**Aspects of Intradisciplinary Research in Physically, Biologically and Chemically Motivated Equations**

25 March 2022

About

On the 25th of March 2022 we will be hosting a one-day meeting revolving around mathematical aspects of interacting systems. Such systems are at the heart of many (if not most) equations and models that pertain to physical, biological and chemical phenomena.

This meeting is supported by the London Mathematical Society (via Scheme 9 grants - Celebrating New Appointments) and the Department of Mathematical Sciences at Durham University.

The current planned format of the meeting is *hybrid* - The meting will be streamed on Zoom to allow participation without physical attendance. The format of the meeting, however, is subject to any potential changes to Government and University regulations regarding the ongoing coronavirus pandemic. We will keep all registered participants informed on the status of the event via email.

If you wish to attend this meeting then please complete the registration form here by Friday 11 March 2022. Please use a university-affiliated email address where possible.

Participants are required to take care of their own travel and accommodation arrangements.

Speakers

- José A. Carrillo, University of Oxford.
- Amit Einav, Durham University.
- Clément Mouhot, University of Cambridge.
- Michela Ottobre , Heriot Watt University.

Provisional Schedule

We plan to hold the meeting in the Scott Logic Lecture Theatre (MCS0001) in the Department of Mathematical Sciences. In case you have been to Durham University in previous years, note that we have now moved into a new building (together with Computer Science). We are now on Upper Mountjoy. You find the building on the map here. In the mornings and evenings there is a bus connection up to Upper Mountjoy, see the information and timetable here. The Scott Logic Lecture Theatre is located next to the main entrance on floor 0.

Schedule:

09:00 - 09:50: Arrival

09:50 - 10:00: Welcome

10:00 - 11:00: Talk by Clément Mouhot - * Quantitative Hydrodynamic Limits of Stochastic Lattice Systems *

11:00 - 12:00: Talk by Amit Einav - * Interacting Systems – The Melting Pot of Analysis, PDEs and Probability *

12:00 - 14:00: Lunch break

14:00 - 15:00: Talk by José A. Carrillo - * The Landau Equation: Particle Methods & Gradient Flow Structure *

15:00 - 16:00: Talk by Michela Ottobre - * Non Mean-Field Vicsek Type Models for Collective Behaviour *

16:00 - 17:30: Reception

Abstracts

Clément Mouhot

Title: * Quantitative Hydrodynamic Limits of Stochastic Lattice Systems *

Abstract: In this talk, I will present a simple abstract quantitative method for proving the hydrodynamic limit of interacting particle systems on a lattice, both in the hyperbolic and parabolic scaling. In the latter case, the convergence rate is uniform in time. This "consistency-stability" approach combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates à la Kruzhkov in weak distance, and consistency estimates exploiting the regularity of the limit solution. It avoids the use of “block estimates” and is self-contained. We apply it to the simple exclusion process, the zero range process, and the Ginzburg-Landau process with Kawasaki dynamics. This is a joint work with Daniel Marahrens and Angeliki Menegaki (IHES).

José A. Carrillo

Title: * The Landau Equation: Particle Methods & Gradient Flow Structure *

Abstract: The Landau equation introduced by Landau in the 1930’s is an important partial differential equation in kinetic theory. It gives a description of colliding particles in plasma physics, and it can be formally derived as a limit of the Boltzmann equation where grazing collisions are dominant. The purpose of this talk is to propose a new perspective inspired from gradient flows for weak solutions of the Landau equation, which is in analogy with the relationship of the heat equation and the 2-Wasserstein metric gradient flow of the Boltzmann entropy. Moreover, we aim at using this interpretation to derive a deterministic particle method to solve efficiently the Landau equation. Our deterministic particle scheme preserves all the conserved quantities at the semidiscrete level for the regularized Landau equation and that is entropy decreasing. We will illustrate the performance of these schemes with efficient computations using treecode approaches borrowed from multipole expansion methods for the 3D relevant Coulomb case. From the theoretical viewpoint, we use the theory of metric measure spaces for the Landau equation by introducing a bespoke Landau distance dL. Moreover, we show for a regularized version of the Landau equation that we can construct gradient flow solutions, curves of maximal slope, via the corresponding variational scheme. The main result obtained for the Landau equation shows that the chain rule can be rigorously proved for the grazing continuity equation, this implies that H-solutions with certain apriori estimates on moments and entropy dissipation are equivalent to gradient flow solutions of the Landau equation. We crucially make use of estimates on Fisher information-like quantities in terms of the Landau entropy dissipation developed by Desvillettes.

Amit Einav

Title: * Interacting Systems – The Melting Pot of Analysis, PDEs and Probability *

Abstract: The goal of the talk is to introduce the audience to the fascinating field of interacting systems, which are the basis of many models and equations we use to describe everyday physical, biological, and chemical phenomena and have been utilised in recent years to try and explore some aspects of social behaviour.
Our talk will give a broad overview on mathematical aspects of such systems, and how it acts as a meeting point of Analysis, PDEs and Probability in many situations.
We plan to discuss the notions of microscopic, macroscopic and mesoscopic scaling and the connection between them (hydrodynamical limits and mean field limits), and to introduce several recent (and not so recent) ideas, techniques and approaches that are used to explore a plethora of models and equations which pertain to these systems.

Michela Ottobre

Title: * Non Mean-Field Vicsek Type Models for Collective Behaviour *

Abstract: We consider Interacting Particle dynamics with Vicsek type interactions, and their macropscopic PDE limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). It was observed by Motsch and Tadmore that in this non-mean-field regime the influence between agents (i.e. the interaction term) can be scaled either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting at time t (local scaling). We compare the behaviour of the globally scaled and the locally scaled system in many respects; in particular we observe that, while both models exhibit multiple stationary states, such equilibria are unstable (for certain parameter regimes) for the globally scaled model, with the instability leading to travelling wave solutions, while they are always stable for the locally scaled one. This observation is based on a careful numerical study of the model, supported by formal analysis.
Based on soon to be preprint in collaboration with P. Butta', B. Goddard, T. Hodgson, K.Painter.

Organisers:

Amit Einav (amit.einav (at) durham.ac.uk)