Statistical Physics (2016)
Statistical mechanics enables us to model the behavior of macroscopic objects, which are made up of large numbers of constituents for which we only have incomplete descriptions, using probability theory and a microscopic theory of the constituents. It bridges the disconnect between mechanics which require complete knowledge of initial conditions, and the real world where such information is not available. These ideas therefore find widespread application.
Target Audience
This is course SP-1 or P-204 in the TIFR graduate school. Students who have joined after their B.Sc. or M.Sc may both take this course.
This page will be updated regularly with course-related information. Please check frequently.
Administrivia
Time: M, W at 1130 hrs
Venue: AG 69
First lecture: 3 Feb
Credit policy: 5 problem sets (25%), mid-term exam (25%) and end-term exam (50%)
Instructor: Basudeb Dasgupta
Tutors: Anirban Das and Abhisek Samanta
Course Contents
1. Preliminaries: Motivation and review of thermodynamics (2 lectures)
2. Probability and statistics: Counting, distributions, large numbers (2 lectures)
3. Kinetic theory and approach to equilibrium (5 lectures)
4. Classical statistical mechanics: Formalism and simple systems (5 lectures)
5. Classical statistical mechanics: Interactions, approximations, phase transitions (3 lectures)
6. Quantum statistical mechanics: Formalism, ideal Bose/Fermi gases, phase transitions (7 lectures)
7. Modern outlook, and review of the course (3 lectures)
+ 4 problem solving sessions
References
1. Statistical Mechanics of Particles, Kardar (Main text, but very terse. Videos are awesome.)
2. Statistical Physics (Berkeley Physics Course Vol.5), Reif
3. Statistical Mechanics, Huang (I personally find it most readable)
4. Statistical Mechanics Part-I (Course of Theoretical Physics Vol.5), Landau and Lifshitz
5. Lecture notes by David Tong, Cambridge Univ. (Kinetic Theory, Statistical Mechanics)
Problem Sets
1. PS1 (thermodynamics) (on 3 Feb., due 15 Feb)
2. PS2 (probability & kinetic theory) (on 22 Feb, due on 22 Mar)
3. PS3 (classical stat. mech) (on 14 March, due on 30 March)
4. PS4 (interacting classical stat. mech) (on 4 April, due on 25 April)
5. PS5 (quantum stat. mech) (on 18 April, due on 18 May)
Exams
1. Mid-term on 2 April
2. End-term on 28 May
Lecture Summaries
Lecture 1 (3 Feb): PS1 handed out
Motivation & administrivia for the course
Definitions needed in thermodynamics
Relationships: 0th, 1st, 2nd and 3rd laws
Significance of these laws (i.e., definitions of T, Q, S , importance of quantum mechanics, etc.)
Tutorial 1 (8 Feb): Problem solving session for thermodynamics (by Abhisek Samanta)
Lecture 2 (10 Feb):
Consequences (e.g., S increases, vanishing of heat capacity, expansion coefficients, etc., at zero T)
Extensivity of E and Gibbs-Duhem relation between intensive variables
Thermodynamic potentials E, F, G, H, and derivation of Maxwell relations
Conditions for equilibrium and stability (e.g., S maximized, Hessian negative semi-definite)
Gibbs phase rule and phase coexistence
Lecture 3 (15 Feb): PS1 due
Definition of probability and its interpretations
Counting (basic counting, overcounting & correction, partitions, generating functions)
CPF, PDF, expectation, CF = characteristic function as the Fourier transform of PDF, moments from power series of CF = MGF, cumulants from power series of log(CF) = CGF
Graphical trick relating cumulants and moments
Explicit computation for Gaussian, Binomial, Poisson PDFs
Lecture 4 (17 Feb):
Probability distribution of many variables
CPF, PDF, CF, MGF, CGF of many variables
Graphical trick for many variables
Joint Gaussian ~ Product of gaussians
Central limit theorem ~ sum of random variables has Gaussian PDF
Rules for large numbers and sum over exponentials ~ max-term
Entropy as information in a PDF (Caution: discrete/continuous)
Lecture 5 (22 Feb): PS2 handed out
Macrostates and microstates
rho(p,q) as the full 6N dimensional probability dist. for N-particles
measurement O as an expectation value
Liouville's theorem: uncompressibility of rho
Consequences: T-reversal, evolution of O, equilibrium exists
Lecture 6 (24 Feb):
Notion of Equilibrium
BBGKY hierarchy of equations for s-particle densities f_s
Simplifying BBGKY with "physical insight" to get Boltzmann equation
Lecture 7 (29 Feb):
BBGKY -> Boltzmann equation using dilute approx. + molecular chaos + coarse-graining
Interpreting the Boltzmann equation and irreversibility
H-theorem: Entropy tends to increase, Loschmidt's paradox, Poincare recurrence
Consequence: equilibrium is probable
Lecture 8 (2 March): Problem solving session on probability and kinetic theory (by Anirban Das)
Lecture 9 (7 March)
Properties of local equilibrium (Boltzmann-like but position and time dependent)
Approach to full equilibrium (f_2 -> f_1*f_1 at intrinsic time-scale, local equilibrium at t_mfp, global equilibrium at t_eq >> t_mfp, e.g., t_ballistic^2/t_mfp)
Collision conserved quantities
Hydrodynamic equations
Lecture 10 (9 March)
0th order hydrodynamics (waves, no shear stress, no heat flow)
1st order hydrodynamics (damping, velocity diffusion, heat flow)
Consequence: global equilibrium is reached
Lecture 11 (14 March) PS3 handed out
Phase space flows, ergodicity, mixing, equilibrium, density of states
Maximum entropy method to assign probabilities
Microcanonical ensemble and Fundamental Postulate of Stat. Mech.
Microcanonical derivation of 0th, 1st, and 2nd Laws of thermodynamics
Lecture 12 (16 March)
Recipe:
Decide on M(E,X,N), microstates_i(p,q), and H(p,q)=E.
Find Omega(E,X,N) by doing an integral of dGamma between surfaces at E-dE and E
Assign probability p_i = 1/Omega
Calculate S = k_B ln(Omega)
Calculate thermodynamics from S and microscopic info from p_i
2-level systems: thermodynamics, negative temperatures (b'cos no way to absorb more energy. More about this.)
Ideal gas: hyperspherical integrals, thermodynamics, mixing entropy/Gibbs Paradox resolution and intensive mu from 1/N!, Maxwell velocity distribution, equipartition
Lectures 13 (21 March)
Canonical (E<->T) ensemble from microcanonical ensemble of system and reservoir
Sum over microstates exp[E_i/(k_B T)] = Sum over energies exp[(E_i-TS(E_i))/(k_B T)]
Free energy F(T,X,N) as a central thermodynamic quantity
Averages, fluctuations, heat capacity as a measure of fluctuation
Recipe:
Decide on M(T,X,N), microstates_i(p,q), and H(p,q).
Find Z(T,X,N) = exp(-E_i/k_BT) sum over states i is the CGF
Assign probability p_i = exp(-E_i/k_BT)/Z
Calculate F = -k_B T lnZ
Calculate thermodynamics from F and microscopic info from p_i
2-level systems
Ideal gas
Lecture 14 (23 March) PS2 due
Gibbs canonical (V<->P or M<->B) ensembles with Z(T,F,N)
Gibbs free energy G=E-TS+PV=-k_B T lnZ as the central quantity
Grand canonical (N<->mu) ensemble with Q(T,V,mu)
Grand potential/Landau free energy L=E-TS-muN=-k_B T lnQ as the central qty
Examples: Z(T,P,N) for ideal gas, Z(T,B,N) for magnet, Q(T,V,mu) for ideal gas
When are fluctuations important? T large, "C_x" diverging.
Gibbs-Duhem relationship and No triple Legendre transforms (1 extensive variable needed)
Lecture 15 (28 March)
Review before mid-term exam
Lecture 16 (30 March) PS3 due
PS2 and PS3 discussion
See Shannon's 1950 paper on information and language. See also page 12-14 of his 1948 paper.
See the note on priors and Bayesian posteriors sent by email.
Mid-term Exam (2 April at 2PM in AG69)
Everything taught up to 28 March will be tested
Closed-book but a cheat-sheet is allowed (1 A4 sheet both sides)
Lecture 17 (4 April) PS4 handed out
Interactions and approximation techniques
Cumulant expansion and its failure for hard core type interactions
Lecture 18 (6 April)
Cluster expansion for dilute gas
B_2 and B_3 from diagrammatic approach
Lecture 19 (11 April)
Mean field approach
van der Waal's equation and interpretation of corrections
Gibbs canonical ensemble and understanding phase coexistence
Saddle point justification to Maxwell's construction
No universal eqn. of state for real gases (see Kardar for proof)
Mention of critical point/exponents (universality around critical point)
Mention of Monte Carlo methods (Why importance sampling?)
Lecture 20 (13 April)
Failures of classical stat. mech. (advent of old Q.Mech)
C_V of diatomic molecules (exponential suppression of dofs at k_B T< h w)
C_V of solid at small T (T^3, not exponential, because only low-w modes imp, where w=c_s k.)
Blackbody radiation (Planck's spectrum, Stefan-Boltzmann)
Lecture 21 (18 April) PS5 to be handed out
From classical to quantum: microstates, macrostates, probability density (matrix)
The ensembles: microcanonical, canonical, and grand canonical
Canonical Z for 1 free particle in box and the density matrix in position basis
Lecture 22 (25 April) PS4 due
Many identical noninteracting particles:
Symmetrization/antisymmetrization for bosons/fermions
Normalized many fermion/boson microstate |{n_k}>
Canonical Z and density matrix in position basis
Second virial coefficient from Z, as a consequence of single-exchanges
Grand canonical ensemble: Q and p[{n_k}] have all macro and microscopic info
Lecture 23 (27 April)
Dilute ideal nonrelativistic quantum gas at high T limit = Ideal classical + virial corrections
Lecture 24 (2 May)
Degenerate nonrelativistic Fermi gas, Sommerfeld expansion
mu(T), P(T), C_V(T), etc at low T regime
Idea of a Fermi-sphere and the relevant dofs
Lecture 25 (4 May)
Degenerate nonrelativistic Bose gas
Condition for condensation, and condensed fraction
mu(T), P(T), C_V(T), etc at low T regime and the first order phase transition T_c
Lecture 26(9 May)
Beyond ideal nonrelativistic gases: Qualitative impact of interactions, dimensionality, dispersion relations
Electrons in Copper (Image: Fermi surfaces of metals)
Superfluid Helium (Video: Alfred Leitner's Demo)
Photons, and other examples
Lectures 27-28 (11, 16 May) Problem solving sessions
Lecture 29 (18 May) PS5 due
Things that we did not cover: beyond SP-1
Lecture 30 (23 May)
Review
Lecture 31 (25 May)
Review
End-term Exam (28 May, 2-5 PM in AG69) and end of the course
Open notes (your own handwritten notes only). No solved problems please.
Final Grades
Congratulations!