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### 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.

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

#### 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.

#### 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:
1. Decide on M(E,X,N), microstates_i(p,q), and H(p,q)=E.
2. Find Omega(E,X,N) by doing an integral of dGamma between surfaces at E-dE and E
3. Assign probability p_i = 1/Omega
4. Calculate S = k_B ln(Omega)
5. 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:
1. Decide on M(T,X,N), microstates_i(p,q), and H(p,q).
2. Find Z(T,X,N) = exp(-E_i/k_BT) sum over states i is the CGF
3. Assign probability p_i = exp(-E_i/k_BT)/Z
4. Calculate F = -k_B T lnZ
5. 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!