Program of the main courses
Coagulation-Fragmentation Models
Klemens Fellner (Graz, Austria)
Abstract
Coagulation-Fragmentation models describe the formation and break-up of clusters/polymers
in such diverse areas of applications as chemistry of polymers, blood clotting in biology,
formation of aerosols and sprays in physics, development of raindrops and smoke,
formation of galaxies in astrophysics e.t.c. Going back to the pioneering works of
Smoluchowski in the early 19th century, we distinguish discrete and continuous
coagulation-fragmentation models describing the size of clusters either by an integer
or by a real, positive number.
This course wants to constitutes a introduction in the modelling and the mathematical
analysis of coagulation-fragmentation models. It shall show that some coagulation
processes are expected (and intended) to violate the formal conservation of mass
by yielding the formation of clusters of infinite size: a phenomenon which is
called gelation. We shall see that gelation occurs also in Coagulation-Fragmentation
models and consider the discrete Becker-Döring model for an asymptotic analysis
of the large-time behaviour. Next we shall consider spatial inhomogeneous
coagulation-fragmentation models with diffusion.
We shall present an elegant way for proving existence of solutions
in a natural L^1 setting as done by Laurencot and Mischler and compare
this approach with existence of solutions via duality arguments.
Moreover, we shall explore the decay of entropy by means of a so-called
entropy entropy dissipation estimate and quantify rates of convergence towards equilibrium.
Finally we study the fast reaction limit.
Outline
1. Modelling: Derivation of Coagulation-Fragmentation models
1.1. Some kernels modelling arising from different application backgrounds
2. The Smoluchowski equation: a model for coagulation
2.1. Formal properties
2.2. An example featuring gelation
3. Coagulation-Fragmentation Models
3.1. The Becker-Döring model
3.2. Saturation phenomena and large-time asymptotics
4. Inhomogeneous Coagulation-Fragmentaion models with diffusion
4.1. An excursion to systems of reaction-diffusion equations
4.2. The existence theory of Laurencot, Mischler
4.3. An duality argument
4.4. An entropy method and explicit large-time behaviour
4.5. The fast-reaction limit
Asymptotic-Preserving schemes for the numerical simulation of magnetically confined plasmas
Claudia Negulescu (Toulouse, France)
Abstract
The central theme of this presentation is the mathematical modelling, analysis and numerical simulation of magnetically confined plasmas. Magnetically confined plasmas are charged particles evolving in a strong electromagnetic field. The evolution of this system of particles can be described either by kinetic or fluid models. The mathematical as well as numerical difficulties one has to handle with when treating plasmas, are the multiple scales occurring in their evolution.
The aim of this course shall be to introduce and analyse so-called Asymptotic-Preserving numerical schemes for the resolution of the plasma evolution, which have the essential advantage of being equally precise in both eps=O(1) and eps<<1 regimes. The perturbation parameter eps occurring in the different equations we shall treat, is a ratio of characteristic lengths and describes the different time or space scales of the problem.
Outline
1.Introduction/Motivation
1.1. Magnetically confined plasma physics
1.2. Mathematical description of plasmas via kinetic or fluid models
1.3. Multiple scale problems
1.4. Asymptotic-Preserving schemes
2.Vlasov equation in the drift-diffusion scaling
Perturbation parameter stands for the mean free path (Knudsen number)
2.1. Micro-macro reformulation
2.2. Construction of the AP-scheme
3.Vlasov-Poisson system in the quasi-neutral limit
Perturbation parameter stands for the rescaled Debye-length
3.1. Determination of the Limit problem
3.2. Construction of the AP-scheme based on a reformulation of the Poisson equation
4.Vlasov equation in the high field limit
Perturbation parameter stands for the rescaled gyro-period
4.1. Asymptotical analysis in order to get the guiding-centre approximation
4.2. Construction of the AP-scheme based on a micro-macro reformulation
5.Highly anisotropic elliptic equations
Anisotropy comes from the strong magnetic fields occurring in the plasma
5.1. Construction of different AP-schemes
5.2. Numerical analysis
6.Euler-Lorentz system in the high-field limit
Perturbation parameter stands for the rescaled gyro-period as well as rescaled Mach number
6.1. Construction of an AP-scheme
Statistical mechanics and dynamics of long-range interacting systems
Stefano Ruffo (Firenze, Italy)
Abstract
An account of the recent developments in the study of the microscopic and
macroscopic behavior of systems, for which the interactions decay so slowly
with the distance that additivity is violated, will be given.
This implies that the standard thermodynamic limit cannot be straightforwardly
applied and one is forced to resort to a different scaling of thermodynamic
potentials with volume and particle number. Both equilibrium and out-of-equilibrium
properties will be discussed. As for the former the main concept is the one of
ensemble equivalence (e.g. microcanonical and canonical), while concerning the
latter in somewhat detail the process of slow relaxation towards equilibrium,
implying a time scale which diverges with system size, will be discussed.
Outline
1. Extensivity and additivity in statistical mechanics. Definition of long-range
and weak long-range systems. Convexity in the space of macroscopic parameters
and broken ergodicity. An overview of physical systems with long-range interactions:
gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity,
Coulomb systems, dipolar systems, systems of finite size.
2. Equilibrium statistical mechanics and ensemble equivalence for short-range systems.
Phase separation and Maxwell's construction. Concavity/Convexity of thermodynamic potentials.
Ensemble unequivalence: negative specific heat and temperature jumps.
Example: the Blume-Emery-Griffiths spin model, canonical vs. microcanonical solution.
Negative susceptibilities. The Hamiltonian Mean Field(HMF) model and its generalizations.
3. Large deviations for equilibrium statistical mechanics. Computing entropies as
large deviations rate functions. An illustration of the method for the three-state mean-field
Potts model: the appearance of convex entropy and negative specific heat.
Dealing with continuous microscopic variables when computing entropies.
A generalized HMF model that shows ergodicity breaking.
The mean-field phi^4 model and negative susceptibilities.
The Colson-Bonifacio model for the free-electron laser.
The min-max method: relating free energies to entropies.
4. Violent relaxation in long-range systems. Quasistationary states.
How to interpret quasi-stationary states using kinetic equations
(Klimontovich, Vlasov and Lenard-Balescu). Stationary stable solutions of the Vlasov
equation for the HMF model. Linearized Vlasov equation and Landau damping for the HMF model.
Using the linearized Vlasov equation to derive a Fokker-Planck equation that describes anomalous diffusion. Kubo linear response theory of quasi-stationary states.
5. Lynden-Bell's theory of violent relaxation for gravitational dynamics. Application of Lynden-Bell theory to the HMF model and to the Colson-Bonifacio model of the free electron laser. Phase transitions out of equilibrium. The core-halo theory of Levin.BGK modes and self-consistent steady states. The Miller-Robert-Sommeria theory for the evolution of vorticity in the 2D Euler equation. Applications to geophysical flows.
6. Decaying interactions: the alpha-HMF model and the Ising-Dyson model. Systems with both short and long-range interactions.The transfer integral method. The Vlasov equation for systems defined on lattices. One-dimensional gravitational models. Dipolar interactions.