Statistical Mechanics.

The goal of this course is to teach students the fundamentals of statistical physics, both classical and quantum. It will focus on the application of the methods to specific physical problems and play down the justification part. In this sense it is a statistical physics course. Briefly, the course will start with the foundations of statistical mechanics and derivation of the laws of thermodynamics and proceed to their applications to diverse simple physical systems: ideal and slightly non-ideal classical and quantum gases, charged gases. Then it will cover the basic physics of collective effects: phase transitions, fluctuations, critical phenomena.

Plan of the course (advanced topic lectures or lecture parts are indicated by italics).

0.       Jan 21. Course overview.  Basic assumptions of statistical mechanics: ergodicity and chaos. Liouville equation and microcanonical distribution.

1.       Jan 26. No lecture.

2.       Jan 28.  Density matrix. Master equation. Quantum canonical and microcanonical distributions. Density matrix evolution in open systems (Lindblad equation). [1]

3.       Feb. 2.  Main concepts of statistical mechanics: entropy, partition function and free energy. Main postulates of thermodynamics and their justification by statistical mechanics.

4.       Feb. 4. Thermodynamics potentials, chemical potentials and their usage. Maxwell relations, Joule-Thompson process.

5.       Feb. 9. Thermodynamics: gas expansion, ideal engine, maximal work.

6.       Feb. 11. Brief review of the foundations of statistical mechanics: the formalism of the ergodicity proofs, the problem of chaos onset in almost integrable systems, KAM theorem. [2,3]

7.       Feb. 16.  Ideal gases: partition function and equation of state for ideal gas.  Free energy and specific heat of a monoatomic gas. Free energy of an ideal gas with constant specific heat. 

8.       Feb. 18. Ideal gases: free energy of a generic ideal gas, rotational and vibrational degrees of freedom and their contributions to specific heat.

9.       Feb. 23.  Non-ideal gases. Virial expansion.

10.    Feb. 25. Non-ideal gases, Van-der-Waals law.

11.    Mar. 2. Cancelled due to weather. Note however a new homework assignment below.

12.    Mar. 4. Van-der-Waals gases and a general theory of first order phase transitions.

13.    Mar. 9. Plasma and electrolytes: a special case of non-ideal gases.

14.    Mar. 11 Mid Term Exam.

15.    Mar. 23. Mid Term Exam solution. Ideal Bose gas.

16.    Mar. 25. Bose-Einstein condensation of ideal bose gas. Applicability of ideal Bose gas formulas. Black body radiation.

17.    Mar. 30. Ideal Fermi gas.

18.    Apr. 1 Fermi gas: effect of weak interaction.

19.    Apr. 6 Fermi liquid (strongly interacting Fermi gas): Landau theory.

20.    Arr. 8. Fermi liquid: main properties. Zero sound [4].

21.    Apr. 13. Magnetic properties of Fermi gases and Fermi liquids. Failure of Fermi liquid theory: superconductivity of metals.

22.    Apr. 15. Fluctuations of thermodynamic quantities. Response function. Dissipation.

23.    Apr. 20. Fluctuation-Dissipation Theorem.

24.    Apr. 22. Phase transitions of the first and second order. Examples: magnetic transitions, superconductivity, superfluidity.

25.    Apr. 27. Landau theory of phase transition. Fluctuations in the vicinity of the phase transitions.

26.    Apr. 29. Critical behavior, basic ideas of renormalization group approach.

 

Textbooks:

More details and examples can be found in

Additional reading for advanced topics lectures:

[1] S. Haroche, J.-M. Raimond, “Exploring the quantum”, 4.2-4.3.

[2] V. Arnold, “Mathematical methods of classical mechanics”, Ch. 10 (50-52), A8.

[3] M. Tabor, “Chaos and integrability in nonlinear dynamics”, 3.3-3.4.

[4] L.D. Landau and E.M. Lifshitz "Statistical Physics II" (Volume IX, chapter 1)

 

 

Prerequisites:

            Classical mechanics, quantum mechanics.

MidTerm exam: March 11.

Final Exam: May 7th, 2:00 pm. ARC 205.

Subjects covered by Final Exam: Lectures 1-23, except those indicated by italics in the course syllabus above. 

 

Homework Problems:

·         Hw 1 (due Feb 9). 

·         Hw 2 (due March 11).

·         Practice problems (Thermodynamics).

·         Practice problems (Quantum gases).

 

Time and location: Monday and Wednesday at 3:20 in ARC 205.

Office hours: after the classes on Mondays and Wednesday or by email appointment.

Click here to send e-mail to Lev Ioffe

Important: put the words “New Sender” (without quotes) in the subject of your email if you are doing it for the first time.

 

 

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