Special Session 22: Models of emergence and collective dynamics

On weak solutions to self-organized systems of Euler-type with non-constant interaction kernel

Cleopatra Christoforou University of Cyprus Cyprus Co-Author(s):    Debora Amadori

Abstract: We present recent results on the existence and long-time asymptotic behavior of solutions to a hydrodynamic model of flocking-type with pressure, within the framework of weak solutions. The analysis is carried out in one spatial dimension, considering both all-to-all interaction kernel and non-constant kernel in a periodic domain. For entropy weak solutions on the torus, we establish exponential decay in time toward a flocking state in the $L^2$ norm, under the assumptions of an integrable interaction kernel and a density uniformly bounded away from vacuum. The approach relies on the front-tracking approximate solutions, the relative entropy and a suitable energy functional.

Finite speed of propagation in the one-dimensional pressureless Euler alignment system

Szymon Cygan University of Heidelberg Poland Co-Author(s):    Szymon Cygan, Bartosz Drzygala, Grzegorz Karch

Abstract: In this talk, I will present recent results on the one-dimensional pressureless Euler alignment system with a strongly singular interaction kernel and initial data allowing vacuum \begin{align*} \rho_t + (\rho u)_x &= 0 , \ u_t + u u_x &= - \Lambda^\alpha (\rho u) + u\,\Lambda^\alpha \rho. \end{align*} Building on a previously developed framework for the case $\alpha \in (0,1)$, we extend the analysis to the full range $\alpha \in (0,2)$. We work under the structural condition imposed on the initial data, \begin{align*} 0 \leqslant G_0(x) \leqslant a\,\rho_0(x) \quad \text{where} \quad G_0 = \partial_x u_0 - \Lambda^\alpha \rho_0, \end{align*} which determines an admissible class of initial velocity profiles and is propagated by the dynamics. I will discuss the construction of global weak solutions corresponding to compactly supported initial data and show that the support remains compact for all times. Moreover, I will explain how finite speed of propagation can be established by adapting barrier arguments and contact analysis techniques from the theory of nonlocal porous medium equations.

On the incompressible limit for a viscoelastic tumour growth model.

Tomasz Debiec University of Warsaw Poland Co-Author(s):

Abstract: I will discuss some approaches to mathematical modelling of living tissues, with application to tumour growth. In particular, I will describe recent results on the incompressible limit (or stiff-pressure limit) for a compressible model, building a bridge between density-based description and a geometric free-boundary problem by passing to the singular limit in the pressure law. We set out from a two-species advection-reaction system --- the cell densities are advected by the gradient of a chemical potential which satisfies the so-called Brinkman law, while the growth rate of each population is governed by a function of the joint population pressure. In the limit problem the total population density is limited to a critical value and the pressure vanishes on unsaturated regions.

Stick with me, naturally: from Cucker--Smale to Euler-alignment through gradient flows

Sondre T Galtung SINTEF Norway Co-Author(s):

Abstract: This talk presents a new perspective on the one-dimensional pressureless Euler-alignment system, which can be seen as a hydrodynamic limit of the Cucker--Smale model for collective behaviour. While previous analyses of the Cucker--Smale model typically allowed particle trajectories to cross, recent works by Leslie and Tan suggest that such solutions are not suitable for hydrodynamic limits, and instead advocate for sticky particle dynamics. Building on this perspective, we characterise the sticky particle Cucker--Smale dynamics as an $L^2$-gradient flow of a convex functional, providing a new variational framework for the system. This approach not only recovers the unique entropy solution of the associated scalar balance law in Leslie and Tan's framework, but also clarifies how (non-)monotonicity properties of the model's so-called natural velocities determine cluster formation. Our results place the Euler-alignment system within a rigorous gradient flow framework in one dimension, already established for the pressureless Euler and Euler--Poisson systems.

A collective dynamics perspective on transformer dynamics beyond gradient flows

Nicolas Garcia Trillos University of Wisconsin Madison USA Co-Author(s):    Sixu Li (UW-Madison), Jan Peszek (Warsaw), Trevor Teolis (Rice), Konstantin Riedl (Oxford), Jake Maranzatto (Maryland), Semih Akkoc (Maryland), and Sennur Ulukus (Maryland)

Abstract: In this talk, I will discuss a collective dynamics perspective on transformers, the architecture at the heart of modern large language models. In particular, we will discuss how dimensionality reduction techniques akin to those used in the study of the Kuramoto model can be employed to explore the rich structure that the evolution of the distributions of tokens (particles) can have when selecting different values for the key, query, and value matrices parameterizing a transformer model consisting of compositions of multiple self-attention layers. This perspective will allow us to explore token dynamics beyond the gradient flow setting obtained by very specific choices of model parameters and to uncover parameter choices inducing cyclical behavior, consensus formation without stability, and Hamiltonian dynamics. While our theoretical discussion will focus exclusively on 2-dimensional token embeddings, I will also discuss numerical experiments that suggest that our theoretical findings can be extrapolated to general multi-dimensional settings.

A new formula for the Wasserstein distance between solutions to (nonlinear) continuity equations

Piotr Gwiazda Institute of Mathematics Polish Academy of Sciences Poland Co-Author(s):

Abstract: Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent m, we prove that solutions are Lipschitz continuous with respect to m, providing a quantitative version of the result of Benilan and Crandall. This result can be extended to a general aggregation-diffusion equation.

On a Cross-Diffusion System with Independent Drifts and no Self-Diffusion: Mixing Dynamics

Alpar R Meszaros Durham University England Co-Author(s):

Abstract: We present a global existence theory for weak solutions to a two-species cross-diffusion system, in one space dimension, in which the evolution of each species is governed by two mechanisms: a diffusion which acts only on the sum of the species with a logarithmic or fast diffusion type pressure law, and a drift term, which can differ between the two species. We will discuss two types of phenomena, regarding qualitative properties of solutions. First, we will show that if the initial densities are `totally mixed', this property will propagate over time. Second, we will show how to remove the this condition on the initial data, resulting in weak solutions which can be only partially mixed. All these solutions are in contrast with the segregated solutions, known previously in the literature. The talk will be based on joint works with Guy Parker (Durham).

Self-Attention as a Multi-Agent System: a Dynamical Perspective on Transformers

Jan Peszek University of Warsaw Poland Co-Author(s):

Abstract: Transformers, the architecture underlying modern large language models, (specifically its self-attention component) was recently recast as the multi-agent system \begin{equation} \dot{x}_k(t) = \operatorname{P}^{\perp}_{x_k(t)} \left( \sum_{j=1}^n \exp \left(\langle Qx_k(t), K x_j(t) \rangle \right) V x_j(t) \right), \quad k=1, \dots, n, \end{equation} where $x_k(t) \in {\mathbb S}^{d-1}$ represents the state of the $k$-th token after the $t$-th attention layer and $Q,K,V \in \mathbb{R}^{d\times d}$ are the trained query, key and value matrices and $\operatorname{P}^{\perp}_{x_k(t)}$ is the projection onto ${\mathbb S}^{d-1}$. The output of the model is determined by the long-time state of the system, making its asymptotic behavior a central object of study. The goal of this talk is to illustrate how methods from collective dynamics can help analyze large language models. In this talk I present a dynamical systems perspective on a 2D linearized transformer architecture, where the evolution of tokens across attention layers can be reformulated as a Kuramoto-type interacting particle system. Using the Ott-Antonsen ansatz, the high-dimensional dynamics admits a low-dimensional reduction describing the evolution in a reduced vocabulary regime. I will focus on the stability of the reduced dynamics and how their behavior persists beyond the Ott-Antonsen resulting in the stability of the full particle system.

Mean field limit of non exchangeable interacting diffusions on co-evolutionary networks

David Poyato University of Granada Spain Co-Author(s):    Juli\`an Cabrera-Nyst

Abstract: Traditional models of interacting particle systems often assume a fixed network of connections, which simplifies the analysis but fails to capture many real-world phenomena. Indeed, interacting particles where the network structure and particle states co-evolve in mutual influence, are increasingly recognised as essential in diverse fields. For instance, they appear in neuroscience, where learning is encoded through the strengthening and weakening of synaptic connections. In this talk I will present the rigorous mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The main challenge arises from the coupling between the network dynamics and the agents` states, which results in a non-Markovian dynamics where the system`s future depends on its entire history. Consequently, the mean-field limit is not described by a partial differential equation, but by a system of non-Markovian stochastic integrodifferential equations. A second difficulty stems from the non-linear weight dynamics, which requires a careful choice for the limiting network structure. Due to the limitations of the classical theory of graphons (Lov\`asz and Szegedy, 2006), in our mean-field limit we employ for the first time K-graphons (Lov\`asz and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025), as they provide a natural framework compatible with non-linear structures.

Evolution of Correlation in Simple Models

Keith Promislow Michigan State University USA Co-Author(s):    Adam Petrucci and Vinh Nguyen

Abstract: We present particle-based models for cell cycle evolution that possess asymmetrical forcing and for collections of water molecules that interact through a dipole moment (Stockmayer model). We show that both systems have some degree of mean-field limit. In the cell cycle model, we examine the temporal evolution of measures of correlation starting from weakly correlated initial data and compare particle evolution to solutions of the corresponding Vlasov system. In the Stockmayer model we examine the scaling of dipole length with particle number N that yields a mean-field scaling, and show that the dipolar properties of water uncouple from the fluid properties in the mean field.

Graph-induced rotational twisted states in systems of identical oscillators

David N Reynolds University of Warsaw Poland Co-Author(s):    Nastassia Pouradier Duteil, David Poyato

Abstract: The study of fish behavior and the phase-transition between milling (rotations) and schooling (velocity alignment) has given rise to the current study. Milling behavior in second order systems has been successfully modeled, but largely induced by the combination of self-propulsion and friction forces coupled with attraction and repulsion forces. We investigate the feasibility of producing phase-transitions between milling and schooling via heterogeneities in an alignment operator, as opposed to the attraction and repulsion forces used previously. In this talk we present results on first-order oscillatory models which under certain graph structures produce bi-stability of the rotational states indicative of milling, and fully synchronized states indicative of schooling.

Global solutions to cross-diffusion systems with independent advections in one dimension

Jakub Skrzeczkowski University of Oxford England Co-Author(s):

Abstract: We consider cross-diffusion systems describing evolution of two species $u$ and $v$ moving according to Darcy`s law with the pressure law $p(s) = \frac{1}{\alpha-1} s^{\alpha-1}$ where $s=u+v$. One of the most challenging questions in the field is the construction of solutions to the problem in the presence of additional advection fields, without imposing any artificial structure on the fields or the initial conditions. Although advection arises naturally in these models, it breaks the symmetry of the system and prevents application of techniques developed in recent years. Here, we provide a new approach to construct solutions in one space dimension that works in a unified manner for all pressure exponents $\alpha \in (0,\infty)$ and for arbitrary initial data. In~particular, in the regime $\alpha > 1$, this yields the first existence result of its kind, obtained without any structural assumptions. We construct the solutions as a limit of a vanishing viscosity approximation $(u_{\varepsilon}, v_{\varepsilon})$. The main challenge is to identify the limit of $u_{\varepsilon} \, \partial_x p(s_{\varepsilon})$, where $s_{\varepsilon} = u_{\varepsilon} + v_{\varepsilon}$. The key new insight is that possible oscillations of $u_\varepsilon$ and $\partial_x p(s_\varepsilon)$ are correlated, simplifying the Young measure analysis in the compensated compactness argument and allowing identification of the limit. Somewhat surprisingly, in contrast to the theory of $2\times2$ hyperbolic systems, the argument relies on only three entropy-entropy flux pairs. This is particularly useful for $\alpha>2$, where it is unclear whether additional entropies are available.

Relative entropy method for non-local models

Agnieszka Swierczewska-Gwiazda University of Warsaw Poland Co-Author(s):

Abstract: We will discuss Euler-Poisson and Euler-Korteweg systems with friction and exponential pressure. Our interest is directed to strong solutions of the Keller-Segel system and Cahn-Hillard, which are high-friction limits of the dissipative measure-valued solutions to the hydrodynamic systems. For the passage to the limit system, we use the highly efficient approach of the relative entropy method, which found use in a various different fields, ranging from weak-strong uniqueness problems, to stability studies, asymptotic limits and dimension reduction problems.

Collective dynamics with nonlinear velocity alignment

Changhui Tan University of South Carolina USA Co-Author(s):

Abstract: I will discuss recent developments in flocking models with nonlinear velocity alignment, focusing on the role of nonlinearity in the emergence of collective behavior, with quantitative estimates and rates.

From Nanbu Particle Systems to Transformer Approximations of Boltzmann Solutions

Trevor Teolis Rice University USA Co-Author(s):    Trevor Teolis, Maarten de Hoop

Abstract: We show that solutions of the Boltzmann equation can be uniformly approximated, in suitable Wasserstein metrics, by measure-theoretic transformer models. Our approach is based on a pushforward representation of Boltzmann dynamics induced by the Nanbu stochastic particle system, in which the evolution of a tagged particle depends on an underlying context consisting of the initial velocity distribution and a Poisson random measure encoding collision events. In this formulation, the fundamental objects (or tokens) consist of velocity-path pairs, where the paths are Radon measures describing jump sequences. By exploiting this structure, we place the Boltzmann pushforward representation within the scope of recent universality results for measure-theoretic transformers.

On Models of Self-Organized Dynamics and the Euler Equations

Konstantina Trivisa Trivisa University of Marylad USA Co-Author(s):

Abstract: In this lecture I will present results on models describing self organized dynamics and discuss their relation to the Euler equations for compressible fluids.

On integrability properties of Euler-Riesz systems

Athanasios Tzavaras KAUST/Applied Mathematics and Computational Science Saudi Arabia Co-Author(s):

Abstract: We consider the system of compressible Euler equations augmented with nonlocal term associated to a Riesz potential. This system emerges as a critical point of an action functional. Also it is equipped with a stress tensor that may be expressed in two distinct representations and gives rise to an associated bilinear fractional integral operator. We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and Marcinkiewicz interpolation. The estimate leads to integrability analysis of the associated stress-tensor. In addition, using the theory of compensated integrability it yields a gain in integrability for finite-energy solutions. Finally, for smooth periodic solutions of the reformulated system, we derive a stability result (joint work with N. Alves (KAUST) and L. Grafakos (Univ. of Missouri)).

On the weak solution of the uni-directional Euler-alignment system with strongly singular communication

Liutang Xue Beijing Normal University, China Peoples Rep of China Co-Author(s):

Abstract: We talk about the recent development on the weak solution for the Euler-alignment system in d dimensions with unidirectional flows and strongly singular communication protocols. A main part is via the modulus of continuity method.