# Statistical Physics (2024)

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 930 hrs

Venue: AG 69

First lecture: 6 Feb

Credit policy: Tentatively Midterm (35%) + Endterm (35%) + Assignment (30%)

Instructor: Basudeb Dasgupta

Tutors: Sandeep Jangid and Soumya Pal

Course Webpage: Moodle [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)

## Exams

1. Droptest on 18 Feb

2. Midterm (4 April, AG69)

3. Endterm (30 May, 9am-1pm, AG69)

## Lecture Summaries (tentative)

Lecture 1 (Feb 6)

Why Statistical Mechanics?

Introduction to Thermodynamics

Basic notions / terms used in Thermodynamics

Lecture 2 (Feb 8)

Zeroth Law

First Law

Lecture 3 (Feb 13)

Second Law

Potentials

Third Law

Lecture 4 (Feb 15)

Probability

Bayesian and Frequentist

Generating functions

Lecture 5 (20 Feb)

Moments, Cumulants, and MGF, CGF,

Examples: Gaussian, Binomial

Many Random Variables

Lecture 6 (Feb 22)

Central Limit Theorem for iids; and extensions

Summing large exponentials, Saddle point, Stirling's approximation

Lecture 7 (Feb 27)

Information and Entropy

Lecture 8 (Feb 29)

Phase space and Liouville eq.

Equilibrium and its features

Lecture 9 (Mar 5)

Liouville to BBGKY for dilute gas

N particle density

Time-scales

Lecture 10 (Mar 7)

BBGKY to Boltzmann

Coarse-graining and molecular chaos

Boltzmann eq. and local equilibrium

H Theorem

Lecture 11 (Mar 12)

Boltzmann eq. to Hydrodynamics via moments

Hydrodynamics at zeroth order: Waves

Lecture 12 (Mar 14)

First order: Diffusion

Approach to global equilibrium and relevant timescales

Lecture 13 (Mar 19)

Equilibrium

Time average vs Phase average

Main goal of Stat Mech : Assign probability to microstates

Lecture 14 (Mar 21)

Density/Number of states

Lecture 15 (Mar 26)

Microcanonical ensemble

Microcanonical Example: Ideal gas

Derivations of 0th, 1st Law

Lecture x (Mar 28)

Lecture cancelled

Midterm Week (1-5 April)

Midterm on 4th April

Lecture x (April 9)

No class [Gudi Padwa]

Lecture x (April 11)

No class [Eid ul Fitr]

Lecture 16, 17, 18, 19, 20 (April 16, 18, 23, 30, May 2)

2nd Law of Thermodynamics

Canonical and Examples

Grand Canonical and Examples

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 21, 22 (April 22 at 4pm, 25 at 10am)

Stochastic Processes [Guest Lectures by Prof. Shamik Gupta]

Lectures 23 onwards (May 7, 9, 14, 16, 21, 23)

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)