# Statistical Physics (2017)

Statistical mechanics enables us to probabilistically model the behavior of macroscopic systems, which are made up of large numbers of identical microscopic constituents for which we only have incomplete description. It bridges the disconnect between mechanics which requires 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.

Any comments and/or suggestions about special interests/requests, pre/during/post-course are highly welcome. I am available over email (ALWAYS include the tag "SP2017" in the subject line to ensure my spam filter doesn't reject it). You can send anonymous emails if you prefer. No promises to act on any of them unless I think it will be useful for a majority of the class.

## Administrivia

**Time:** 11:30 AM, Tuesdays and Thursdays

**Venue:** AG 69

**First lecture: **Thursday, 2 Feb 2017

**Credit policy:** 25% from 5 problem sets + 25% from mid-semester exam + 50% from end-term exam

**Instructor:** Basudeb Dasgupta

**Tutors:** Subhajit Ghosh (DTP) and Mayank Narang (DAA)

## Target Audience

This is the core 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 (3 lectures)

2. Probability and statistics: Counting, distributions, large numbers (3 lectures)

3. Kinetic theory and approach to equilibrium (5 lectures)

4. Classical statistical mechanics: Formalism and simple systems (4 lectures)

5. Classical statistical mechanics: Interactions, approximations, phase transitions (4 lectures)

6. Quantum statistical mechanics: Formalism, ideal Bose/Fermi gases, phase transitions (6 lectures)

7. Beyond SP-1 (1 lecture)

+ 4 problem solving sessions & review lectures

## References

1. Statistical Mechanics of Particles, Kardar (Main text, but very terse. The videos are highly recommended.)

2. Statistical Mechanics, Huang (I personally find it most readable)

3. Statistical Physics (Berkeley Physics Course Vol.5), Reif (easier than others)

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)

6. Statistical Mechanics, R. P. Feynman (especially for solids, density matrices, and the later chapters)

## Problem Sets

PS1 (Thermodynamics) (on 7 Feb, due 21 Feb)

PS2 (Probability and Kinetic Theory) (on 21 Feb., due 21 March)

PS3 (Classical Stat. Mech.) (on 21 March, due on 11 April)

PS4 (Interacting Classical Stat. Mech.) (on 11 April, due on 4 May)

PS5 (Quantum Stat. Mech.) (on 27 April, due on 23 May)

## Exams

1. Drop-test on 11 Feb (10 AM, A 304)

2. Mid-term on 1 April (2 PM, AG 69)

3. End-term on 27 May (2 PM, AG 69)

## Lecture Summaries

Lecture 1 (2 Feb): Orientation

Syllabus

Logistics

Calibration

Lecture 2 (7 Feb): PS1 handed out. Thermodynamics I

Motivation

Definitions and basic notions in thermodynamics

0th Law: Statement; Consequence = T exists; Application = Thermometer

1st Law: Statement = E exists if walls adiabatic, and Q defined by energy conservation

Lecture 3 (9 Feb): Thermodynamics II

2nd Law: Kelvin and Clausius statements. Equivalence.

Carnot cycle, Carnot theorem

Clausius theorem

Definition of entropy S

Potentials and how to get appropriate derivatives

Extensivity (Gibbs-Duhem relation between intensive variables) and Maxwell relations

Applications: dmu/dP|_T = V/N or dv/dN|_T,P ; dS/dJ|_X = dX/dT|_S etc.

Lecture 4 (14 Feb): Thermodynamics III

Third Law: Weak and Strong Statement, Justification, Consequences (dS/dX,C,etc vanish)

Relations from equilibrium (T,P,mu,.. same)

Relations from stability (Second derivatives of S are negative)

Application: Critical point in PV plane of real gas (dP/dV, d^2P/dV^2 is zero, Third derivative -ive, Maxwell construction for van der Waals EOS)

Clausius-Clapeyron equation for P(T)

Gibbs Phase Rule: # of intensive variables needed = N_work + 1 + C_components - P_phases

Lecture 5 (16 Feb): Probability I

Definitions of probability: objective (occurrence frequency) and subjective ("reasonable" expectation)

Counting methods (Basic, Permutation, Combination, Partition, Generating function)

Bayes theorem p(A&B) = p(A|B)p(B) = p(B|A)p(A)

Lecture 6 (21 Feb): PS1 due. PS2 handed out. Probability II

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 7 (23 Feb): Probability III

Probability distribution of many variables

CPF, PDF, CF, MGF, CGF, Conditional and Unconditional PDFs of many variables

Graphical trick for many variables

Joint Gaussian ~ Product of gaussians

Statement of Weak law of large numbers, Statement and Proof of Central Limit Theorem (without assuming iid) ~ sum of random variables has Gaussian PDF

Sum of exponential(N*x) ~ exp(N*x_max)

Saddle point approximation to integrals and Stirling's approximation

Entropy as information in a PDF

Information I[{p_i}] = ln(M) - S, and Entropy of a PDF S[{p_i}]= - sum_i p_i ln(p_i)

Maximum Entropy Method as a formalization of subjective assignment of probability.

Claude Shannon and encoding info (See Shannon's 1950 paper on information and language. See also page 12-14 of his 1948 paper)

Lecture 8 (28 Feb): Kinetic Theory (Hamiltonian dynamics)

Last bit of Probability theory: Saddle point, Shannon information and entropy, MEM

Macrostates and microstates

Hamilton's equations

Lecture 9 (2 Mar): Kinetic Theory (BBGKY)

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 and is defined by rho = rho(E)

1 particle density, s-particle density

BBGKY hierarchy for N particles with pairwise interactions

Lecture 10 (7 Mar): Kinetic Theory (Route to Equilibrium I: BBGKY to Boltzmann eqn.)

BBGKY -> Boltzmann equation using dilute approx. + molecular chaos + coarse-graining + separation of scales

Interpreting the Boltzmann equation and irreversibility

Lecture 11 (9 Mar): Kinetic Theory (Route to Equilibrium II: H theorem, local equilibrium, Hydrodynamics)

Statement and proof of H theorem, Loschmidt paradox, Poincare recurrence

See this Article by Cedric Villani (much of it is easy to read) on the mysteries of the Boltzmann equation

Local equilibrium for all except conserved qtys.

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)

Hydrodynamic equations for collisionally conserved quantities

Deriving the hydrodynamic equations (Continuity, Navier-Stokes, Heat-flow)

Physical meaning of the terms in the equations

Lecture 12 (14 Mar): Kinetic Theory (Route to Equilibrium III: Hydrodynamics)

Zeroth order: Sound waves, but no dissipation, Shear, Heat-flow

First order: Shear, Heat-flow

Global equilibrium

Lecture 13 (16 Mar): Problem solving and review

Steady Atmosphere

Benford's Law

Lecture 14 (21 Mar): PS3 handed out. PS2 due. Classical Stat Mech (Foundations, Microcanonical)

From Hamilton's eqns to Phase space: Recap

Volume in phase space, introduce 1/h^3N, number of microstates W, density of states

Ergodicity and Mixing

Fundamental Postulate of Statistical Mechanics and its justification via i) MEM, ii) Dynamics

Lecture 15 (23 Mar): Classical Stat Mech (Microcanonical)

Macrostate = M(E,X,N), microstate = mu(p,q)

p(mu) = 1/W(E), where W(E) = # of microstates within energy E and E-dE

1/T=dS/dE|_V,N

Derivation of 0th, 1st, 2nd Law

Ideal gas thermodynamics and "microscopic" info

Fixing non-intensive mu, Mixing entropy using 1/N! factor

2 level systems and negative temperatures (Observation of Negative T)

Holiday (Gudi Padwa) (28 Mar)

Lecture 16 (30 Mar): Problem solving and review

Midterm Exam (1 April): AG 69, 2 PM onwards. Open book.

Score Distribution

Lecture 17 (4 April): Classical Stat Mech (Canonical)

Canonical Macrostate M(T,X,N), microstate = mu(p,q)

p(mu) = exp(-E/kT)/Z

Canonical Partition Function Z = W(E) exp(-E/kT) sum over E

E -> F = E - TS and Z = exp(-F/kT) sum over F

Energy fluctuations in CE: Why E_most-prob = <E> + O(1/sqrtN)

Z_N = Z_1^N ; include 1/N! if indistinguishable

Ideal gas and variations in D, dispersion relation, multiple dofs

Lecture 18 (6 April): Classical Stat Mech (Gibbs and Grand Canonical)

Gibbs Canonical Macrostate = M(T,J,N), microstate = mu(p,q)

E -> G = E - TS - JX, and Z = exp(-G/kT) sum over G

Ideal gas in isobaric ensemble

Grand Canonical Macrostate = M(T,X, mu), microstate = mu(p,q)

Number fluctuations in GCE

Ideal gas in GCE

Lecture 19 (11 April): PS4 handed out. PS3 due. Approximations (Cumulants)

Interacting systems and approximation techniques

Series expansions / perturbation theory

Cumulant expansion and failure for hard core type interactions

Lecture 20 (13 April): Approximations (Cluster)

Cluster expansion

Virial series for P

Comments

Lecture 21 (18 April): Approximations (Mean field)

Mean Field Theory of Condensation

Proof & Meaning of Maxwell Construction

The search for Universality

Lecture 22 (20 April): Approximations (Critical Phenomena and Monte Carlo)

Weiss Mean Field Theory for Ising Model

Critical Exponents, Universality, and Heuristic Justification

Ordinary Monte Carlo and its Failure

Basic Markov Chain Monte Carlo

Metropolis Algorithm

Lecture 23 (25 April): Problem solving and review

Lecture 24 (27 April): PS5 handed out. Quantum Stat Mech (Old QM)

Issues with Classical Stat Mech

Quantization a la Old Quantum Mechanics

Vibrations, Rotations, and Blackbody Radiation

Lecture 25 (2 May): Quantum Stat Mech (Formalism and Density matrix)

From classical to quantum: microstates, macrostates, probability density (matrix)

The ensembles: microcanonical and canonical

Canonical density matrix of 1 particle

Interpretation in terms of classical and quantum uncertainty

Lecture 26 (4 May): PS4 due. Quantum Stat Mech (Anti/Symmetrization and multi-particle states)

Exchange symmetry of wavefunction of 2 identical particles

Hilbert space of N identical particles and normalization

See this

**link**for a nice primer on quantum mechanics of many particlesCanonical density matrix of N particles

Lecture 27 (9 May): Quantum Stat Mech (Canonical and Grand Canonical Ensembles)

N particle Hilbert space, density matrix, product state basis, entangled states, reduced density matrix

Interpretation of canonical density matrix in terms of statistical attraction/repulsion

Grand Canonical Ensemble

Lecture 28 (11 May): Quantum Stat Mech (Ideal Quantum Gas at High T)

Ideal gas in GCE

High T limit of ideal gas and virial expansion

Lecture 29 (16 May): Quantum Stat Mech (Degenerate fermions)

Sommerfeld expansion in small T limit, i.e. z >> 1

Why fermions are simple: Fermi surface

mu, P, E, C_v, etc.

Lecture 30 (18 May): Quantum Stat Mech (Degenerate bosons)

Why z cannot exceed 1 : Bose Einstein condensation

What happens to mu close to z = 1?

mu, P, E, C_v, etc.

Lecture 31 (23 May): PS5 due. Beyond SP-1

More examples, interactions, bands, superfluidity

Lecture 32 (25 May): Problem solving session and review

**27 May: Endterm exam (AG 69, 2 PM onwards. Open notes) and End of Course**

**Grades and Break-Up into Assignment, Midterm, and Endterm**

**Congratulations!**