Statistical Physics

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• Contents
• List of Problems

The following sections are included:

• Introduction
• Combinatory analysis
  • Number of permutations
  • Number of arrangements
  • Number of combinations
• Probability
  • Definition
  • Fundamental properties
  • Mean values
• Statistical distributions
  • Binomial distribution
  • Gaussian distribution
  • Poisson law
• Microstates — Macrostates
  • Microstates — Enumeration
  • Macroscopic states
  • Statistical averages — Ergodic hypothesis
• Statistical entropy
• Conclusion
• Problems

The following sections are included:

• Introduction
• Fundamental postulate
• Properties of an isolated system
  • Spontaneous evolution of an isolated system toward equilibrium
  • Exchanges of heat and volume
  • Exchange of particles
  • Statistical distribution of an internal variable
• Phase space — Density of states
  • Density of states
  • Density of states of free quantum particles
  • Density of states of free classical particles
• Applications of the micro-canonical method
  • Example 1: two-level systems
  • Example 2: Classical ideal gas
• Conclusion
• Problems

Most of experiments to study properties of materials are conducted at a given temperature. The system under study is maintained at a constant temperature by a thermostat. The thermostat is considered as a heat reservoir very large with respect to the system so that its temperature T does not vary by the contact with the system. The thermostat imposes its temperature on the system. This situation is called “canonical situation”. One shows in this chapter that in such a situation, one can use the fundamental postulate to calculate the probability of a microstate of the system in contact with a thermostat. Microstates obeying such a “canonical” probability law form a canonical ensemble. This ensemble is used to calculate properties of the system at a constant temperature. There is of course an exchange of heat between the thermostat and the system but due to the huge energy reservoir of the thermostat this heat exchange does not change the temperature of the thermostat as said earlier. The description using the canonical ensemble is called “canonical description”.

The following sections are included:

• Introduction
• Grand-canonical probability
• Grand partition function Z — Grand potential J
• General properties of grand-canonical systems
• Spontaneous evolution of a grand-canonical system
• Systems of identical, independent particles
  • Factorization of Z
  • Bose-Einstein distribution
  • Fermi-Dirac distribution
  • Maxwell-Boltzmann distribution
• Applications of the grand-canonical method
  • Classical ideal gas
  • Two-level systems
• Conclusion
• Problems

The following sections are included:

• Introduction
• Fermi-Dirac distribution
• General properties of a free Fermi gas
  • General formulas
  • Formulas for large systems
• Properties of a free Fermi gas at T = 0
  • Fermi energy
  • Total average kinetic energy
• Properties of a free Fermi gas at low temperatures
  • Sommerfeld's expansion
  • Chemical potential, average energy and calorific capacity
• Free Fermi gas at the high-temperature limit
• Applications
  • Paramagnetism of conducting electrons in metals
  • Thermo-ionic emission
• Conclusion
• Problems

The following sections are included:

• Introduction
• Bose-Einstein distribution
• General properties of a free boson gas
  • General formulas
  • Formulas for large systems
• High-temperature limit
• Bose-Einstein condensation
• Properties at temperatures higher than T_B
• Applications
  • Photons: black-body radiation
  • Helium-4
• Conclusion
• Problems

The following sections are included:

• Introduction
• First quantization: symmetric and antisymmetric wave functions
• Representation of microstates by occupation numbers
• Second quantization: the case of bosons
  • Hamiltonian in second quantization
  • Properties of boson operators
• Second quantization: the case of fermions
• Field operators
• Hartree-Fock approximation
• Conclusion
• Problems

In this chapter, one briefly introduces some elements on the crystalline symmetry which are necessary in the calculation of physical properties of crystals. The periodicity of positions of atoms in a solid allows one to simplify the calculation not only in the real space but also in the reciprocal space as seen for example in chapters 9 and 12.

Elementary excitations of interacting particles such as atoms or spins on a lattice obey the Bose-Einstein statistics developed in chapter 6. In this chapter, one shows in some details the case of phonons which are the quantization of atom vibrations in crystals. Some other popular subjects such as magnons (collective motions of interacting spins), plasmons (oscillations of charges in plasmas) and excitons are very important in condensed matter where statistical physics demonstrates its efficiency. The theory of magnons, or quantized spin waves, which is very important in magnetism, deserves a large space in chapter 12.

The following sections are included:

• Introduction
• Gas of interacting electrons
  • Kinetic and exchange energies
  • Effective mass
• Gas of interacting electrons by second quantization
  • Kinetic energy
  • Energy at first-order perturbation
  • Energy at second-order perturbation
• Fermi Liquids
• Kondo effect
• Conclusion
• Problems

So far, on the one hand, we have examined free electrons in metals in chapter 5 and effects of their interaction in chapter 10. On the other hand, we have examined how ions vibrate due to their own ion-ion interaction in chapter 9. However, we did not taken into account the interaction between electrons and the potential created by the ions in a crystal. This chapter fills the gap…

This chapter is devoted to systems of interacting spins. On the one hand, spin systems describe magnetic materials which have been extensively studied both theoretically and experimentally for decades due to numerous applications. On the other hand, spin models can be also used to describe systems where parameters and symmetry can be mapped into a spin language. Such systems are not limited to physics, they are found in other domains such as biology, chemistry, and even social sciences. Spin systems constitute one of the central activities in statistical physics. As for other systems of interacting particles, systems of interacting spins can be exactly solved only in a very limited number of particular cases such as systems with short-range interaction in one and two dimensions. Otherwise, one has to use various approximations to obtain their properties…

The following sections are included:

• Introduction
• Generalities
  • Order parameter
  • Order of the phase transition
  • Correlation function — Correlation length
  • Critical exponents
  • Universality class
• Ferromagnetism in mean-field theory
  • Mean-field equation
  • Critical temperature
  • Specific heat
  • Susceptibility
  • Validity of mean-field theory
  • Improved mean-field theory: Bethe's approximation
• Landau-Ginzburg theory
  • Mean-field critical exponents
  • Correlation function
  • Corrections to mean-field theory
• Renormalization group
  • Transformation of renormalization group — Fixed point
  • Renormalization group applied to an Ising chain
  • Migdal-Kadanoff decimation method and Migdal-Kadanoff bond-moving approximation
• Transfer matrix method applied to an Ising chain
• Phase transition in some particular systems
  • Exactly solved spin systems
  • Kosterlitz-Thouless transition
  • Frustrated spin systems
• Conclusion
• Problems

The following sections are included:

• Introduction
• Properties of conventional superconductors
• Ginzburg-Landau theory of superconductivity
• Superconductors of type II
• Bardeen-Cooper-Schrieffer theory
  • Electron-phonon interaction
  • Cooper electron pairs — BCS Hamiltonian
  • Ground-state wave function
• High-temperature superconductivity
• Conclusion
• Problems

The following sections are included:

• Introduction
• Boltzmann's equation
  • Classical formulation
  • Quantum formulation
• Linearized Boltzmann's equation
  • Explicit linearized terms
  • Relaxation-time approximation
• Applications in general transport problems
  • Heat current
  • Thermo-electric current
  • Electric conductivity
  • Thermal conductivity
  • Seebeck effect
  • Peltier effect
• Resistivity
• Spin-independent transport in metals — Ohm's law
• Transport in strong electric fields — Hot electrons
• Transport in semiconductors
  • Motion of electrons in applied fields — Hall effect
  • Calculation of the diffusion coefficient by the Boltzmann's equation
  • Transport in semiconductors: Gunn's effect
  • Conductivity in extrinsic semiconductors — Doping effects
  • Doped semiconductors: generation, recombination, equation of continuity
  • p − n junctions—Diodes
• Spin transport in magnetic materials
• Conclusion
• Problems

The following sections are included:

• Solutions of Problems of Part 1
  • Solutions of problems of chapter 1
  • Solutions of problems of chapter 2
  • Solutions of problems of chapter 3
  • Solutions of problems of chapter 4
  • Solutions of problems of chapter 5
  • Solutions of problems of chapter 6
  • Solutions of problems of chapter 7
• Solutions of Problems of Part 2
  • Solutions of problems of chapter 8
  • Solutions of problems of chapter 9
  • Solutions of problems of chapter 10
  • Solutions of problems of chapter 11
  • Solutions of problems of chapter 12
  • Solutions of problems of chapter 13
  • Solutions of problems of chapter 14
  • Solutions of problems of chapter 15

The following sections are included:

• Appendix A: Mathematical Complements and Table of Constants
• Appendix B: Sommerfeld's Expansion at Low Temperatures
• Appendix C: Origin of the Heisenberg Model
• Appendix D: Hubbard Model: Superexchange
• Appendix E: Kosterlitz-Thouless Phase Transition
• Appendix F: Low- and High-Temperature Expansions of the Ising Model

The following section is included:

• Bibliography
• Index