# C10 - Numerical analysis for nonlinear SPDE models of particle systems

**Head(s):** Prof. Dr. Ana Djurdjevac (FU Berlin), Prof. Dr. Nicolas Perkowski (FU Berlin)**Project member(s): **Xiaohao Ji **Participating institution(s):** FU Berlin**Research area:** Mathematics

## Project Summary

Our aim is to derive and study continuum models for the mesoscopic behavior of interacting particles systems (below the hydrodynamic limit), following a new paradigm: We will introduce nonlinear and non-Gaussian models that provide a more faithful representation of the spatiotemporal distribution of particles than the usual linear Gaussian perturbations around

the hydrodynamic limit models.

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as biology, sociology (agents), molecular dynamics (proteins) etc. Unfortunately, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive

effective equations that describe the time evolution of the empirical particle density on various space-time scales. If the individual particles live in R^{d }(say *d* = 3) then these effective equations will be PDEs or SPDEs on the low-dimensional space R_{+ }x R^{d}.

The behavior on macroscopic space-time scales and as the system size goes to infinity is described by the hydrodynamic limit, which is often given by a nonlinear PDE.A common strategy to study the behavior at mesoscopic scales and at large but finite system size is to linearize the empirical particle density around the hydrodynamic limit and to derive a central

limit theorem for the fluctuations, which typically are given by linear, Gaussian SPDEs.

A drawback of this approach is that the linearization may and typically will violate physical constraints of the original system. For example, the original dynamics might be restricted C10 346 3. Project details to a sub-manifold of the state space or they might have some conservation laws, and these properties would be lost after linearization. Also, in applications such as molecular dynamics and social sciences, the finite size system may exhibit dynamic phenomena such as metastable transitions that are not seen in the hydrodynamic limit or in the linearized fluctuations.

The goal of our project is to derive and study nonlinear SPDE models which more faithfully reproduce the evolution of the empirical density of a given particle system. In particular, we want to study the well-posedness of these nonlinear SPDE models, to control the weak error of the SPDE approximation, and to perform a rigorous numerical discretization of the

SPDE. A prototypical example is the formal identification of a finite system of mean-field interacting diffusions with the singular Dean–Kawasaki SPDE. But also exclusion processes or zero range processes can be formally mapped to singular, nonlinear SPDEs, plus error terms. We will study the weak error that is made by this mapping. We will also introduce structurepreserving discretization methods for these SPDEs and study their weak error. And we will

investigate dynamical properties such as metastability for our nonlinear SPDE models. In particular we will focus on the reaction-diffusion equations that appear in C03 and equations that are analyzed in the hybrid dynamics approach for approximation of the target measure considered in A02. Furthermore, we will investigate many particle limits of open systems that are studied in C01.