Basic Statistical Mechanics (PH 3201),  Spring 2015
Indian Institute of Science Education and Research Kolkata

Instructor: Bhavtosh Bansal (bhavtosh)

Teaching Assistant : Swarnadeep Acharyya (sa13rs042)

Syllabus for Final Exam: Lecture 19-40

Summary of Lectures:
1. Lecture 1 (January 1, Thursday) -- Axiomatic Thermodynamics: Length and time scales of measurement, essence of the spirit of thermodynamics, example of coupled harmonic infinite chain in long wavelength limit [Callan], Postulates of thermodynamics, the equilibrium state, Extensive variables, Fundamental relations for entropy S (U, V, N) or U (S, V, N). [Pal or Callan]. [Pal: 1.1, 1.2, 1.3.1, 1.3.2]
2. Lecture 2 (January 5, Monday) -- Fundamental relations and the definition of intensive variables (T, P, \mu), notion of temperature, homogeneous functions and Euler relations (see this link for a better proof), Gibbs-Duhem relation, derivation of the entropy formula for an ideal gas using two equations of state and why it is incomplete. [Pal: 1.3.4, 1.3.5, 1.4, 1.5.1, 1.5.3]
3. Lecture 3 (January 6, Tuesday) --Legendre transforms, example of parabola and passage from Lagrangian to Hamiltonian formalism in mechanics, thermodynamic potentials and Legendre transforms of the fundamental relation, Helmholtz free energy, enthalpy and Gibb's free energy. extensivity property of thermodynamic potentials and consequence for Gibb's free energy, thermodynamic work and heat, why dQ and dW are not exact differentials, Three types of fundamental relations, Extrema principles (qualitative).  [Pal: 1.6.1, 1.6.2, 1.7.1, 1.7.2, 1.12.1, 1.12.2] [Callan's book has a beautifully written leisurely discussion on all these topics. You may want to read it sometime later].
4. Lecture 4 (January 8, Thursday) -- Statistical Mechanics: Introduction and the task of statistical mechanics. Example of distribution of N particles equally likely to be in one of two boxes. Binomial distribution, notion of microstates and macrostates, mean and variance of the binomial distribution using the generating function. Equivalence of this and the coin-tossing problem. [The idea that even though each of the particles can be in either of the two boxes with equal likelihood, for a large number of particles the combinatorial factor ensures that it is overwhelmingly likely for the two partitions to have equal number pf particles.]: [See Reif]. [You may also see also V. Balakrishnan's online lecture.]  Idea of phase space and ergodicity.
5. Lecture 5 (January 12, Monday) -- Microcanonical Ensemble: Discussion on microstates and macrostates, notion of phase space, trajectory in phase space, and ensemble. Hypothesis that the time average for a physical system is equal to the ensemble average. Phase space density function \rho(\gamma, t). Liouville theorem. Equilibrium and stationary ensembles (\del \rho \over \del t=0 implies that its Poisson bracket should vanish and this implies constraint on the possible form for \rho). Choice 1: \rho=const (Microcanonical ensemble)=postulate of equal a-priori probability. Definition of the microcanonical probability density. Boltzmann entropy formula. S=k ln \Omega.
6. Lecture 6 (January 12, Monday) [Extra Class in place of Dr Mitra's Advanced QM class]---Discussion on the Boltzmann entropy formula and its plausibility, notion of temperature. Generalization of the entropy formula to probabilities: S=-k\sum p_i log (p_i). Proof that this formula implies p_i equal to constant in equilibrium under the condition of entropy maximization (using Lagrange multipliers). Introduction to density matrix.
7. Lecture 7 (January 13, Tuesday) --Density matrix (cont). Prove quantum Liouville theorem [H.W.]. Postulates of quantum statistics. Entropy in the microcanonical ensemble in terms of the density matrix. Density of states.
[Reading for Lecture 4--7: Full chapter 2 of Pal (except 3.3), Pathria: Chapter 5, pg 115 to 121]
8. Lecture 8 (January 14, Wednesday) [Extra Class in place of Advanced QM class]--Thermodynamics of two level system in microcanonical ensemble. (Note that the example is a little different from the one discussed by Pal section 3.2). Discussion of negative temperature. [I mostly followed Kardar section 4.3 pg 102 and some other online material].
9. Lecture 9 (January 15, Thursday) ---Entropy, internal energy, and Helmholtz free energyof the ideal gas in microcanonical ensemble. Volume integrals in higher dimensional space. Gibb's paradox and its resolution. [Kardar section 4.4 and 4.5], [Pal: section 3.3.1, 3.3.2, 3.3.3]
10. Lecture 10 (January 19, Monday)
---Entropy, internal energy and Helmholtz free energy of a system containing harmonic oscillators (microcanonical ensemble). Volume of a generalized ellipsoid in d dimensional space. [Pal section 3.4]
11. Lecture 11 (January 20, Tuesday) ---Canonical Ensemble: Subsystem in contact with a large reservior, probability density function, calculation of averages in the Canonical ensemble, partition function, relation of the partition function to Helmholtz free energy (Fundamental relation), internal energy and entropy. Density matrix in the canonical ensemble (quick introduction) and one particle partition function (Q). Relation between Q and Z. [Pal 4.1, 4.2, 4.3]
Class on 22 January Thursday was taken by Dr Mitra. 26th Jan, Monday: Holiday.
12. Tuesday 27 January: [Tutorial by Swarndeep Acharya]

13
. Lecture 12 (January 29, Thursday) ---Examples: Two level system. Classical and quantum oscillator in canonical ensemble. Partition function, free energy and internal energy. [Pal Section 4.4, 4.6]
14. Lecture 13 (February 2, Monday) ---Classical ideal gas in Canonical ensemble. Partition function and magnetization of magnetic dipoles in an external field. [Pal 4.5.1, 4.7]
15. Lecture 14 (February 3, Tuesday): Density matrix in the Canonical Ensemble. Introduction to Density Matrix for free particle in a box. [Also see example 5.3.A pg 122 in Pathria]. [Pathria pg 121]
16. Lecture  15 (February 06, Friday) [In lieu of thursday class]: Quantum particle in a box. Single particle density matrix. In momentum (energy) basis and in position basis. Physical interpretation of off-diagonal and diagonal components of the density matrix in position basis. Equivalence of trace in position and momentum basis and comparison of the partition function with the classical partition function. [Pathria: pg 123-125]
-----------------------------------------Syllabus for Midsem Ends Here--------------------------------
17. Lecture 16 (February 09, Monday) --Equipartition Theorem [Pal 4.8.1, 4.8.2, Pathria pg 61-63].
18. Lecture  17 (February 10, Tuesday) --Virial Theorem. Virial theorem for Statistical averages (proved for the canonical ensemble). Virial V=-3NkT. [Aside: Proof of Virial theorem for classical systems for time averages.] Derivation of equation of state of ideal gas [see Pathria]. Estimation of the average temperature of the sun [Q. Assume that the sun is in thermodynamic equilibrium and its radius is determined by the balance of the potential and kinetic energies. Estimate its average temperature assuming that the kinetic energy obeys equipartition (this assumption seems wrong because equipartition does not hold for 1/r potential energy..let us use it anyway). See this link for a solution] [Pal 4.8.3, Pathria: pg 63-64]
19. Lecture 18 (February 12, Thursday) --Quantum mechanical treatment of heat capacity of diatomic gas: Total Hamiltonian as sum of translation, rotational, vibrational, electronic, nuclear,..contributions. Single particle partition functions and the total partition function. Spirit of the Born-Oppenheimer approximation (qualitative argument that different contributions can be treated independently due to different energy/time scales). Detailed discussion of the rigid rotator. Calculation of the specific heat in the low and high temperature limits (show that Cv ->0, as T-->0 and Cv--> equipartition result at high temperature.). [Section 9.1 Pal, for a discussion of rigid rotator, see Cohen Tannaudji (Quantum Mechanics -I) or I. Levine (Quantum Chemistry)].
21. Tutorial session (Monday 16, February 5pm)
20. Midesem Exam (February 20, Friday)
[No classes on Monday 16/2, Tuesday 17/2, Thursday 19/2, Monday 23/2 due to midsem exams]

Final Exam Syllabus Starts Here

21. Lecture 19 (February 24, Tuesday)
The grand canonical ensemble.-- Derivation of the grand partition function and probability function for the system in a grand canonical ensemble. The grand potential and derivation of various thermodynamic quantities from the grand partition function. Connection between the Canonical and the grand canonical ensemble (to be continued).
22. Lecture 20 (February 26, Thursday)  Grand Canonical Ensemble continued: General discussion on the equivalence of various ensembles. energy fluctuations in the canonical ensemble, number and energy fluctuations in grand canonical ensemble [Read Pal's book for a more detailed discussion]. Correspondence with thermodynamics: how the calculations in different ensembles just amounts calculating the entropy, Helmholtz free energy and the grand potential respectively by taking the largest term in the summation over possible energies (Canonical ensemble) and the possible number of particle (grand canonical ensemble) [see Kardar's book].
Alternative (and simpler) derivation the probability density functions in the three using the Gibb's entropy expression and Lagrange multipliers invoking the information input we are in a position to give [This approach, based on 'information theory' is not standard and was pioneered by E. T. Jaynes. This was for 'entertainment' only and is not in the syllabus.]
23. (March 02, Monday): No class due to Inquivesta, make up class on Saturday March 14.
24. Lecture 21 (March 03, Tuesday): Classical Ideal Gas in the Grand Canonical Ensemble, fundamental relation in terms of intensive variables, equation of state, entropy and internal energy for classical ideal gas. One orbital partition function in the Grand Canonical ensemble.  [Pal and Pathria]
25. Lecture 22 (March 05, Thursday): Expression for Fermi-Dirac and Bose-Einstein distributions in the Grand Canonical ensemble. Calculation of the Bose and Fermi distributions in the  microcanonical ensemble: counting of states for Bose and Fermi particles (to be continued) [Pal and Pathria].
Class test : Thursday March 12, 2015
(Syllabus: Lecture 11 to lecture 21)
26. Lecture 23 (March 09, Monday): Bose and Fermi gases in microcanonical ensemble. Calculation of entropy and the distribution functions for Fermi, Bose and Maxwell-Boltzmann statistics [Pal and Pathria].
27. Lecture 24 (March 10, Tuesday): Fermi-Dirac and Bose Einstein distribution functions [Pal/Pathria]. Meaning of the classical limit (ratio of thermal wavelength and density). The connection between the grand canonical and the canonical partition function [Hermann pg 149-]. Chemical potential for an ideal gas. Qualitative remarks on the physical meaning of the chemical potential. Some online material
28. Lecture 25 (March 12, Thursday): Qualitative introduction to the Bose gas and Bose distribution. Restriction on the values the chemical potential can take, concept of the critical temperature [Bowley pg 229]
29. Lecture 26 (March 16, Monday): Formal thermodynamics calculations via the grand partition function for a Bose gas. Fundamental relation for pressure. Expression for average number density, internal energy, specific heat, entropy, Helmoltz free energy. Bose integrals and their properties. [Pal/Pathria/Greiner]
30. Lecture 27 (March 17, Tuesday): Expression for average N, Bose condensation, condensate fraction as a function of temperature. Qualitative discussion on Bose condensation and superfluid helium (which is actually not a Bose condensate)   [Pathria]
31. Lecture 28 (March 19, Thursday): Thermodynamics of the Photon gas. Dispersion relation for photons. Grand partition function and fundamental relation for photons. Expression for energy density and Planck distribution [Pal/Pathria].
Elementary discussion on what is a photon and connection with Harmonic oscillators. [Number of photons (with wave vector k) = energy content of a Fourier component (kth cavity mode) of electric field in units of \hbar \omega].   [I more or less followed Gerry and Knight. For a more comprehensive introduction, see Sakurai's Advanced Quantum Mechanics]
[Not in syllbus, for "fun"].

[I seem to have messed up the order of classes somewhat, in particular there was a class on Saturday 14 March]
32. Lecture 29 (March 23, Monday): Phonons Elastic waves in solids and decomposition in terms of normal modes (what you know from 2nd year physics class). Similarity between phonons and photons. [not in syllabus]
33. Lecture 30 (March 24, Tuesday): Einstein and Debye theories of specific heat.[Pathria and Greiner]
34. Lecture 31 (March 26, Thursday): Fermi Gas: Introduction via the distribution function. Zero temperature properties of the Fermi distribution. Concept of Fermi energy, Fermi wave vector, Fermi momentum, Fermi velocity. Estimation of these parameters for a 'typical' metal like copper. [Hermann and Bowley have a very readable elementary introduction].
34. Lecture 32 (March 30, Monday): Thermodynamics of Fermi Gas: Fundamental relation from the grand partition function, expression for average N, internal energy, Helmholtz free energy. Properties of Fermi integrals: Recurrence relation, high temperature expansion. Low temperature (Sommerfeld expansion is not in syllabus). [Greiner/Pathria
35. Lecture 33 (March 31, Tuesday): Thermodynamics of Fermi Gases (cont.):  Power series expansion for the high temperature limit of the Fermi integrals Fn(z). Low temperature limit and the Sommerfeld expansion (just the power series was mentioned but not derived). Expression for the chemical potential at low but finite temperature.
36. Lecture 34 (April 06, Monday): Review of thermodynamics of Fermi Gases. Hamiltonian for free charged particle in magnetic fields. Bohr van Leuween Theorem (for absence of magnetism in classical equilibrium stat mech) For Fermi gases please consult Pathria and Greiner's book.
37. Lecture 35 (April 07, Tuesday): Quantum mechanics of free charged particle in magnetic field (Landau levels), Energy eigenvalues and degeneracy. [Huang and Griener's Quantum Mechanics book]
37. Lecture 36 (April 09, Thursday): Calculation of the free electron susceptibility in magnetic field (Pauli paramegnetism and Landau diamagnetism). High temperature result only. (I mostly followed the discussion in Ashoke Sen's lecture notes).
38. Lecture 37 (April 13, Monday):
Thermodynamics of white dwarf stars, (approx) mass-radius relationship for non-relativistic and ultrarelativistic Fermi gases.  (See class notes which were mostly based on the discussion in Padmanabhan's Astrophysics book, v1. You can also consult Bowley-Sanchez. You don't have to bother with the calculation of the grand partition function for a relativistic gas, like it is done in Pal.]
Tuesday, April 14: Holiday
39. Class Test 2: 16th April, Thursday. Syllabus: Lecture 22-Lecture 34.
40. Lecture 38 (April 20, Monday):
The Ising Model: Introduction to the Ising model. Exact solution for 1D chain with periodic boundary conditions using the transfer matrix [Huang Chapter 14, pg 341-342 and section 14.6, pg 361-363].
41. Lecture 39 (April 21, Tuesday): Solution for 1D chain (cont.). Reason for absence of magnetization in 1D at any finite temperature (entropy gain vs energy cost of a domain wall) [Huang, pg 349].
42. Lecture 40 (April 23, Tuesday): Introduction to Mean Field Theory. Mean field solution of the Ising model. Calculation of transition temperature in mean field. Introductory remarks on spontaneous symmetry breaking [see e.g. page 348 and 300 in Huang and David Tong's notes).
[I followed the method in David Tong's notes., the discussion in Huang (Bragg-Williams Approx.) is more complicated and you may skip that.].

43. Lecture 41 (April 27, Monday):
Summary of the Course. Discussion, comments, doubts.
44. Final class to be taken by students (April 28, Tuesday).
Two short seminars on (i) Thermodynamics of black holes. [Lesson: A deep link between statistical thermodynamics and gravity is conjectured or rather seems inescapable.]  (ii) Elasticity of rubber  [Lesson: This is a nice example where essentially entropy and not the energy of the physical system seems to determine its mechanical properties.]

Syllabus for Final Exam: Lecture 19-40

References:

What we'll cover is pretty standard and is discussed in most introductory books on Statistical Mechanics and there are at least 25 such good books. You are encouraged to go to a real or virtual library and discover the one(s) to your taste. I have decided to more of less follow
• Palash B. Pal, Introductory course of statistical mechanics (Narosa 2008). [This is a crisp, clear, and intelligently written book (from Kolkata!). The problem with this book is that glosses over many of the subtleties and makes the subject look simpler than it is (which may not be a bad thing, but beware!).] I will try to sequentially follow Pal's book (more or less).
• R. K. Pathria and P. D. Beale, Statistical Mechanics (Academic Press 3rd Ed, 2011). This is going to be the supplementary text and it is recommended that you also look through Pathria. Most topics discussed in Pal's book are also discussed in Pathria in greater detail (and sometimes with greater clarity).
The following will be used as supplementary references.
• H. B. Callen, Thermodynamics and an introduction to thermostatistics (Wiley, 2nd Ed). The best book on thermodynamics [imho]. You can also check out the chapters on Stat Mech to get a quick qualitative introduction to many of the topics we have covered.
• W. Greiner, L. Neise, H. Stöcker, Thermodynamics and Statistical Mechanics (Springer 2000). [Mostly Pathria worked-out in a notes format, with a few typos. But generally very useful. Covers a lot of interesting  material not found elsewhere.
• L. Landau and E. M. Lifshitz, Statistical Physics 1,
• K. Huang, Statistical Mechanics (Wiley)
• M. Kardar, Statistical physics of particles (Cambridge 2007). A very nice book though arguments tend to be quite advanced sometimes. Has a lot of solved problems.
• C. Hermann, Statistical Physics: Including applications to Condensed matter (Springer 2006).
• R. Bowley and M. Sanchez, Introductory Statistical Mechanics (Oxford 1999).
• Online notes by David Tong are also very nice.