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 goodbooks.
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. StÃ¶cker, 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.
BowleyandM.
Sanchez, Introductory Statistical
Mechanics (Oxford 1999).