Dynamics of complex systems

Applied Mathematics Münster

Institute for Analysis and Numerics

Orléans-Ring 10

48149 Münster

Room 130.026

University of Münster

Tel.: +49 251 83-35091

a.schlichting@uni-muenster.de

## News and Updates

- Preprint: Variational convergence of exchange-driven stochastic particle systems in the thermodynamic limit
Together with Chun Yin Lam we consider the thermodynamic limit of mean-field stochastic particle systems on a complete graph. The evolution of occupation number at each vertex is driven by particle exchange with its rate depending on the population of the starting vertex and the destination vertex, including zero-range and misanthrope process. We show that under a detailed balance condition and suitable growth assumptions on the exchange rate, the evolution equation of the law of the particle density can be seen as a generalised gradient flow equation related to the large deviation rate functional.

We show the variational convergence of the gradient structures based on the energy dissipation principle, which coincides with the large deviation rate function of the finite system. The convergence of the system in this variational sense is established based on compactness of the density and flux and Γ-lower-semicontinuity of the energy dissipation functional along solutions to the continuity equation. The driving free energy Γ-converges in the thermodynamic limit, after taking possible condensation phenomena into account.

The preprint is on arXiv:2401.06696. - Preprint: Modelling and numerical analysis for a durotaxis model in “Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels”
In this interdisciplinary project with a group from the Institute of Physiology II in Münster and the Department of Medicine at Duke, together with Angela Stevens we contributed a mathematical model and numerical analysis for a phenomenological model of durotaxis with mechanosensitive ion channels.

For details see the preprint bioRxiv:2023.12.22.572956 and the supplementary material about the numerics. - Preprint: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport:
Together with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré we propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies an a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives. arXiv:2312.13284

- Conference announcement: PDE & Probability in interaction: functional inequalities, optimal transport and particle systems
Together with Pierre Monmarché (Sorbonne Université), Julien Reygner (École des Ponts ParisTech), and Marielle Simon (Université de Lille), we are delighted to announce the upcoming workshop “PDE & Probability in interaction: functional inequalities, optimal transport and particle systems”.

The event will be held from January 22 to 26, 2024, at CIRM in Marseille.

Registrations are now open on the website: https://conferences.cirm-math.fr/2988.html

This workshop will also feature two courses, delivered by J. LEHEC (Université de Poitiers) and M. GOLDMAN (Université Paris Cité) on functional inequalities in high dimensions and random matching problems, respectively.

Invited talks will be given by:

Nathalie Ayi (Sorbonne Université)

Roland Bauerschmidt* (University of Cambridge)

Maria Bruna (University of Cambridge)

Kleber Carrapotoso (École Polytechnique, Palaiseau)

Giovanni Conforti (École Polytechnique, Palaiseau)

Alex Delalande (Lagrange Center, Paris)

François Delarue (Université Côte d’Azur)

Alex Dunlap (NYU Courant)

Rishabh Gvalani (MPI MIS Leipzig)

Martin Huesmann (University of Münster)

Jean-Francois Mehdi Jabir (HSE Moscow)

Jasper Hoeksema (TU Eindhoven)

Bo’az Klartag (Weizmann Institute of Science)

Daniel Lacker (Columbia University)

Jean-Christophe Mourrat (ENS Lyon)

Emanuela Radici (University of L’Aquila)

Milica Tomasevic (École Polytechnique, Palaiseau)

Dario Trevisan (Pisa University)

Isabelle Tristani (ENS Paris)

Haava Yoldas (TU Delft) - Preprint: Graph-to-local limit for the nonlocal interaction equation
Together with Antonio Esposito and Georg Heinze, we study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

See arXiv:2306.03457