Physics 504

Statistical Physics

Academic Year 2012/2013

Spring Semester 2013

Instructor: Professor Eduardo Fradkin

Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 Engineering Sciences Laboratory MC-704,
1110 West Green Street, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-244-7704
E-mail efradkin@illinois.edu
http://webusers.physics.illinois.edu/~efradkin/

Time: 10:30 am-11:50 am MW
Place: Rm. 158 Loomis Laboratory
Call Number: 36782
Credit: 1 unit.
Office Hours: Tuesdays 4:00pm-5:00pm, Rm 2119 ESB
TA: Mr. Lei Xing,
Address: 4133 ESB
Office Hours: Fridays 2:00-3:00 pm, 3rd. floor Commons area in ICMT (ESB building) (the office hours will be held by Mr. Michael LeBlanc)
Phone: 721-8476
e-mail: leixing2@illinois.edu

Announcements

Homeworks

Homework Set No. 1

posted on Sunday January 27, 2013; Due date Monday February 4, 2013, 9:00 am

Solutions to Homework Set No. 1

Homework Set No. 2

posted on Friday February 1, 2013; Due date Monday February 18, 2013, 9:00 am

Solutions to Homework Set No. 2

Homework set No. 3

posted on Monday February 18, 2013; Due date Monday march 4, 2013, 9:00 am

Solutions to Homework Set No. 3

Homework set No. 4

posted on Tuesday March 5, 2013; Due Saturday March 16, 2013

Solutions to Homework Set No. 4

Homework set No. 5

posted on Monday March 25, 2013 ; Due date Sunday April 7, 2013

Solutions to Homework Set No. 5

Homework set No. 6

posted on Sunday April 7, 2013; Due date Sunday April 28, 2013

Solutions to Homework Set No. 6

Term Paper

Term Papers: The Final Exam for this class consists of written Term Paper and an Oral Presentation on the Term Paper on the day of the Final. Students must do both the term paper and the oral presentation to be able to pass this course. Each student must choose a term paper subject before Friday April 5, 2013 . Please click here to see a list of suggested Term Papers. The term papers are due on the day of the Final Exam. Each term paper should be written in TeX/Latex format and they should be at least 10 pages long in paper format (draft RevTeX). On the day of the Final Exam each student registered in this class will give a 15 minute long oral presentation for the benefit of the rest of the class. I will arrange for the term papers to be printed into a single pdf file so that you can learn from each other's work.

Final Exam

The Final Exam will take place on May 8, 2013 from 9:00 am to 12:00 pm and from 1:00 pm to 3:00 pm. The exam will be in Rm 190 ESB which is not our usual classroom. On that date the students will give their Oral Presentations and we will have a "mini-workshop''. Students must send me by email the pdf file of their term paper no later than 8:00 pm on the daye before the final. I will bring my laptop to the final with all the files loaded. Each student will have 15 minutes for his/her presentation. We will have a one hour lunch break.

Course Plan

Fundamental Principles of Statistical Mechanics.

Classical Statistical Mechanics. Phase space and ergodicity. Distribution functions. Postulate of equal a-priori probabilities. Statistical independence. Liouville's theorem. Microcanonical Ensemble.
Quantum Statistical Mechanics. The role of the density matrix. Entropy and the law of increase of the entropy.

Thermodynamics and Statistical Mechanics.

Temperature. Adiabatic Processes. Pressure. Work and Heat.
Functions of State. Thermodynamic potentials. The Nernst Theorem. Systems with a variable number of particles. Inequalities and thermodynamic stability.

The Gibbs Distribution

The density matrix. The partition function and the free energy. The Maxwell distribution. The gibbs distribution for quantum mechanical systems. Derivation of Thermodynamics from Statistical Mechanics.

Classical Ideal Gases and Kinetic Theory

Ideal gases. The Boltzmann distribution. Classical systems.
Real gases and the role of collisions. Kinetic theory of dilute gases. The H Theorem.Conservation Laws. Hydrodynamics. Transport and the Boltzmann equation.

Random Walks, Langevin and Fokker Planck Equations

Simple models of stocahstic processes. Random walks and the diffusion equation. Relation with equilibrium statistical distributions.
Density matrices for quantum systems at finite temperatures. Relation with the Fokker-Planck equation. Path integral picture of the density matrix. Simple approximations to the path integral. Applications to simple quantum systems and to Fokker-Planck processes.

Classical Gases in Equilibruim

The free enrgy and the equation of state of an ideal classical gas. The law of Equipartition. Monoatomic and polyatomic gases. Angular momentum and other internal degrees of freedom.
Non-ideal gases. The van der Waals equation of state. Mayer expansions and virial expansions.

Quantum Statistical Mechanics

Density Matrix. Oscillators at finite temperature. Path integrals. Variational approximations.
Systems of bosons and fermions. Fermi-Dirac and Bose-Einstein distributions. Bose Condensation. Blackbody radiation. degenerate electron gases. Electrons in magnetic fields. Relativistic systems.

Statistical Mechanics and Phase Transitions: Spin Systems

Elementary picture of a magnet. Heisenberg and Ising models. Symmetry and the concept of Order Parameter.
Independent spins. Interacting spin systems.High temperature and low temperature expansions. Duality transformations in spin systems.
The one-dimensional Ising model. Peierls' argument. Continuous symmetries and the Mermin Wagner Theorem. Spontaneous symmetry breaking.
Mean field theory and phase transitions. Curie-Weiss theory and variational approximations.
Elementary theory of Superfluidity and Superconductivity.
Fluctuations and Correlation functions.
First order and continuous phase transitions. Landau-Ginzburg phenomenological theory of phase transitions.
The one-dimensional Ising model. The Onsager solution to the two-dimensional Ising model.
Critical fluctuations. Scale Invariance and Universality. The renormalization group.

Bibliography

M. Kardar, "Statistical Physics of Particles", and "Statistical Physics of Fields", first edition, Cambridge University Press.

L. D. Landau and E. M. Lifshitz, "Statistical Physics, Third Edition Part 1", Course of Theoretical Physics, Volume 5. Pergamon Press.

Kerson Huang, "Statistical Mechanics", Second Edition, Wiley.

Richard P. Feynman, "Statistical Mechanics, A set of Lectures", Frontiers in Physics , Benjamin (Perseus Books).