## An introduction to spin

**Unusual Starting Time:** 2:30pm

Suresh will lead the discussion on how spin appears from the representation theory of the Poincare group. All necessary background will be provided and the talk will dynamically adjust to the level of the audience present.

## Markov Processes

**Unusual Starting Time:** 1:45pm

Akarsh will lead tomorrow's discussion on "Markov Processes". We will have a sharp time-cutoff at start-time + 1 hour (<= 3 PM).

Although we have used them heavily in the past few Boltzmann meetings, we will do a formal introduction followed by some results on Markov processes. The discussion will closely follow Chapter 5 titled "Markov Processes" from Prof. Balakrishnan's book "Elements of Non-Equilibrium Statistical Mechanics". We will attempt to finish all sections of that chapter that are relevant to Continuous Markov Processes in 1 hour :)

Since the talk will mostly be rigorous and rather mathematical, if there is excess time, I'll summarize an application of these to a simple situation we solved during my internship.

For those who don't have the book, here's the list of sections I plan to "uncover":

- Continuous Markov Processes [Introduction]
- Chapman Kolmogorov Equation and the Master Equation. I'll also state the discrete analogue (which is much easier to understand)
- Kramers-Moyal expansion of the Master Equation (I need to figure out what this is!)
- The Forward Kolmogorov Equation (a.k.a The Fokker-Planck equation)
- Backward Kolmogorov Equation

The purpose of this talk is to put all those terms that've been floating around in place.

**Pre-requisites**: Notion of a random process will be a good thing have, but is not really necessary.

The discussion will be led by Naveen.

We shall be discussing rudiments of Group Representations. The following shall be the course of the talk:

- Motivation for representations of groups.
- The notion of a G-invariant form(This is the same as the more familiar "inner product").
- Unitary representations. We shall prove a theorem which says that every representation of a finite group is conjugate to a unitary representation.(Pretty remarkable, at face value!).
- If time permits, we shall try to extend the above to continuous groups(also compact). The extension is almost direct except that we have to worry about some subtle existence issues.

A good prerequisite would be a working knowledge of action of finite groups on a set. This is not necessary though.

We shall be essentially discussing Chapter 9 in *Algebra* by M. Artin.

## The Central Limit Theorem

The discussion will be led by Naveen.

We shall first look at how the Normal distribution comes about naturally in the binomial distribution when we take the number of trials to be very large. We shall then state the Central Limit Theorem for distributions with a finite variance and look at some familiar examples. We shall then move on to Stable / Levy distributions in general and try to characterize the distributions in terms of the parameter $\alpha$ and again look at some examples. A particularly nice example we shall be going to discuss is the Holtsmark distribution which was discovered well before Levy came up with the distributions.

A good reference for this would be "An Introduction to Probability Theory and Its Applications", Vol I and Vol II, Feller.

## FPE from SDE The Feynman-Kac Formula

The discussion will be led by Suresh.

The aim of the talk is to connect up the last two talks — the one by Akarsh on the Langevin equation and the one by Siva on the Feynman path integral. We start with stochastic differential equation (SDE)

(1)where $\zeta(t)$ is white Gaussian noise with zero mean and $\langle \zeta(t)\zeta(t')=\delta(t-t')$. Let ${p}(x,t|x_0,t_0)$ probability of finding the system at $x$ at time $t$ given that it was at $x_0$ at earlier time $t_0$. We will show that ${p}(x,t|x_0,t_0)$ can be formally represented as a path-integral. It can also be shown that ${p}(x,t|x_0,t_0)$ solves the Fokker-Planck equation (FPE):

(2)**References**

- M. Kardar,
*Statistical Physics of Fields*, Chapter 9, Cambridge Univ. Press. - V. Balakrishnan,
*Elements of Nonequilibrium Statistical Mechanics*, Chapter 6, CRC Press. - Mark Kac, On Distributions of Certain Wiener Functionals , Trans. of the AMS,
**65**, No. 1 (Jan., 1949), pp. 1-13 — this is the original reference.

## Path Integrals in Quantum Mechanics

The discussion will be led by Sivaramakrishnan.

**Outline:**

**Introduction to Path Integrals**

**The Path integral approach to Quantum Mechanics**

- A "derivation" from Heisenberg Mechanics
- Probability amplitude and not probability
- Sum over paths of the probability amplitude
- The "Kernel" and its meaning

**Examples**

- A free particle
- The harmonic oscillator

**Comments on the beauty/advantages of the Path Integral Approach**

The discussion shall be mainly based on the first 3 chapters of *Quantum Mechanics and Path Integrals* by Feynman & Hibbs.

## Langevin Equation and the Einstein relation

The discussion will be lead by Akarsh.

**Random Processes** - a very brief introduction

**Langevin Equation**

- Empirical argument leading to the Langevin Equation for Diffusion
- Solution of Langevin Equation with no potential force
- Connection to diffusion equation

**Diffusion under a spring potential**

- Solution to the Langevin Equation
- Einstein's Relation
- Finding the Avagadro Number; The Test of Molecular Hypothesis
- A (mere) mention of the fluctuation-dissipation theorem

The theory of Brownian Motion, given by Einstein in 1905, played an important role in verifying the molecular hypothesis. I will derive

the Einstein Relation relating the diffusion coeffecient to the mobility of a Brownian particle in a fluid and explain how the

Avagadro number can be found out experimentally using Brownian Motion.

This topic is pretty simple, and I think the only prerequisite is an idea of random variables, and ordinary differential equations!

I will be, as I mentioned earlier, following Mehran Kardar's *Statistical Physics of Fields*, Chapter 9, Section 9.1, where this is

an example used to develop the Langevin equation for fields.

## Random Walks

Shuddhodan on Random walks

## Introduction to Random Processes

The discussion will be a very basic, introductory discussion on random variables and random processes.

Ramya will lead the discussion. The following is a rough scheme of what is going to be covered in the course of the discussion:

Random Variables, Distribution Functions: Discrete and Continuous, Expectation Values, Variance, Moments, Probability generating functions, Characteristic functions, Moment Generating Functions, Gaussian Distribution, and Random vectors. Random processes.