Special Session 105: Dynamics of Many-Particle Systems and Mean-field Equations

Pair approximation for continuous-valued dynamics on networks

WEIQI CHU University of Massachusetts Amherst USA Co-Author(s):

Abstract: Dynamics on networks arise in diverse contexts such as information spreading, disease transmission, epidemiology, opinion dynamics, and statistical physics. In these systems, agents (nodes) interact with their neighbors through network edges, and their states evolve in time depending on the states of their neighbors. The network structure, therefore, plays an essential role in shaping the overall dynamics. When networks are large or highly heterogeneous, direct computations become infeasible and mean-field approximations often fail to capture essential correlations. Pair approximation offers a powerful reduction framework by incorporating network heterogeneity through degree-based correlations, but most existing analyses focus on systems with discrete-valued states, where agents take values from a finite set. In this work, we extend the framework to continuous-valued dynamics. We introduce a family of degree-based density functions to characterize the state distributions of nodes with the same degree and derive the corresponding pair approximation for these density functions.

Data-driven Learning of Interaction Laws in Multispecies Particle Systems

Jinchao Feng Great Bay University Peoples Rep of China Co-Author(s):    Charles Kulick, Sui Tang

Abstract: We develop a Gaussian process framework for learning interaction kernels in multi-species interacting particle systems from trajectory data. Such systems provide a canonical setting for multiscale modeling, where simple microscopic interaction rules generate complex macroscopic behaviors. We formulate the learning problem in a nonparametric Bayesian setting and establish rigorous statistical guarantees. Our analysis shows recoverability of the interaction kernels, provides quantitative error bounds, and proves statistical optimality of posterior estimators, thereby unifying and generalizing previous single-species theory. Numerical experiments confirm the theoretical predictions and demonstrate the effectiveness of the proposed approach, highlighting its advantages over existing kernel-based methods. This work contributes a complete statistical framework for data-driven inference of interaction laws in multi-species systems, advancing the broader multiscale modeling program of connecting microscopic particle dynamics with emergent macroscopic behavior.

Mass splitting in the generalized Euler equations: a new explanation via discretization

Gero Friesecke Technical University of Munich (TUM) Germany Co-Author(s):

Abstract: Arnold made the celebrated observation that solutions to the incompressible Euler equations of fluid dynamics correspond to geodesics in the group of volume-preserving diffeomorphisms. A nontrivial fact is that minimizers of the corresponding variational principle may not exist. Brenier introduced a relaxation which he showed to be well-posed. Physically this formulation allows mass splitting, i.e., a fluid particle can move from A to B via an ensemble of trajectories. Mathematically this formulation is an instance of multi-marginal optimal transport; for a simple introduction to this formulation see my recent textbook on optimal transport [1]. After revieving the different formulations of the Euler equations, we - show that mass splitting still occurs after discretizing Brenier`s relaxation either in time or in both space and time - provide a new argument for the mass splitting which is much simpler than previous analyses of the continuous case and reveals a transparent underlying mechanism. We close with a brief discussion from a modeling point of view: what is physically more correct, Euler (no mass splitting) or Brenier (mass splitting)? [1] G. Friesecke, Optimal Transport: a Comprehensive Introduction to Modeling, Analysis, Simulation, Applications, SIAM, 2025

Large-Corridor Pedestrian Flow and Non-Separable Mean-Field Games: A Stability Analysis

Mohamed Ghattassi NYU Abu Dhabi United Arab Emirates Co-Author(s):

Abstract: We investigate the existence and stability of small perturbations of constant states of the generalized Hughes model for pedestrian flow in an infinitely large corridor. We show that constant flows are stable under a condition on the density. Our findings indicate that when the density is less than half of the maximum density, which is the Lasry-Lions monotonicity condition, we can control the perturbation and prove positive stability results for the nonlinear Generalized Hughes model.

Propagation of chaos for multi-species moderately interacting particle systems up to Newtonian singularity

Shuchen Guo University of Oxford England Co-Author(s):    Jose Carrillo and Alexandra Holzinger

Abstract: We derive a class of multi-species aggregation-diffusion systems from stochastic interacting particle systems via relative entropy method with quantitative bounds. We show an algebraic L^1-convergence result using moderately interacting particle systems approximating attractive/repulsive singular potentials up to Newtonian/Coulomb singularities without additional cut-off on the particle level. The first step is to make use of the relative entropy between the joint distribution of the particle system and an approximated limiting aggregation-diffusion system. A crucial argument in the proof is to show convergence in probability by a stopping time argument. The second step is to obtain a quantitative convergence rate to the limiting aggregation-diffusion system from the approximated PDE system. This is shown by evaluating a combination of relative entropy and L^2-distance.

Optimal control of diffusive mean-field models for swarming particles on the sphere

Dohyun Kim Sungkyunkwan University Korea Co-Author(s):    Jinwook Jung

Abstract: In this presentation, we consider a mean-field optimal control problem for a consensus dynamics of high-dimensional Kuramoto-type with diffusion on the unit sphere. The control acts through a prescribed drift field and an interaction gain, and the cost functional is given to track a given target density while penalizing the control effort. At the microscopic level, we formulate the corresponding controlled $N$-particle Liouville problem and establish the existence of optimal controls. For fixed controls, we obtain a quantitative stochastic mean-field limit showing that the one-particle marginal converges to the solution of the mean-field equation with the convergence rate $\mathcal O(1/\sqrt{N})$. Finally, we show that microscopic optimal controls approximate a mean-field optimal control: any weak limit of particle-level minimizers is optimal for the mean-field problem.

Analysis of an iterative scheme for computing the Kantorovich problem

Hicham Kouhkouh University of Graz Austria Co-Author(s):    Likhit Ganedi

Abstract: In the study of multi-particle dynamics and mean-field equations, the Wasserstein distance plays a central role in quantifying the evolution of particle distributions. With Likhit Ganedi (University of Utah), we introduce a simple algorithm for the computation of the Wasserstein metric based on a gradient ascent flow for the Monge-Kantorovich dual formulation, and using a discrete approximation of the double c-transform. The method is computationally efficient and does not require entropic regularization, which can otherwise introduce approximation errors. In addition, we develop a framework for analyzing its global convergence by studying the discrete-to-continuous limit of the underlying dynamics. This approach yields convergence guarantees and highlights a connection with elliptic regularity.

Convergence Rates of Mean-Field Fluctuations in the 2D Viscous Vortex and Coulomb Models

Paul Nikolaev TU Berlin Germany Co-Author(s):    Alekos Cecchin

Abstract: We investigate how fluctuations behave in large systems of interacting particles when the interaction is given by the Biot--Savart kernel, a key model from fluid dynamics. Our main result provides the first quantitative convergence rates for these fluctuations, and remarkably, the rates are optimal. The key idea is to compare the generators of the particle system and of the limiting fluctuation process in an infinite-dimensional setting. This comparison allows us to derive a sharp error bound for the fluctuations. Beyond the Biot--Savart case, the method is versatile and can also be applied to other singular interactions, such as the repulsive Coulomb kernel or general interactions of mean-field type.

On the Diffusive-Mean Field limit of Kinetic Interacting Particle Systems

Grigorios A Pavliotis Imperial College London England Co-Author(s):    R. Gastaldello, G. Stoltz, and U. Vaes

Abstract: We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck PDE.

New results on the critical Keller-Segel system

Filippo Santambrogio Institut Camille Jordan France Co-Author(s):    Charles Elbar, Alejandro Fernandez-Jimenez

Abstract: The talk will be concerned with aggregation diffusion equations and in particular with the well-known Keller-Segel model for chemotaxis, where a particle population is subject to diffusion and to advection in the direction of the gradient of the concentration of a substance produced by the particles themselves. Such a concentration is defined at each point in terms of a non-local integral involving the whole distribution of particles. I will briefly recall the main facts about the Keller-Segel system with particular attention to the critical cases, in terms of diffusion exponent and mass and depending on the dimension, and then present some recent results obtained in collaboration with Elbar and Fernandez-Jimenez. In our recent work we obtained a new estimate on the Laplacian of the pressure associated with the solution, which allows one to obtain or recover global existence results, regularity and regularization, and information of the asymptotic behavior and decay.

Particle-Based Stochastic Reaction-Diffusion Models: Mean field limits and fluctuation corrections.

Konstantinos Spiliopoulos Boston University USA Co-Author(s):

Abstract: Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock Space representation of the corresponding forward equation. We then prove, (a): the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e. thermodynamic) limit, and (b): the next order fluctuation correction to the mean field limit, which satisfies systems of stochastic PIDEs with Gaussian noise. Numerical examples are presented to illustrate how such approximations can enable the accurate estimation of higher order statistics of the underlying PBSRD model.

MEAN-FIELD CONTROL FOR DIFFUSION AGGREGATION EQUATION WITH COULOMB INTERACTION

Yucheng Wang Shanghai University Peoples Rep of China Co-Author(s):    Li Chen, Zhao Wang

Abstract: The mean-field control problem for a multidimensional diffusion--aggregation system with Coulomb interactions (the so-called parabolic elliptic Keller--Segel system) is considered. The existence of optimal control is proven through the $\Gamma$ convergence of the corresponding control problem of the interacting particle system. There are three building blocks in the overall argument. First, for the optimal control problem at the particle level, instead of using the classical method for stochastic systems, we directly study the control problem of high-dimensional parabolic equations, specifically the Liouville equation. Second, we obtain strong propagation of chaos results for the interacting particle system by combining the convergence in probability and relative entropy methods. Owing to this strong mean-field limit result, we avoid imposing compact support requirements for control functions, which have often been used in the literature. Third, because of the strong aggregation effect, additional difficulties arise from the control function in obtaining the well-posedness theory of the diffusion--aggregation equation, making known methods inapplicable. Instead, we use a combination of local existence results and bootstrap arguments to obtain the global solution in the subcritical regime.

Optimal Control for Kuramoto Model: from Many-Particle Liouville Equation to Diffusive Mean-Field Problem

Valeriia Zhidkova University of Mannheim Germany Co-Author(s):    Li Chen, Yucheng Wang

Abstract: We investigate the mean-field optimal control problem of a swarm of Kuramoto oscillators. Using the notion of wrapped distribution, we explain the connection between the stochastic particle system and the mean-field PDE on the periodic domain. In the limit of an infinite number of oscillators, the collective dynamics of their density is described by a diffusive mean-field model in the form of a nonlocal PDE, where the nonlocal term arises from the synchronization mechanism. We prove that the macroscopic optimal control problem admits a solution by using $\Gamma$-convergence strategy of the cost functional corresponding to the Liouville equation on the particle level.