See the general timetable for the dates and times of the various lectures.
- oscillators, phase and amplitudes
- isochrons and phase response curve
- phase dynamics under small forcing
- phase locking and frequency entrainment
- beyond phase approximation
- effects of noise
- phase and its synchronization for chaotic oscillators
- Kuramoto model, self-consistent solution
- Watanabe-Strogatz and Ott-Antonsen theories
- effects of nonlinear coupling and partial synchronization
- multifrequency ensembles
- effects of multiharmonic coupling
- compactons at Hamiltonian coupling
- long-term coupling and chimera states
4) Synchronization by common noise
- elementary theory
- interplay of common noise and coupling
We consider resetting a stochastic process by returning to the initial condition with a fixed rate. Resetting is a simple way of generating a nonequilibrium stationary state in the sense that the process is held away from any equilibrium state and a non-vanishing steady-state probability current is directed towards the resetting position. The nature and properties of nonequilibrium stationary state are questions of fundamental importance within statistical physics. Thus, the resetting paradigm provides a convenient framework within which to study such nonequilibrium properties.
In these lecture we shall study some interesting properties of the resetting paradigm focussing mainly on the case of diffusion with stochastic resetting
Lecture I: Diffusion with Stochastic resetting
Motivation from search strategies; stationary state; survival probability with absorbing target; mean first passage time; renewal process and extreme value statistics interpretation.
Lecture II: Many particle system
Average and typical survival probabilities of a diffusing target; Relaxation dynamics and equilibration front.
General resetting-induced nonequilibrium stationary states; Generalised resetting and Optimisation Problems e.g. minimisation of mean time to locate a target; non Markovian resetting; open problems
In the first lecture I will discuss the notion of hydrodynamic limit and as example we will present the proof for the symmetric simple exclusion process in contact with slow or fast reservoirs. In this process, particles jump in the bulk to one of its nearest neighbor sites with rate 1/2. At the boundaries particles are injected or removed at a rate which depends on a parameter that can be chosen in such a way that the boundary dynamics is either slow or fast. We will explore the hydrodynamic limit scenario and we will see a phase transition for the heat equation with different types of boundary conditions. This lecture is based on the article at arxiv.1407.7918.
In the second lecture we will consider a model similar to the one of the first lecture but in which particles can jump in the bulk according to a symmetric transition probability rate which allows long jumps, but has finite variance. Moreover, infinitely many reservoirs are added at each end point of the bulk and now particles can be injected or removed from any site of the bulk. As in the previous lecture, the rates of injection and removal depend on a parameter which turn the boundary dynamics slowed of fast depending on the range of this parameter. We will analyze the hydrodynamic limit and we will see a phase transition of a reaction diffusion equation with different types of boundary conditions. This lecture is based on the article arxiv.1702.07216.
In the third and last lecture we will analyze the model of the previous lecture, for a certain choice of the transition probability rate which has infinite variance and in the hydrodynamic limit we will get a fractional heat equation with Dirichlet boundary conditions written in terms of the regional fractional Laplacian.
I will discuss how noise can arise in deterministic systems with strong instability with respect to the initial conditions. Starting with a discussion of the Central limit theorem for deterministic systems with random initial conditions and trying to get to the derivation of stochastic processes as limit theorems.
Lesson 1 - Chaos and randomness
I will describe how establish a CLT or an invariance principle for observables of a deterministic system. To avoid technical details I will describe various approaches in the simplest possible case: smooth expanding maps.
Lesson 2 - Gaussian or not Gaussian
I will discuss the large deviations for a deterministic system and discuss (again in the simplest possible setting) how to compute the rate function and its relations with the rate function of a Gaussian process. Also I will briefly comment on the class of systems to which the previous results can be generalised.
Lesson 3 - Non-equilibrium
I will discuss (deterministic) systems with a locally (almost) conserved quantity, explain how averaging theory applies and discuss how to prove that the evolution of the almost conserved quantity might be very close to the solution of an SDE for a very long time.
The lectures will discuss the following topics:
- Scalar Conservation Laws and theirs Markovian solutions
- Conservation laws with stochastic external force
- Hamilton-Jacobi PDE, Hamiltonian ODEs and Mather Theory
- Homogenization for Hamilton-Jacobi Equations and Stochastic Growth
When a large deviation result is proved there is some topology involved in the statement because it affects the class of sets for which the estimates hold. Often the choice is natural and obvious. There may be a stronger topology in which the theory holds. This requires additional work and we may or may not be inclined to do it. However the application we have in mind might require it, in which case we have no choice. It is also possible that the the principle fails in the familiar strong topology and the weak topology is not sufficient to prove the result. Then we have to invent a new topology. We will look at some examples to illustrate these points.
Laurent Desvillettes (Univ. Paris Diderot) [videos: first lecture]
Balanced chemical reaction networks and reaction-diffusion systems
Chemical reaction networks appear in many industrial devices and natural processes (combustion, photosynthesis, etc.). When the spatial structure is not taken into account, they give rise to complex system of ODEs with polynomial terms, and a lot of results were obtained lately in the study of the large time behavior for the solutions of those systems.
When the chemical species are supposed to be traces in a solvent, they are diffusing (each with its diffusion rate), so that their concentrations are solution to a system of reaction-diffusion with polynomial reaction rates. The specificity of such systems is, at least for a large class of them (called systems with balanced equilibria) the existence of a Lyapunov functional related to the physical entropy.
Recently, the systems of reaction-diffusion coming out of chemical networks with balanced equilibria have attracted a lot of attention. The study of existence of weak and strong solutions to those systems uses a lot of subtle tools of analysis: renormalized solutions, duality methods, etc. Their large time behavior is also an interesting issue. It is possible to develop entropy methods and convexity arguments which enable to show convergence to equilibrium for some systems, but many interesting systems are still puzzling.
In this series of lectures, we shall present (using specific examples) the main features of reaction-diffusion systems related to chemical networks, and explain how to get typical existence results as well as large time behavior results.
Stefano Olla (doctoral course)
Thermodynamics from Scaling Limits: equilibrium and non equilibrium [slides]
This course is an elementary introduction to Thermodynamics and Statistical Mechanics. The aim is to explain the main ideas in few concrete models, more than constructing a general theory.
In the first part of the course, the `principles of thermodynamics' are introduced, and it is explained how these principles can be deduced from the microscopic dynamics of molecules through space-time macroscopic limits. Thermodynamics is obtained as macroscopic theory, i.e., valid on macroscopic space-time scales, while statistical mechanics provides the microscopic model. This means that the objects of thermodynamics are those macroscopic complex systems that satisfy the thermodynamic principles, while statistical mechanics explains how this complexity arise from dynamics of systems with a very large number of components. The central point of this connection is the identification of the thermodynamic entropy, a function of the thermodynamic macroscopic equilibrium states, introduced by Clausius using Carnot cycles, with the Boltzmann definition of entropy in statistical mechanics, as logarithm of the number of microstates corresponding to the given macroscopic state. Boltzmann and Planck discovered this identification at the end of 19th century, but in order to understand it, we need to obtain the thermodynamic transformations constituting the Carnot cycle, isothermal and adiabatic, from the microscopic dynamics through a scaling limit procedure. This aspect differentiates this course from more classical courses in statistical physics, where only the equilibrium properties of the systems are studied.
As classical thermodynamics concerns transformations from an equilibrium state to another, in the second part of the course I will illustrate how some ideas generalize to transitions between non-equilibrium stationary states.