Advanced Statistical Physics: February - July, 2021

Course structure

Updated, July 14, 2021

Instructor: Tridib Sadhu

Teaching assistant: Aman Kumar

Class timings: Wednesday 16h and Friday 17h45. Each class 1h 30min long. Total around 56 hours of lectures.

Course outline

1. Stochastic processes:

1. Master equation, Langevin and Fokker Planck description, Path integral for stochastic processes. Few important results for Brownian motion, Levy flight, Anomalous diffusion. Extreme events and Large deviations.
2. Concepts of equilibrium and non-equilibrium, Time reversibility, Stochastic trajectories.

2. Equilibrium:

1. Revision of basic equilibrium concepts, Phase transitions, Lee-Yang theory.
2. Question of order, Peierls argument, Graphical expansion and duality, Mermin-Wagner theorem, Entropic forces.
3. Exact solution of lattice models: Ising on 1d and Bethe lattice. Ising on 2d by transfer matrix and graphical methods, Potts model, Dimer covering, Heisenberg model (Bethe ansatz).
4. Critical phenomena: Ginzburg criterion, Scaling, Universality, Landau theory.
5. Renormalization group: Idea and phenomenology, Real-space RG, Epsilon expansion.
6. Dynamics: Phase ordering kinetics, Model A and B dynamics.

3. Out of equilibrium:

1. Fluctuation theorems (Jarzynski relation, Gallavotti-Cohen symmetry), Linear response outside equilibrium, Onsager-Machlup theory .
2. Non-equilibrium processes: Interface growth (EW and KPZ), Disordered medium, Exactly solvable models (zero range process, exclusion process using Matrix algebra and Bethe ansatz), Transport phenomena, Active matter, Non-equilibrium phase transitions.
3. Hydrodynamic approach for non-equilibrium systems.

Books followed

1. M. Kardar: Statistical Physics of Fields.
2. G. Mussardo: Statistical Field Theory
3. N. Goldenfeld: Lectures on phase transitions and the renormalization group.

Influenced by other courses

1. Joachim Krug: Several related courses.
2. David Tong: Statistical Field Theory
3. Kay Wiese: Advanced Statistical Field Theory

Relevant reading materials

1. Lecture note of Abhishek Dhar (ICTS). These are basic concepts of probability theory. I expect that these concepts are known to all of you from Stat Phys I course, and they will not be covered. If you have difficulty, please let me know and I can help.
2. A chapter from notes of Dauchot and Demery.
3. Analyticity of cumulant generating function.

Second Lecture

Levy stable distributions,

Lectures

Relevant reading materials

1. Slides from A. Chechkin on Stable distributions.
2. An article on Levy flight.
3. An article on Levy stable distribution in stock price
4. Article diffusion on a comb network.
5. Article on Levy flight in Living polymer.
6. Article on Intesteller scintillation
7. Article on Levy flight in Lorentz gas
8. Article on flight of an Albatros

Third Lecture

Renormalization group approach for Stable distributions,

Lectures

Relevant reading materials

1. An article about renormalization group approach for stable distributions.
2. Second article about RG for stable distributions.
3. Third article on RG for stable distributions.

Fourth Lecture

Large deviations,

Lectures

Relevant reading materials

1. An article about Legendre transformation.

Fifth Lecture

Extreme value statistics,

Lectures

Relevant reading materials

1. A lecture note by Sanjib Sabhapandit on extreme value and record statistics.

Sixth Lecture

RG approach for universal extreme value statistics,

Lectures

Relevant reading materials

1. An article on RG for extreme value statistics.
2. A second article on RG for extreme value statistics.

Seventh Lecture

Stochastic processes, Markov chains,

Lectures

Relevant reading materials

1. An article on Perron-Frobenius theorem and phase transitions.

Eigth Lecture

Continuous time Markov processes,

Lectures

Relevant reading materials

1. Kolmogorov criteria .

Ninth Lecture

Time reversed process and Entropy,

Lectures

Relevant reading materials

1. Self adjoint operators in Quantum mechanics. .
2. Second article on self adjoint operators. .

Tenth Lecture

FokkerPlanck to Schrodinger equation,

Lectures

Relevant reading materials

1. Perrin's Nobel prize citation. .
2. Atomistic world and Perrin's work. .
3. Criteria for bound states in Schrodinger equation. .

Eleventh Lecture

Langeving to Fokker Planck,

Lectures

Relevant reading materials

1. A very readable note on stochastic processes .

Tweelvth Lecture

Path integrals for stochastic processes,

Lectures

Thirteenth Lecture

Properties of Brownian motion, Brownian functionals,

Lectures

Relevant reading materials

1. A Note by David Tong .

Fourteenth Lecture

Feynman-Kac formula, Kramers escape problem,

Lectures

Relevant reading materials

1. Cramers escape problem for a non-Markovian process .

Fifteenth Lecture

Kramers escape problem using path integrals,

Lectures

Relevant reading materials

1. Cramers escape problem for a non-Markovian process .

Sixteenth Lecture

Time reversibility and fluctuation theorems,

Lectures

Relevant reading materials

1. Review article by Seifert on fluctuation theorems .

Seventeenth Lecture

Jarzynski work relation, Entropy production, Gallavotti-Cohen relation,

Lectures

Relevant reading materials

1. A very good article on Entropy, Fluctuation theorems, time reversibility, etc. .
2. A short presentation about fluctuation theorems. .
3. An experimenal verification of fluctuation theorems. .
4. Entropy as a measure of distance from equilibrium. .
5. Basics of Entropy. .

Eighteenth Lecture

Few basic topics of equilibrium statistical mechanics,

Lectures

Relevant reading materials

1. Generalized Gibbs Ensemble .

Nineteenth Lecture

Yang-Lee zeros, transfer matrix

Lectures

Relevant reading materials

1. An article about Yang-Lee zeros for equilibrium and theri non-equilibrium extensions.
2. A more detailed article about Yang-Lee zeros.

Twentieth Lecture

Ising model on Bethe lattice,

Lectures

Relevant reading materials

1. A chapter from the book of Baxter.

Twenty first Lecture

Thermodynamic stability and Landau Peierls argument,

Lectures

Relevant reading materials

1. Thermodynamic stability .
2. Peierls-Griffiths argument .

Twenty second Lecture

Well-known Models in equilibrium statistical mechanics,

Lectures

Relevant reading materials

1. A note on method of steepest descent .

Twenty third Lecture

Gaussian model, Spherical model, Graphical enumeration methods,

Lectures

Relevant reading materials

1. An article by Deepak Dhar.

Twenty fourth Lecture

Duality of statistical mechanics models, XY and SOS model, Potts model and Percolation.

Lectures

Relevant reading materials

Twenty fifth Lecture

Dimer model and solution of 2d Ising model,

Lectures

Relevant reading materials

Twenty sixth Lecture

Solution of 2d Ising model by fermionization,

Lectures

Relevant reading materials

1. Article by Schultz, Mattis, and Lieb.
2. Chapter of a book by Plischke and Bergersen.

Twenty seventh Lecture

Ginzburg Criteria, Goldstone mode, and Mermin-Wagner theorem,

Lectures

Relevant reading materials

1. Limitations of Mermin-Wagner theorem.

Twenty eigth Lecture

KT-transition and 2d melting,

Lectures

Relevant reading materials

1. More details on 2d melting. See section 1.5 of this thesis. Link to download is on the top right corner of the page.

Twenty ninth Lecture

Real space RG for Ising model,

Lectures

Relevant reading materials

Thirteath Lecture

Niemeijer van-Leewen RG for triangular lattice Ising model,

Lectures

Relevant reading materials

Thirty first Lecture

Widom scaling theory,

Lectures

Relevant reading materials

Thirty second Lecture

Basics of RG,

Lectures

Relevant reading materials

1. An article on importance of RG.
2. An article by Weinberg on importance of RG.

Thirty third Lecture

Momentum space RG for Gaussian model,,

Lectures

Relevant reading materials

Thirty fourth Lecture

RG for phy4 model and epsilon expansion,

Lectures

Relevant reading materials

1. Review artile by Wilson and Kogut.
2. Lecture note by David Tong.

Thirty fifth Lecture

Continuation on epsilon expansion,

Lectures

Relevant reading materials

1. A small note on Borel summation.

Thirty sixth Lecture

RG for O(n) model and KT transition,

Lectures

Relevant reading materials