Preparatory Lectures
Lectures by Abhishek Dhar:
1. Random Walks [Notes1] [Probset1]
2-3. Hydrodynamics [Notes2][ASEP and Burgers comparison (Ritwik Mukherjee) - movie]
4. Dynamical systems: linear stability analysis [Notes3]
Lectures by Sanjib Sabhapandit:
1-6. Stochastic processes
Lecture Notes - Link
Course 1: Statistical Physics of Ecosystems -Akshit Goyal
Course Outline:
Day 1: Introduction to ecological dynamics + low-dimensional ecology (with few species)
Day 2: High-dimensional ecology (many-species); ecosystems as disordered systems; cavity method
Day 3: Phase transitions in many-species ecosystems; cavity method and perturbent resolvent methods from random matrix theory
Day 4: Chaotic ecological dynamics; dynamical mean-field theory; high-dimensional deterministic chaos as an effective stochastic process
Day 5: Ecological dynamics with many trophic levels; consumer-resource models; open questions
The following reading materials would be useful:
https://arxiv.org/abs/2403.
https://arxiv.org/abs/2507.
https://journals.aps.org/prx/
Preparatory Material:
- Nonlinear dynamics and bifurcations
- Probability (moments, Gaussian integrals)
- Stochastic processes
Course 2: Introduction to fluid dynamics and turbulence - Samriddhi Sankar Ray
Course Outline:
Lecture 1: Brief introduction to turbulence; Navier-Stokes equation; Properties of the Navier-Stokes equation; Symmetries and invariants; Energy budgets; Introduction to chaos.
Lecture 2: Two key experimental results; Interpretation of them through 1D Burgers equation; Phenomenological model; Kolmogorov theory; Multiscaling; Random Cascade; Intermittency.
Lecture 3: Beta model; Bifractal Model; Multifractal model; Cascade models.
Lecture 4: Coarse-grained approach; Onsager's conjecture; anomalies.
Lecture 5: Closure models; Direct Interaction Approximation; Eddy-Damped, Quasi-Normal, Markovian Approach; Overview of recent developments.
Tutorial Topics [by Ritwik Mukherjee]: Numerical solution of chaotic ODEs and chaotic maps;
Introduction to the computation of Fourier transform using the Fast Fourier Transform (FFT); Discussion on Navier-Stokes equations in Fourier space; Calculations of distributions of velocity, velocity differences, gradients etc from a given dataset to get a feel for intermittency; Introduction to fractal-dimension, generalized dimension and singularity spectrum. Explanation of how generalized dimension is related to the singularity spectrum; Computing the generalized dimension and singularity spectrum from a dataset;. Creating multifractal signals using cascade models with different degree of multifractality; Shell models as a toy model for Navier-Stokes equations.
Course 3: Introduction to Information Theory - Jaikumar Radhakrishnan
Course Outline:
Day 1: Shannon entropy
Day 2: Conditional entropy and mutual information
Day 3: Entropy and its applications in combinatorics
Day 4: Relative entropy in statistics
Day 5: Communication complexity of Boolean functions
For more information and resources click here
Course 4: Disordered Systems: Statics and Dynamics - Alberto Rosso
This course deals with systems in which the presence of impurities or amorphous structures (in other words, of disorder) influences radically the physics, generating novel phenomena. These phenomena involve the properties of the system at equilibrium (freezing transitions, glassy phase and glassy system), as well as their dynamical evolution out-of-equilibrium (pinning, avalanches).
- The simplest spin-glass: solution of the Random Energy Model.
- Interface growth and Directed polymers in random media.
- Scenarios for the glass transition: the glass transition for directed polymers in d>2.
- Depinning and avalanches.
- Bienaymé-Galton-Watson processes.
More information - click here
Course 5: First passage problems: classical and quantum - Abhishek Dhar
First passage problems: classical and quantum
Course 6: Introduction to Generalized Hydrodynamics - Benjamin Doyon
Tutorial problems - Link