Basic Statistical Mechanics (PH
3201), Spring 2016
Indian Institute of Science Education and Research Kolkata
Instructor: Bhavtosh
Bansal (bhavtosh)
Graders:
Subhadip Roy (sr14ip015@iiserkol.ac.in)
BasebendraRoy (br14ip001@iiserkol.ac.in)
Syllabus
for Class Test 2: Lecture
24-Lecture 32
Class Test 2:
19-April-2016, Tuesday 8am
Syllabus for final exam: From Lecture 10-
End
Summary of Lectures:
Part 1. Axiomatic Thermodynamics
Lecture 1 (January 4, 2016,
Tuesday): Time and length scales of
measurement, basic idea that thermodynamic description encapsulates
the hidden degrees of freedom that carry energy, lessons form normal
mode description (time scale ->infinity and wavelength -->
infinity). Postulates of Equilibrium Thermodynamics. Fundamental
relation E(S, V, N). Notion of temperature, derivation of the second
law (heat flows from hotter to cooler body) (Callen)
Lecture 2 (January 6, 2016, Wednesday):
Homogeneous functions and Euler relations, exact and inexact
differentials, intensive and extensive variables, generalized forces
and displacements. Equivalence of energy minimum and entropy
maximization principle (stated pictorially, without proof)
Lecture 3 (January 7, 2016, Thursday):
Legendre transforms, various thermodynamic potentials (Helmoltz free
energy, Enthalpy, Gibb's free energy, Grand Potential), prescription
for expressing thermodynamic variables as the partial derivatives of
the thermodynamic potentials, Maximization of entropy of the
'universe' => minimization of the Helmoltz free energy of the
system alone.
Lecture 4 (January 12, 2016, Tuesday): Further discussion of ideas in Lecture 3.
Thermodynamic stability and its implications on
the response functions [Pal]
Lecture 5 (January 13, 2016, Wednesday):
Discussion on stability continued.
Qualitative discussion on the Third Law of Thermodynamics.
Part 2. Classical Statistical Mechanics of
Non-Interacting Systems (General formalism. Calculations in the
microcanonical, canonical, and grand canonical
ensembles)
Lecture 6 (January 14, 2016, Thursday):
Microstates and macrostates. definition of temperature and zeroth
law of thermodynamics, Boltzmann entropy formula, thermodynamic
phase space, ensemble and time averages, Liouville's theorem
(without proof), possible forms of the probability density function
for equilibrium states. Microcanonical ensemble. Phase space
counting and introduction of the Planck's constant via the
correspondence principle.
Lecture 7 (January 19,
2016, Tuesday): Approximating exponential
integrals via Taylor expansion around the maximum ('saddle point
method'). Sterling Approximation for log N!
Lecture 8 (January
20, 2016, Wednesday): Entropy
calculation of ideal gas in the microcanonical ensemble. Volume and
area of a d-dimensional hypersphere
Lecture 9 (January 21,
2016, Thursday): Gibb's
paradox and its resolution (inclusion of 1/N! factor in the phase
space integral), the correct entropy formula (Sackur-Tetrode
equation).
Gibb's entropy formula S=\Sum p_i log p_i
January 26, Tuesday.
Holiday (Republic Day)
Lecture 10 (January 27,
2016, Wednesday):
Canonical Ensemble: Derivation of the probability density
function and the partition function. Average of physical quantities
in the canonical ensemble. Relationship between the partition
function and the Helmoltz free energy.
Lecture 11 (January 28,
2016, Thursday):
Energy fluctuations in the canonical enesemble. Detailed discussion
about the equivalence of the Canonical and the Microcanonical
ensembles.
Single particle partition functions for non-interacting systems.
Lecture 12 (February 2,
2016, Tuesday): Equipartition
theorem. (Viral theorem was not discussed. You will most probably
see it in your astrophysics course next semester).
Lecture 13 (February 3,
2016, Wednesday):
Partition function for the ideal gas and N-one dimensional harmonic
oscillators. Calculation of various thermodynamic properties of the
classical ideal gas and harmonic oscillators.
Other examples of non-interacting systems with discrete energy
levels: (i) Two Level system. Calculation in the microcanonical and
the canonical ensembles and the equivalence of the results (the
problem was not completely solved, HW)
Lecture 14 (February 4,
2016, Thursday):
(ii) Quantum harmonic oscillators. Calculation both in the
microcanonical and the canonical ensembles. (iii) Rigid Rotor
(classical and quantum). Calculation in the Canonical Ensemble
Breakdown of the equipartition theorem for quantum systems and how
it is recovered in the classical limit.
Lecture 15 (February 9,
2016, Tuesday):
Monoatomic and diatomic gases with internal (rotational,
vibrational, electronic, and nuclear) degrees of freedom
(qualitative, will not be asked in exam) and step-like behaviour of
specific heat with temperature.
--------------------------------------Syllabus
for MidSem Ends Here (but includes examples discussed in
Lecture 17)------------------------------
Statistics of Paramagetism: Gyromagnetic ratio, Bohr
magneton, g-factor
Lecture 16 (February 10,
2016, Wednesday):
Difference between H, B and M fields. Thermodynamics in magnetic
field. Poynting theorem. Different ways of the writing the Helmoltz
free energy. Magnetization and magnetization density. See [Bowley-Sanchez Appendix
A] and optionally Griffiths'
Electrodynamics for a detailed discussion of the Poynting theorem
and magnetic fields in matter. Poynting theorem-related topics are
not in the syllabus.
Midsem Exam Thursday
18/02/16, 10am-11am
Lecture 17 (February 11,
2016, Thursday): Some examples. (i) Thermodynamics of N
non-interacting spin 1/2 particles in magnetic field (and
equivalence of the problem to N 2-level systems). Negative
temperatures. [Pathria] (ii) Model for Rubber Elasticity
[Bowley-Sanchez pg78]
Lecture 18 (February
23, 2016, Tuesday): Statistics of Paramagnetism.
I could not complete the discussion on quantum case (Brillouin
function)
Lecture 19 (February
24, 2016, Wednesday): Grand Canonical Ensemble
(T, V, \mu): Derivation of the probability density function for
a microstate. Calculation of averages. The
grand partition function and its connection with thermodynamics.
Classical Ideal Gas.
Part 3. Quantum statistical mechanics of
non-interacting particles
Lecture 20 (February 25, 2016,
Thursday): Introduction to
Density Matrices: Writing the quantum
mechanical expectation value as trace. Statistical averaging
over and above quantum expectation. Properties of density
matrices. Pure and mixed states.
Lecture 21 (March 01,
2016, Tuesday): Density
Matrices in quantum statistical mechanics. Quantum Liouville theorem
and acceptable forms of the density operator. Density operator for
microcanonical ensemble. Postulates of equilibrium quantum
statistics--(i) Equal a-priori probability and (ii) random phases
[=> density operator is diagonal in any representation, not just
energy basis].
Lecture
22 (March 02, 2016, Wednesday):
Density matrix in the canonical ensemble. Example of density matrix
of free spin half particle in magnetic field.
Lecture 23
(March 03, 2016, Thursday):
Single particle density matrix for free
particle. Physical meaning of the thermal wavelength.
Many-particle wave functions for indistinguishable
particles. The permutation operator. Construction of
symmetric and antisymmetric wave functions (Only worry
about the 2-particle case). Slater determinant Exclusion
principle. Diagonal
components of two particle density matrix for free particles.
Correlations due to symmetrization/antisymmetrization.
Lecture 24 (March 08,
2016, Tuesday): Quantum
gases in the microcanonical ensemble (Pathria 6.1).
Lecture 25 (March
09, 2016, Wednesday):
Quantum gases in the microcanonical ensemble (Cont). Distribution
function for fermions and bosons. Connection with thermodynamics.
How the method of Lagrange multipliers works (See Reif)
Lecture 26 (March 10, 2016, Thursday):
Ideal quantum gases in
the grand canonical ensemble. Decomposition of the grand
partition function in terms of single orbital partition
functions for non-interacting particles. Recovering the
expression for the grand partition function for fermi and
bose gases (the expression was derived in the
microcanonical ensemble).
Distribution
functions and density of states. [Not in syllabus: Use of Lagrange
multipliers to derive the microcanonical, canonical, and
grand canonical probability density functions.]
Class Test 1: 15
March, Tuesday [Syllabus
for Class Test 1: Lecture 16-Lecture 25]
Tutorial/discussion
of solutions 16 March
No class on 17 March
Thursday
Just to help you navigate through Pathria
[Sections from Pathria not covered in class and which
can hence be left out:
Chapter 3: Section 3.2, 3.7 Eq. 10 onwards Virial theorem was
not done but equipartition was.
Chapter 4: 4.2, 4.6, 4.7
Chapter 5: 5.3C (Density of matrix for Harmonic oscillator), 5.5
was simplified
Chapter 6: 6.2 (Equations 1-10 not discussed-Canonical ensemble.
The rest of the section is), 6.3, 6.4, 6.5 is very qualitative
(just read through it once and get the basic idea). 6.6 not
done.] Spring Break
From This Point Onwards you may consider
C. Hermann's Statistical Physics book to be the primary
reference.
Lecture
27 (March
29, 2016,
Tuesday):Recap
of previous few classes. Calculation of density of states (Pathria
section 1.4. Hermann chapter 6. In Hermann there is also a
discussion of the periodic boundary conditions). General
expression for
the chemical for
the ideal
quantum gases
and its
classical limit
and proof of
PV=(2/3)U
Lecture
28 (March
30, 2016,
Wednesday): Bose Statistics Expression
for the grand
potential and
number of
particles in
terms of Bose
integrals,
importance of
the zero
energy states,
bose
condensation.
Lecture
29 (March
31, 2016,
Thursday):
Bose Condensation (cont). Thermodynamics of a
photon gas in
a cavity. Energy
density of a
photon gas (Planck distribution/black
body
spectrum). Wein
displacement
law,
Stefan Law [Hermann:
Chapter 9 up to section 9.2.3.
Pathria: Relevant portions of
section 7.1 and 7.3 only]
Lecture 30 (April
05,
2016, Tuesday):
Fermi-Dirac
Statistics: Formal solution of the expression for
the grand partition
function and the carrier
distribution function
in terms of Fermi integrals (Pathria).
Lecture 31 (April 06,
2016, Wednesday): Fermi
Gases (cont): Specific
heat close to zero
temperature (qualitative
argument, see Hermann
7.2.2, pg 163).
Bohr-van
Leuween theorem, quantum
mechanical problem of
free electron
in
magnetic field, Landau
levels, degeneracy
of Landau levels, extreme
quantum limit (good
discussion in
Huang and
Greiner's
Quantum Mechanics
book).
Lecture 32 (April 07, 2016, Thursday): Pauli
spin paramagnetism (Pathria 2.2.A, up to
equation 7 only). High temperature expression for Landau diamagnetism and
Pauli paramagnetism (Pathria 8.2.B, eq 28-43) [Also check out Ashok Sen's
Notes]
Pathria:
Chapter 8: 8.2.A (up to equation 7), 8.2.B (equation 28-43),
8.3 (up to the start of 8.3.A). Hermann Chapter 7 (Except
Thermionic emission). Landau Levels (Greiner QM Book)
Lecture 33 (April 12, 2016, Tuesday): Ising Model.
The Model and its physical motivation. The mean field solution.
Transcendental equation for magnetization.
Lecture
34 (April 13,
2016, Wednesday):
Ising Model (cont.) Argument that there is no
spontaneous
magnetization
at finite
temperature in
1D. Expansion
of
the free
energy in
powers of magnetization
(order
parameter) and
its shape
below and
above the
transition
temperature. Spontaneous
symmetry
breaking
(qualitative).
[Here
is a mix
of printouts
from three
different
books (Huang,
Greiner and
Salinas)...]
Class
Test 2 (April
19, 2016,
Tuesday):
Syllabus:
Lecture
24-Lecture 32
Textbooks
1. [Callen]
Herbert B. Callen, Thermodynamics
and an Introduction to Themostatistics. second
edition. (1985).
New York: John Wiley & Sons.
2.
[Pathria] Raj K. Pathria and Paul
D. Beale, Statistical Mechanics, third edition. (2011)
Elsevier
3. [Hermann] Claudine Hermann,
Statistical Physics
(2005), Springer
Other Recommended Texts:
1. [Reif] F. Reif, Fundamentals of
Statistical and Thermal Physics, McGraw Hill
2. [Blundell] Stephen Blundell and
Katherine Blundell, Thermal Physics, second edition, Oxford
University Press (2009). [Elementary]
3. [Bowley-Sanchez] R. Bowley and M.
Sanchez, Introductory Statistical Mechanics, second edition,
Oxford University Press (2000). [Elementary]
4. [Huang] K. Huang, Statistical
Mechanics, second edition, Wiley.
5. [Pal] Palash
B. Pal, Introductory course of statistical mechanics (Narosa
2008).