# 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!**