, consisting of N1 objects of one kind, N2 objects of another kind, and so on, then the total number of permutations is N! : N1!N2! ··· In our case, N! N! Ω(N") = = : N"!N#! N"!(N − N")! Then N! Ω(E) = N E N E : 2 + 2b ! 2 − 2b ! Often, we write N N N = + s; N = − s; " 2 # 2 then the total energy is E = −b (N" − N#) = −b(2s); where 2s is called the spin excess. 1.1. Counting States 3 The entropy of the paramagnet is S kT S(E) = k ln Ω = k [ln N! − ln N"! − ln N#!] : In the large N limit, we can apply the lowest order Stirling approximation x ln N! = N ln N − N: If we define Figure 1.1: A plot of the \net N − N N magnetization" x versus inverse x ≡ " # ;N ≡ (1 ± x); N "=# 2 temperature for a paramagnet. then E = −bNx; Tip and the entropy can be written as When counting mi- 1 + x 1 + x 1 − x 1 − x crostates, make sure S = −kN ln + ln : 2 2 2 2 you are counting the states and not just the energy If we plot S=kN as a function of x, we see that it is positive in [−1; 1], symmetric levels. In general, because about x = 0, and it peaks at x = 0 with a value of ln 2. That is of degeneracy, the number of states is greater than the S(x = 0) number of energy levels. = ln 2: kN This implies that S(x = 0) = k ln 2N , but we know
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