# Statistical Physics (2023)

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 in the TIFR graduate school.

This page will be updated regularly with course-related information. Please check frequently.

## Administrivia

Time: Tu, Th at 1130 hrs

Venue: AG 69

First lecture: 17 Jan

Credit policy: TBD

Instructor: Basudeb Dasgupta

Tutors: Rupak Majumdar and Asikur Rahman

Course Webpage: Moodle Link [TIFR only]

Lecture Notes: Dropbox Link [Public]

## Course Contents

1. Preliminaries: Motivation and review of thermodynamics

2. Probability and statistics: Counting, distributions, large numbers

3. Kinetic theory and approach to equilibrium; Brownian walks etc.

4. Classical statistical mechanics: Formalism and simple systems

5. Classical statistical mechanics: Interactions, approximations, phase transitions

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

7. What else is there? and Review of the course

## References

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

2. Statistical Physics (Berkeley Physics Course Vol.5), Reif

3. Statistical Mechanics of Particles, Kardar (Main text, but very terse. Videos are awesome.)

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 and probability)

2. PS2 (kinetic theory)

3. PS3 (classical stat. mech + interactions)

4. PS4 (quantum stat. mech) due by 27 May

## Exams

1. Droptest on 4 Feb (10-12)

2. Midterm (26th March)

3. Endterm (27 May)

## Lecture Summaries

Orientation (17 Jan)

Lecture 1 (19 Jan)

Why Statistical Mechanics?

Introduction to Thermodynamics

Basic notions / terms used in Thermodynamics

Lecture 2 (24 Jan)

Zeroth Law

First Law

Lecture 3 (31 Jan)

Second Law

Lecture 4 (2 Feb)

Potentials

Third Law

Droptest (4 Feb)

Lecture 5 (7 Feb)

Probability

Bayesian and Frequentist

Lecture 6 (9 Feb)

Moments, Cumulants, and Generating Functions

Many Random Variables

Lecture 7 (14 Feb)

Central Limit Theorem for iids; and extensions

Lecture 8 (16 Feb)

Summing large exponentials, Saddle point, Stirling's approximation

Information Entropy

Lecture 9 (21 Feb)

Phase space and Liouville eq.

Equilibrium and its features

Lecture 10 (23 Feb)

Liouville to BBGKY for dilute gas

N particle density

Time-scales

Lecture 11 (28 Feb)

BBGKY to Boltzmann

Coarse-graining and molecular chaos

Lecture 12 (2 Mar)

Boltzmann eq. and local equilibrium

H Theorem

Lecture 13 (7 Mar)

Boltzmann eq. to Hydrodynamics via moments

Lecture 14 (9 Mar)

Hydrodynamics at zeroth order: Waves

First order: Diffusion

Approach to global equilibrium and relevant timescales

Lecture 15 (14 Mar)

Stochastic processes I (Guest lecture by Prof. Shamik Gupta)

Lecture 16 (16 Mar)

Stochastic processes II (Guest lecture by Prof. Shamik Gupta)

Lecture 17 (21 Mar)

Equilibrium

Time average vs Phase average

Main goal of Stat Mech : Assign probability to microstates

Lecture 18 (21 Mar)

Microcanonical ensemble

Density/Number of states

Derivations of 0th, 1st, 2nd Laws of Thermodynamics

Midterm (26 Mar)

Closed book. 1 A4 formula-sheet allowed.

2-5pm

Lecture 19 (28 Mar)

Microcanonical Examples

Lecture 20 (20 Mar)

Canonical and Examples

Lecture 21 (4 Apr)

Grand Canonical and Examples

Lectures 22-26 (6, 18, 20, 25, 27 Apr)

Classical Stat Mech. of Interacting Systems

Mayer cluster expansion technique

Mean Field Theory of Condensation

1st order Phase Transition

Gibbs Canonical Ensemble and Multiple Saddle Points

Meaning of Maxwell Construction

Curie-Weiss Mean Field Theory

2nd Order Phase Transition

Important Results on Phase Transitions (without proof)

Lectures 27-34 (2, 4, 6, 9, 11, 13, 16, 18 May)

Discreteness of energy

C_V of diatomic gas vs T

C_V of a crystal and Debye's T^3 Law

Black body radiation

Symmetrization and Antisymmetrization

Density matrix in energy and position bases

Virial expansion and "effective potential" for ideal quantum gas

Ideal quantum gas at high T (mu vs T, virial expansion again)

Ideal Fermi gas (Fermi energy, mu, C_V, degeneracy pressure)

Ideal Bose gas (BEC, pressure, C_V, mu)

Endterm (27 May)

Closed book, A4 formula sheet allowed (both sides if needed)

2-5 PM in AG 69

All assignments due before exam

Congratulations, everyone!