Advanced Statistical Mechanics

Welcome to the Course on Advanced Statistical Mechanics, oriented to the master students in Physics at Universidad de Pamplona. The course will be taught by Dr. Juan David V. Jaramillo, Postdoctoral Researcher at Universidad del Valle and Full Time Lecturer at Universidad de Pamplona.

The objective of this course is to supply students at the master level in the field of physics with the necessary tools to contribute scientifically within the arena of the most important open problems in statistical mechanics, for instance, multilevel approaches to transport phenomena in solids, finite temperature fluctuations in open quantum systems, field theories in condensed matter at zero temperature, finite temperature and out of equilibrium.

Conceptually, the course is divided in three parts: 1. a review of undergraduate statistical mechanics with a much greater level of detail and integrated with different areas of physics, such as quantum mechanics, classical electrodynamics and social sciences. This review mainly consists in kinetic theory and applications of classical and quantum statistics with rigorous derivations of the Fermi-Dirac and Bose-Einstein Distributions for the case of indistiguisable particles. 2. An exposition of the conventional topics covered at the graduate level in a course in statistical mechanics, such as semiclassical approaches to transport phenomena mainly from the perspective of Boltzmann equation, linear response functions applied to magnetism and optics and the theory of fluctuations along with the Langevin dynamics.3. Illustration of some advanced problems in statistical mechanics that typically are related to open research questions, such as Matsubara Green's functions, coherent state path integrals, Non-Equilibrium Green's functions theory and phase transitions.

I really hope you will enjoy this course and that you will meet of the proposed objectives having enriching and engaging discussions with all the peers.

With Kind Regards,

Juan David V. Jaramillo, Ph.D

Contents

Presentation of the Course

Unit 1: Kinetic Theory of Gases

1. Microstates and Macrostates.

2. Liouville Equation and Ensamble Density.

3. The Maxwell Boltzmann Disitribution.

4. BBGKY Hierarchy.

5. Boltzmann Equation and the H-Theorem.

Teoria Cinetica (PDF)

Espacio Fase (PDF)

Dinamica en el Espacio Fase (PDF)

Densidad del Ensamble en Equilibrio

y Aplicaciones (PDF)

Aplicaciones de La Densidad en Equilibrio (PDF)

Distribucion de Maxwell-Boltzmann (PDF)

Definicion del Marginal de la Funcion densidad (PDF)

Funcion Densidad Marginal (PDF)

Dinamica de la funcioin Densidad Marginal (PDF)

Jerarquia BBGKY (PDF)

Jerarquia BBGKY Para una Sola Particula (PDF)

Jerarquia BBGKY Para Dos Particulas (PDF)

Lagrangian for the Double Pendulum (PDF)

State Space Representations (PDF)

Classical Scattering Theory (PDF)
Rutherford Scattering (PDF)

Unit 2: Classical Statistical Mechanics.

1. Canonical Distributions.

2. Partition Function.

3. Canonical Partition Function.

4. Grand-Canonical Partition Function.

5. Equation of State for the Ideal Gas.

6. Van Der Walls Equation of State.

7. Mixtures of Ideal Gases.

Unit 3: Quantum Statistical Mechanics.

1. Requirement for a Quantum Approach to

Statistical Mechanics.

2. State Operator: Pure vs Mixed States.

3. Density Function vs State Operator.

4. Exchange Symmetry and Permutations.

5. Formal Derivation of the Fermi-Dirac and

Bose-Einstein Distributions.

6. Bose-Eistein Condensation.

7. Fermi Systems and Sommerfeld

Expansion.

8. Applications of Quantum Statistics to

Magnetism, Polarization and Conduction.

9. Second Quantization and the Field

Operator.

Unit 4: Semi-Classical Approach to Boltzmann

Equation.

1. Review of Single Particle BBGKY Hierarchy

2. Boltzmann Equation in the Time

Relaxation Approximation.

3. Application to Electrical and Thermal

Conductivity.

4. The Thermoelectric Tensor and Onsager

Theory

4. Resistivity in Metals and Semiconductors.

5. Electron-Vibration Scattering.

6. Electron-Electron Scattering.

7. Electron-Impurity Scattering.

Bibliography:

1. Statistical Mechanics, Kerson Huang. 2nd Edition

2. Statistical Mechanics of Particles. Mehran Kardar.

3. Statistical Mechanics of Fields. Mehran Kardar.

4. Statistical Physics II: Nonequilibrium Statistical Mechanics. Ryogo Kubo, Morikazu Toda, H. Hashitsume.

5. Thermal and Statistical Physics. Frederick Rief.

6. Modern Quantum Mechanics. J. J. Sakurai and Jim Napolitano

7. Chemical Kinetics and Dynamics. Steinfeld, Francisco, Hase