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Mathematical Physics 2: Classical Statistical Mechanics (SS 2014)

Lectures  Matteo Petrera Tue 12:15 - 13:45 MA 645
Thu 12:15 - 13:45 MA 742
Tutorials Andrea Tomatis Wed 14:15 - 15:45 MA 651

This is a course of the Berlin Mathematical School held in English.


Contents

Thermodynamics, Elements of ergodic theory, Probability spaces, Kinetic theory of gases, Gibbsian formalism for continuous systems at equilibrium, Ising models


Lectures (examinable topics)

  • Week 1. Basic facts on: thermodynamics, measure theory, probability theory, ergodic theory
  • Week 2. Ideal gas, Boltzmann transport equation, Maxwell-Boltzmann equilibrium distributions, thermodynamics of the free ideal gas
  • Week 3. Boltzmann functional, entropy, H-Theorem and its consequences, introduction to Gibbsian formalism for continuous systems
  • Week 4. Gibbsian formalism for continuous systems, definition of Gibbs ensemble, physical definition of ME, CE and GE, ergodic problem and its Hamiltonian formulation
  • Week 5. Ergodic hypothesis, formalism of the ME, free ideal gas in the ME, Sackur-Tetrode formula, Gibbs paradox, formalism of the CE
  • Week 6. Equivalence of ME and CE, equipartition of the energy, free ideal gas in the CE, formalism of the GE, equivalence of CE and GE, free ideal gas in the GE
  • Week 7. Existence of thermodynamic limit, stable and tempered potentials, Van Hove potentials, virial expansion for real gases
  • Week 8. Phase transitions, analytical properties of partition functions, Lee-Yang Theorem, Definition of Ising models, ME and CE for Ising models
  • Week 9. Thermodynamics of Ising models, thermodynamic limit, one-dimensional Ising model: transfer matrix method, partition function and free energy
  • Week 10. Two-dimensional Ising model (Onsager's solution): algebraic tools for the study of the transfer matrix
  • Week 11. Two-dimensional Ising model (Onsager's solution): spinor analysis, diagonalization of the transfer matrix
  • Week 12. Two-dimensional Ising model (Onsager's solution): derivation of thermodynamics, phase transitions
  • Week 13. General overview

Exercise sheets


Homework policy

  • To get a certificate for the tutorial, you need to satisfactorily complete 60% of the homework sheets
  • Homework sheets are to be solved in groups of two students
  • Homework sheets are due weekly at the beginning of the tutorial on Wednesdays

Examinations

An oral exam will be offered at the end of the semester to all students who got the certificate for the tutorial


Literature

  • M. Petrera, Mathematical Physics 2. Classical Statistical Mechanics. Lecture Notes, to be published in Summer 2014
  • K. Huang, Statistical Mechanics, Wiley & Sons, 1987
  • C.J. Thompson, Mathematical Statistical Mechanics, MacMillan, 1972
  • D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin, 1969
  • B. McCoy, Advanced Statistical Mechanics, Oxford University Press, 2010

Office hours

Matteo Petrera Mon   10:00 - 12:00 MA 819
Andrea Tomatis By appointment

Matteo Petrera . 09.07.2014.