**Course 1: Statistical physics of Turbulence — Jérémie Bec**

__Outline: __These lectures aim at familiarising students with the statistical approaches used for the analysis of turbulent flows or of processes taking place in a turbulent environment. The objective is to provide the required background and tools to develop phenomenological approaches and to identify their limits and possible scientific locks.

__Content:__

- Lecture 1: Introduction to turbulence; Phenomenology of turbulent flows; Scales and cascades.

- Lecture 2: Statistical description of homogeneous, isotropic turbulence; Exact results and scaling laws; Intermittency and anomalous scaling.

- Lecture 3: Turbulent transport; Advection of passive scalars and Lagrangian turbulence; Relative dispersion, Lagrangian spontaneous stochasticity and statistical models

- Lecture 4: Exact results in pressure-less Burgers turbulence; Tight relations between singularities, anomalies, scaling, and mass transport..

- Lecture 5: Beyond classical approaches: Statistical mechanics; Variational formulations; Eulerian spontaneous stochasticity.

__Reading material:__

- Turbulence: The Legacy of A.N. Kolmogorov, U Frisch - Cambridge University Press, 1995

- Physics of Turbulence, M K Verma - IIT Kanpur (https://archive.nptel.ac.in/courses/115/104/115104112/)

- Lessons from hydrodynamic turbulence, G Falkovich, KR Sreenivasan - Physics Today 2006 (https://web.archive.org/web/20170809012224id_/http://users.ictp.it/~krs/pdf/2006_008.pdf)

- Turbulence Theory I-II, G Eyink - Johns Hopkins University 2019 (https://www.ams.jhu.edu/~eyink/Turbulence/notes.html)

- Burgers turbulence, J Bec, K Khanin - Phys. Rep. 2007 (https://arxiv.org/abs/0704.1611)

- Non-equilibrium statistical mechanics and turbulence, J Cardy, G Falkovich, K Gawedzki - Cambridge University Press, 2008 (https://web.archive.org/web/20141222011138id_/http://perso.crans.org:80/lecomtev/articles/John-Cardy_Gregory Falkovich_Krzysztof-Gawedzki_Sergey-Nazarenko_Oleg-Zaboronski_Non-equilibrium_Statistical_Mechanics_and_Turbulence.pdf)

**Course 2: Chandan Dasgupta **

**Course 3: Stochastic chemical reaction networks — Supriya Krishnamurthy**

We are often confronted with chemical reactions in chemistry and biology but several familiar models in physics, such as the zero-range process, can also be modeled as chemical reactions or networks of chemical reactions (CRNs). Mathematical models of CRNs either focus on deterministic models via rate equations (systems of nonlinear ordinary differential equations) which quantify how the concentrations of the different species change in time, or as stochastic processes modeled by Master equations.

One of the major theorems pertaining to deterministic models of CRNs is the **deficiency zero theorem** of Feinberg [1]** **which lays out conditions (related to a quantity called deficiency) for when a set of coupled non-linear ODE's can have a unique non-zero solution. The surprising aspect of this theorem is that the conditions to be satisfied are completely related to the network itself, while the implications are for the dynamics. In this set of lectures, we will first begin, after introducing CRN's and the terminology, with explaining the content of this theorem. After this, we will discuss the more recent result of Anderson, Craciun and Kurtz [2], who showed that if the deterministically modeled system satisfies the deficiency zero theorem, then the stochastically modeled version has product-form stationary solutions. After this, time permitting, we can discuss some results and techniques for CRN's which do not have zero deficiency** **[5].

For understanding the deficiency zero theorem, we will mainly use reference [3] , chapters 16-23. This book also has a number of relevant references including [4]. For understanding the stochastic version, we will use [2,3].

**References:**

**[1] **Martin Feinberg, Lectures on Chemical reaction networks, available for download **at https://cbe.osu.edu/chemical-reaction-network-theory**

**[2]**David F. Anderson, Gheorghe Craciun and Thomas G. Kurtz, *Bull. Math. Biol.* ** 72 ,**1947 (2010).

**[3] **John Baez and Jacob D. Biamonte , *Quantum Techniques in Stochastic Mechanics*, World Scientific (2018), also available here** https://arxiv.org/pdf/1209.3632.pdf**

**[4] **Jeremy Gunawardena** **(2003) **, **Chemical reaction network theory for *in-silico* biologists, lecture notes **http://vcp.med.harvard.edu/papers/crnt.pdf**

[5] Supriya Krishnamurthy and Eric Smith, J. Phys. A: Math. Theor. 50, 425002 (2017)

__Lecture 1:__

__Lecture 2:__

__Lecture 3:__

__Lecture 4:__

__Lecture 5:__

**Course 4: Pattern formation in Biology — ****Vijaykumar Krishnamurthy **

Outline of lectures:

Introduction, Tools for pattern formation, Reaction-diffusion patterns, Pattern formation in active materials, Morphogenetic patterns

Reference materials:

1. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation

2. How Active Mechanics and Regulatory Biochemistry Combine to Form Patterns in Development

4. Pattern Formation and Dynamics in Nonequilibrium Systems

**Course 5: Praneeth Netrapalli **

__References:__

- Lectures on convex optimization by Yurii Nesterov
- Convex optimization: algorithms and complexity by Sebastien Bubeck
- Nonconvex optimization for machine learning by Prateek Jain and Purushottam Kar
- Introduction to online convex optimization by Elad Hazan.

__Course 6 (a): Statistical physics of long-range systems — Stefano Ruffo,__

**Lecture 1**

Introduction. Definitions: Long-range interactions, Extensivity vs. additivity. The mean-field limit for long-range interactions: the Kac scaling. Strong and weak long-range interactions. Convexity and ergodicity breaking. Legendre-Fenchel transform and ensemble inequivalence.

**Tutorial**

The Curie-Weiss model. The Ising model in one dimension with long-range interactions. Legendre-

Fenchel transforms of convex and non-convex functions.

**Lecture 2**

The Blume-Capel model: ensemble inequivalence. Short and long-range interactions: the Kardar- Nagel model in the microcanonical ensemble. Broken ergodicity in the Kardar-Nagel model. Metastability and instability in the presence of long-range interactions.

**Tutorial**

Transfer matrix for the 1d Ising model and the Kardar-Nagel model in the canonical ensemble. The Creutz algorithm to simulate the microcanonical esemble. The min-max method to obtain microcanonical entropy.

**Lecture 3**

Entropy and free energy of systems with long range interactions using large deviations. Cramèr’s theorem. Solution of the mean-field Potts model in the microcanonical ensemble. Other models: XY model, free-electron laser, mean-field phi^4 model (negative susceptibility). Quasistationary states in the Hamiltonian Mean-Field (HMF) model.

**Tutorial**

The law of large numbers. Central limit theorem. Large deviations for coin tossing. Solution of the HMF model in the canonical and microcanonical ensembles. Klimontovich and Vlasov equations.

Suggested background readings

**Review papers and Lecture Courses:**

-A. Campa, T. Dauxois and S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions, Physics Reports, 480, 57 (2009).

-H. Touchette, The large deviation approach to statistical mechanics, Physics Reports, 478, 1 (2009)

-Long-range interacting systems, Les Houches Lecture Notes, Oxford (2009), T. Dauxois, L. Cugliandolo and S. Ruffo Eds.

A. Campa, T. Dauxois, D. Fanelli and S. Ruffo, Physics of Long Range Interacting Systems, Oxford University Press (2014).

F. Bouchet, S. Gupta and D. Mukamel, Thermodynamics and dynamics of systems with long-range interactions, Physica A, 389, 4389 (2010)

Y. Levin et al., Nonequilibrium statistical mechanics of systems with long-range interactions, Physcs Reports, 535, 1 (2014)

**Papers:**

M. Antoni and S. Ruffo, Clustering and relaxation in long range Hamiltonian dynamics, Phys. Rev. E, 52, 2361 (1995).

J. Barré, D. Mukamel and S. Ruffo, Inequivalence of ensembles in a system with long range interactions, Phys. Rev. Lett., 87, 030601, (2001).

Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois and S. Ruffo, Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model, Physica A, 337, 36 (2004).

D. Mukamel, S. Ruffo and N. Shreiber, Breaking of ergodicity and long relaxations times in systems with long-range interactions, Phys. Rev. Lett., 95, 240604 (2005).

F. Bouchet and J. Barré, Classification of phase transitions and ensemble inequivalence in systems with long-range interactions, J. Stat. Phys, 118, 1073 (2005).

I. Latella, A. Perez-Madrid, A. Campa, L. Casetti and S. Ruffo, Thermodynamics of nonadditive systems, Phys. Rev. Lett., 114, 230601 (2015).

H. Touchette, Equivalence and nonequivalence of ensembles: thermodynamic, macrostate and measure levels, J. Stat. Phys, 159, 987 (2015).

**Course 6 (b): Quantum Systems with Long-range interactions **__— __**Nicolò Defenu **

**Lecture 1**

Introduction and connection with experiments on quantum systems. The mean-field limit for long-range interactions in quantum systems: the Kac scaling. Ensemble inequivalence goes quantum.

**Tutorial**

Quasi-stationary states of long-range interacting quantum systems: Explicit calculation in the long-range Ising model.

**Lecture 2**

Quantum systems with weak long-range interactions. Universality and effective dimension approach. Holstein-Primakoff transformation and critical exponents. Jordan-Wigner transformation and the domain wall description.

**Tutorial**

Explicit calculation of the phase diagram of long-range theories. Long-range interacting quantum systems”, N. Defenu, T. Donner, T. Macrì, G. Pagano, S. Ruffo, A. Trombettoni; arXiv:2109.01063.

**Papers:**

"Criticality and phase diagram of quantum long-range O(N) models” N. Defenu, A. Trombettoni, S Ruffo; Phys. Rev. B 96 (10), 104432.

"Metastability and discrete spectrum of long-range systems” N. Defenu; Proc. Nat. Acad. Sci. (30) e2101785118 (2021).