# Workshop on Interacting Particle Systems and Non-Local PDEs, June 2024

- Location
- Lecture Theatre A - Watson Building (R15 on the campus map)

- Dates
- Thursday 6 June 2024 (10:00-17:00)

- Contact
Nataliya Balabanova and Hong Duong

Complex systems/phenomena in natural and applied sciences are often described by (stochastically) interacting particle systems at the microscopic scale, and by (non-local) partial differential equations at the macroscopic scale. Understanding these models is the key to the understanding and control of the complex phenomena.

In this workshop, we bring together researchers working in the two fields to foster the interaction between them, aiming to join forces to develop new methods and techniques to advance our understanding of complex systems in physics, biology and social sciences.

## Speakers

- Federico Cornalba (University of Bath)
- Amit Einav (Durham University)
- Alexandra Holzinger (University of Oxford)
- Mabel Lizzy Rajendran (University of Birmingham)
- Julian Tugaut (Jean Monnet University)
- Alexandra Tzella (University of Birmingham)
- Blaine van Rensburg (University of Birmingham)

## Schedule

- 10:00-10: 45, Federico Cornalba, Numerical and analytical aspects for stochastic PDEs from fluctuating hydrodynamics
- 10: 45-11:30, Alexandra Tzella, Active vs passive Brownian particles in the presence of obstacles
- 11:30-12:15, Alexandra Holzinger, Fluctuations around the mean-field limit for attractive Riesz potentials in the moderate regime
- 12:15-13:15, lunch
- 13:30-14:15, Mabel Lizzy Rajendran, Model for cell-cell adhesion via receptor binding
- 14:15-15:00, Julian Tuguat, Long-time behaviour for granular media equation
- 15:00-15:30, coffee break
- 15:30-16:15, Amit Einav, A notion of order in interacting systems
- 16:15-17:00, Blaine van Rensburg, Adaptive dynamics of diverging fitness optima

## Title and abstracts

### Federico Cornalba: Numerical and analytical aspects for stochastic PDEs from fluctuating hydrodynamics

Abstract: Large-scale systems of interacting particles are ubiquitous, with applications ranging from thermal advection phenomena to gradient descent algorithms in machine learning. While the mean-field-limit (MFL) perspective allows to efficiently describe the ‘average’ evolution of many such particles systems (usually via PDEs), it does not incorporate a macroscopic description of the particles’ intrinsic fluctuations. The theory of Fluctuating Hydrodynamics addresses this aspect by instead describing the particle systems via stochastic PDEs which enrich the underlying MFL dynamics.

For a weakly interacting multi-species system, we will discuss the interplay between MFL dynamics and associated stochastic PDE, and show that numerical discretizations of the latter can approximate the particles’ fluctuations in an accurate and efficient way.

Based on joint works with Julian Fischer (IST Austria), Jonas Ingmanns (IST Austria), and Claudia Raithel (TU Dresden).

### Amit Einav: A notion of order in interacting systems

Abstract: Systems that involve many elements, be it a gas of particles or a herd of animals, are ubiquitous in our day to day lives. Their investigation, however, is hindered by the complexity of such systems and the amount of (usually coupled) equations that are needed to be solved. The late 50’s has seen the birth of the so-called mean field limit approach as an attempt to circumvent some of the difficulties arising in treating such systems. Conceived by Kac as a way to give justification to the validity of the Boltzmann equation, the mean field limit approach attempts to find the behaviour of a limiting 'average' element in a many element system and relies on two ingredients: an average model of the system (i.e. an evolution equation for the probability density of the ensemble), and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity. Mean field limits of average models, originally applied to particle models, have permeated to fields beyond mathematical physics in recent decades. Examples include models that pertain to biological, chemical, and even societal phenomena. However, to date we use only one asymptotic correlation relation – chaos, the idea that the elements become more and more independent. While suitable when considering particles in a certain cavity, this assumption doesn’t seem reasonable in models that pertain to biological and societal phenomena. In our talk we will introduce Kac’s particle model and the notions of chaos and mean field limits. We will discuss the problem of having chaos as the sole asymptotic correlation relation and define a new asymptotic relation of order. We show that this is the right relation for a recent animal based model suggested by Carlen, Degond, and Wennberg, and highlight the importance of appropriate scaling in its investigation.

### Alexandra Holzinger: Fluctuations around the mean-field limit for attractive Riesz potentials in the moderate regime

Abstract: In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit.

We will see how a central limit theorem can be shown for moderately interacting particles in the whole space for certain types of interaction potentials. In this talk, the interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proved that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger on fluctuations for the porous-medium equation. To allow for attractive potentials we use a quantitative mean-field convergence in probability in order to explicitly treat aggregation terms in our estimates. This is joint work with Ansgar Jüngel and Li Chen.

### Mabel Lizzy Rajendran: Model for cell-cell adhesion via receptor binding

Abstract: Cancer growth and spread involve nonlocal phenomena such as interactions between cells and their interactions with the surrounding environment. Including these phenomena in the mathematical model results in a partial differential equation (PDE) with non-local operators, which pose interesting challenges in demonstrating their well-posedness and in numerical simulation.

Cell-cell adhesion behaviour, mediated by cell-adhesion molecule (CAM) bindings, plays a key role in cancer migration. Through multiscale modelling, it is demonstrated that including this characteristic results in a system of a PDE with non-local operator in space coupled with a novel non-linear integral equation with two non-local operators.

### Julian Tugaut: Long-time behaviour for granular media equation

Abstract: Starting from the microscopical viewpoint with particles system, we briefly derive the associated Partial Differential Equation that is denoted as granular media equation. The aim is to find out the limiting steady state or at least to find suitable conditions ensuring that the limiting steady state is a given invariant probability measure. We will recall that such measure is unique under convexity assumption and that the convergence holds. Then, we will discuss the nonconvex case, where there is convergence albeit non-uniqueness of the invariant probability measures may occur. In a second time, we will give the main theorem that is a stability type theorem: if the initial measure is sufficiently close to a given stationary measure then this measure is the limiting steady state. Finally, we will give the proof of the theorem. If time allows us, we will discuss on possible generalizations. This talk is based on a work published in Kinetic and Related Models.

### Alexandra Tzella: Active vs passive Brownian particles in the presence of obstacles

Abstract: We consider active particles (microswimmers) moving in an environment with obstacles. These are treated using the Active Brownian Particle model (ABP), where particles move forward at constant speed but in a randomly-varying direction. We use homogenisation theory to predict their coarse-grained dynamics at long times. We present numerical solutions that describe their spatial distribution and effective diffusivity as a function of the strength of swimming and area fraction occupied by the obstacles and contrast these against the long-time behaviour of classical Brownian particles.

### Blaine van Rensburg: Adaptive dynamics of diverging fitness optima

Abstract: Populations adapt to changing environments in a variety of contexts from climate change to chemotherapy to the ageing body. We consider a model of a trait-structured population whose dynamics are governed by non-local parabolic equation accounting for mutation and competition and the changing environment. The main novelty is that there are two time-dependent locally optimal traits, each of which shifts at a possibly different linear speed. We analyse the solution in the long time, small mutation limits. Our results imply that in such a scenario, both the true optimal fitness and the required rate of adaptation for each of the diverging optimal traits contribute to the eventual concentration of the solution on a single trait.

In this talk, I will present some preliminary results on this model, and discuss their biological interpretation.