ScopeThe workshop aims to present and facilitate discussion of approaches to systems of many particles, which at some level of modelling undergo spatial motion and stochastically interact when they collide or at least get very close. Classic applications of such systems include gas dynamics, particle coagulation and chemical reactions, but zoological and other application areas will also be considered. Alongside strong law of large numbers type results the workshop will showcase methods for obtaining further information to complement a characteristic limiting equation.
University of Cambridge, UK
Wasserstein convergence for the Kac process
The lectures will review the use of martingale estimates to give quantitative approximations for large mean-field systems of interacting particles by differential equations.
I will then discuss some ideas which proved useful in a particular model, namely the Kac process for velocities in a dilute gas of hard spheres. These ideas include:
- the use of signed branching processes as a linearization
- the possibility to estimate martingale random measures in Wasserstein distance
- chaining arguments to optimize convergence rates for Wasserstein.
University of Warwick, UK
Functional Central Limit Theorems for Interacting Particle Systems
We present in some detail a (mainly) analytic approach to the study of the dynamic law of large numbers and CLT (central limit theorem) that allows one to get rather precise rates of convergence. We shall pay main attention to the jump-type models of interacting particles. We shall start with the discussion of bounded rates and then consider models of coagulations and collisions under the standard assumptions. In particular, we present the CLT for the Smoluchovskii model solving a problem posed in Aldous' review paper of 1999. Main ideas will be taken from the author's book 'Nonlinear Markov processes', but presented with many improvements.
Université Paris-Dauphine & IUF, France
Quantitative, qualitative and uniform in time propagation of chaos for McKean-Vlasov and Boltzmann models
I will review some old and recent papers on the propagation of chaos issue. I will discuss the different methods and how they may answer the three fundamental problems which are to get quantitative, qualitative and uniform in time propagation of chaos. My talk will be illustrated by application to (possibly singular) McKean-Vlasov and Boltzmann models.
Universidad del País Vasco, Bilbao, Spain
A BBGKY derivation of the coagulation equation with constant kernel
A deterministic coalescing dynamics with constant rate for a particle system in a finite volume with a fixed initial number of particles is considered. It is shown that, in the thermodynamic limit, with the constraint of fixed density, the corresponding coagulation equation is recovered and global in time propagation of chaos holds.
Freie Universität, Berlin, Germany
Modelling Cellular Reaction-Diffusion Kinetics
Accurate modeling of reaction kinetics is important for understanding the functionality of biological cells. Depending on the particle concentrations and on the relation between particle mobility and reaction rate constants, different mathematical models are appropriate. In the limit of slow diffusion and small concentrations, both discrete particle numbers and spatial inhomogeneities must be taken into account. The most detailed model consists of particle-based reaction-diffusion dynamics, where all individual particles are explicitly resolved in time and space, and particle positions are propagated by diffusion equations, and reaction events may occur only when reactive species are adjacent.
For rapid diffusion or large concentrations, the model may be coarse-grained in different ways. Rapid diffusion leads to mixing and implies that spatial resolution is not needed below a certain lengthscale. This permits the system to be modeled via a spatiotemporal chemical Master equation (STCME), i.e. a coupled set of chemical Master equations acting on spatial subvolumes. The talk will discuss these different models; in particular, we will see how the STCME description can be derived from particle-based reaction-diffusion dynamics.
Joint work with Stefanie Winkelmann (FU Berlin)
Technische Universität München, Germany
Coagulation processes with gelling solutions in the gas dynamics of Kac
We consider the classical mean-field model of Kac for collisional kinetic theory. We give a simple notion of cluster as group of particles connected by a chain of interactions. The evolution of clusters of different sizes is driven by a system of coagulation equations where a gelation process is coupled with nontrivial energy transfer. We describe several properties of the solutions to the limiting equations for some specific choice of the interaction kernel.
Universität Zürich, Switzerland
Probabilistic aspects of critical growth-fragmentation equations
(Based on a joint work with Alex Waston). Growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered by Doumic and Escobedo in the homogeneous setting where the rates do not depend on the particle size. Here, we study the general self-similar setting, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous setting, the equation in the non-homogeneous setting has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or +∞, we exhibit unexpected spontaneous generation of mass in the solutions.
Paul Gajewski & Mario Maurelli
WIAS Berlin, Germany
Theory of Many-Particle Electrodes
The smart design of many-particle electrodes of rechargeable lithium-ion batteries is crucial to reduce charging times without decreasing the storage capacity. We describe the storage of lithium in a many-particle electrode consisting of carbon coated storage particles of different size in the range of nano-meters. The particles are in contact to each other and to an electrolyte. On the carbon coated particle surfaces we have adsorption and electron transfer reactions. We show that adsorption can be described by a stochastic process compatible to the second law of thermodynamics. Thus the storage process may be modeled by a system of weakly coupled SDEs. The coupling is induced by an external constraint. The lecture discusses aspects of modeling, analysis and numerical simulations as well.
Uniwersytet Śląski, Katowice, Poland
Particle coagulation-fragmentations in population models. A model of phytoplankton
Populational models of biology differ from physical ones in some points and the most important one is the following: there is no mass conservation - we have always growth/birth and death processes. Phytoplankton, as the first level of food accessible to animals, is the main source of nutrient in the ocean. That is why understanding of its behavior becomes so important. The aim of this talk is to present a model of spatial phytoplankton dynamics that uses stochastic particle systems.
The focus is on the aggregative behavior: phytoplankton cells have the ability of forming aggregates - groups of cells stuck together. The aggregates grow, undergo the diffusion and fragmentation and may coagulate. The microscopic, stochastic dynamics of plankton aggregates will be described by a measure valued stochastic process defined by means of some martingale problem. The convergence of a microscopic model to the macroscopic one, given by an evolution equation, will be shown.
WIAS, Berlin, Germany
The Fokker-Planck and Liouville equations for chemical reactions as large-volume approximations of the Chemical Master Equation
We discuss the reaction kinetics according to a finite set of mass-action type reactions under the additional assumption of detailed balance. This nonlinear ODE system has an entropic gradient structure. The associated Chemical Master Equation is the Kolmogorov forward equation of the Markov process counting the number of particles for each species, where a reaction corresponds to a jump in the discrete state space ℕ0I. We show that this Kolmogorov forward equation has again an entropic gradient structure on the set of probability measures.
Scaling the number of particles by the total volume V, we show evolutionary Γ-convergence for V → ∞ of the discrete Markov process to the continuous Liouville equation which is a gradient flow with respect to a Wasserstein-type metric. We discuss the role of the Fokker-Planck equation as a singularly perturbed Liouville equation with better approximation properties.
This is joint work with Jan Maas, IST Wien.
|W 16-12||T 17-12||F 18-12|
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LocationErhard-Schmidt lecture room at Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany.
(Childcare facilities can be made available to participants, please contact the organisers)
RegistrationTo register for the workshop, please send an email with your details and affiliation to:
Scaling Cascades in Complex Systems