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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, MaxwellBoltzmann equilibrium distributions,
thermodynamics of the free ideal gas
 Week 3. Boltzmann functional, entropy, HTheorem 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,
SackurTetrode 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, LeeYang Theorem,
Definition of Ising models, ME and CE for Ising models
 Week 9. Thermodynamics of Ising models, thermodynamic limit, onedimensional Ising model:
transfer matrix method, partition function and free energy
 Week 10. Twodimensional Ising model (Onsager's solution): algebraic tools for the study of the transfer matrix
 Week 11. Twodimensional Ising model (Onsager's solution): spinor analysis, diagonalization of the transfer matrix
 Week 12. Twodimensional 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
