From microscopic dynamics to continuum limits - ACDC Autumn School

On the 20th-24th of October, the Amsterdam Centre for Dynamics and Computation (ACDC) will host the Autumn School From microscopic dynamics to continuum limits. The school is aimed at Ph.D. students, postdocs and other early career researchers working in areas such as PDEs, Dynamical Systems, Mathematical Biology, and neighbouring fields of mathematics and computer science.

The event will provide tailor-made, hands-on lectures and tutorials for bridging scales from microscopic dynamical systems to continuum limits. The material will cover rigorous results, numerical methods, and applications for which rigorous theory is not yet available, while modelling or numerics provide an effective averaging.

Location

The event is hosted ACDC, and co-funded by NDNS+ and ACDC. It will take place in room NU-6A25 of the NU-VU building (see here for directions)

Confirmed speakers

Schedule

Time/Days Monday Tuesday Wednesday Thursday Friday
9:00-9:30 Welcome
9:30-10:30 B. Lods L1 P. Roux/S. Solem L1 Flash talks (TBA) S. Gomes L1 S. Henkes L1
10:45-11:45 B. Lods L2 P. Roux/S. Solem L2 (TBA) S. Gomes L2 S. Henkes L2
11:45-13:00 Lunch break Lunch break Lunch break Lunch break Lunch break
13:00-14:30 B. Lods L3 P. Roux/S. Solem L3 S. Gomes L3 S. Henkes L3
14:45-17:00 Exercises Exercises Exercises Exercises/Final Discussion (Roeland Merks)

Titles and abstracts of the lectures

Bertrand Lods

Title: Hydrodynamic Limits of the Boltzmann Equation: A Rigorous Derivation of the Navier-Stokes system

Abstract: This course provides a comprehensive introduction to the derivation of hydrodynamic equations from the Boltzmann equation, a central problem in kinetic theory. The lectures will cover the essential mathematical tools, including the functional framework, spectral theory, perturbation methods, and hypocoercivity techniques. We will study how the method can allow to derive the Navier-Stokes-Fourier system from the Boltzmann equation. We hope to provide the students with a deep understanding of the mathematical challenges involved in connecting microscopic and macroscopic descriptions of physical systems.

Susanne Solem and Pierre Roux

Title: Fokker-Planck representation of stochastic neural fields: derivation, analysis and application to grid cells.

Abstract: In neuroscience, noise is more and more investigated not as a bug, but as a feature and a key player in core brain mechanisms. Among other neural phenomena, the hexagonal firing pattern of grid cells, neurons in our internal navigational system, has been the subject of many modelling efforts in the past years. A key question is how noise affects these patterns. We will first explain how to construct a versatile model in the form of a non-local Fokker-Planck equation, that allows to investigate both pattern formation in neurons and the robustness to noise of such patterns. Then, we will explain how to provide a mathematical framework for the analysis of this class of equations: existence theory, local stability of stationary states and bifurcations of stationary states when noise levels change.

Susana Gomes

Title: Mean field limits for interacting particle systems, their inference, and applications

Abstract: In this course, I will discuss interacting particle systems – a mathematical framework used to model collective dynamics in several scenarios, from animal behaviour and cell population dynamics to human dynamics (such as the behaviour of pedestrians to evolution of opinions). Interacting particle systems can be modelled as agent-based models (Markov chains), or systems of Stochastic or Ordinary Differential Equations (S/ODEs) evolving in a confining potential, which can have multiple wells, and in the presence of noise. I will discuss connections between these models, as well as how to obtain the mean-field limit of these systems, and use this to analyse its longtime behaviour. I will then move on to discuss inference for this type of problems, in particular parameter estimation for coefficients of interest. The latter part of the course will focus on specific applications in life and social sciences, where I will discuss modelling, control, data availability, and inference. This will include a framework to estimate parameters on SDEs and PDEs for a population using data from individual trajectories (rather than aggregate data such as density), as well as controlling opinion dynamics by controlling a (co-evolving) social network.

Title: Models of active materials and tissues

Abstract: Active or living materials are made out of agents that can move on their own, such as birds, fish, or artificial systems such as robots or active colloids. Cells are living materials, and the two-dimensional tissues known as epithelial cell sheets have a fundamental role in the developing embryo, and also in adult tissues including the gut and the cornea of the eye. Soft active matter provides a theoretical and computational framework to understand the mechanics and dynamics of these tissues.

I will give an introduction and overview, starting with the simplest model, active Brownian particles (ABPs), which move according to overdamped Langevin dynamics with the activity modelled by internal driving with rotational diffusion. This model is partially solvable, and I will introduce its phenomenology at the single particle level, in the phase-separated MIPS phase and in the active solid phase. The latter includes emergent spatiotemporal correlations, as observed in the cell sheets. This part will include a computational session to write and demonstrate a simulation of ABPs.

I will then introduce Vertex models where the two dimensional cell sheet is mapped to a tesselation, with each cell represented by a polygon. On this planar graph, one then writes an energy as a function of cell areas and perimeters. These models include a rigidity transition as a function of a shape parameter, which also controls the solid to liquid transition when driven by ABP dynamics.

I will then finish by discussing how to introduce mechanochemical feedback, where the activity of the cells couples back to its internal mechanical state, with convergence-extension flows as an example.

Organizers