arXiv subjects

Sara Merino-Aceituno

Publications and source records attributed to Sara Merino-Aceituno.

At least 19 recordsLinked to original sources

arXiv2026

A Spectral-Based Method for Network-Formation PDEs

We propose and study a simple and scalable Fourier-based spectral method for a continuum model of network formation under periodic boundary conditions. The model provides the evolution of the pressure $p$ and the conductivity $m$ over time. The evolution of $p$ is given by an anisotropic Poisson equation, while the equation for $m$ contains three terms corresponding to a diffusion and an activation term of the network -- that depends on the gradient of the pressure -- as well as a relaxation term that acts as a decaying term. This system arises as a formal $L^2$-gradient flow of a non-convex energy functional. Our algorithm combines two ingredients: (i) a splitting method for the equation for $m$, where the activation and relaxation parts are solved analytically, and the diffusion part is solved via Fast Fourier Transform (FFT), and (ii) an FFT combined with the Conjugate Gradient (CG) method applied to the equation for the pressure. This makes the scheme easy to implement compared to implicit schemes and naturally extensible to three dimensions on uniform periodic grids. To showcase the method, we recover the previously documented influence of the activation strength $c$, the diffusion coefficient $D$, and the metabolic exponent $γ$ on the morphology of emergent networks, and report grid convergence results.

math.NA↗

arXiv2026

A framework for continuum modeling of opinion dynamics on a network based on probability of connections

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aims at approximating the distribution of opinions as well as the probability that two given opinions are connected. To illustrate our framework, we focus on a simple model of consensus dynamics on a network and derive a continuum description using techniques inspired by mean-field limits. We also discuss the limitations of this approach and suggest extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

math.AP↗

arXiv2025

The Vicsek-Kuramoto model in collective dynamics: macroscopic equations and pattern formation

In this work, we investigate an individual-based model (IBM) for self-propelled agents interacting locally on a plane. Agents are characterized by their position, the angle determining their direction of motion, and their angular velocity. The dynamics combine features of the well-known Vicsek and Kuramoto models, which describe collective dynamics and synchronization, respectively. The evolution of the directions of motion follows a Vicsek model, where agents align their orientations with the mean orientation of their neighbors, subject to some noise. Similarly, the angular velocities relax towards the average angular velocity of the neighboring agents, also subject to noise. From the IBM we derive the corresponding kinetic equation in the limit of a large number of agents and formally obtain the macroscopic equations through a macroscopic (hydrodynamic) limit. Numerical simulations of the IBM reveal a variety of patterns, including rotating clusters, traveling orientation waves, and globally synchronized rotational motion. A qualitative comparison with simulations of the macroscopic system show the ability of the macroscopic model to reproduce some emergent behavior of the IBM.

math-ph↗

arXiv2025

Modeling Protein Diffusion Across ER-Nuclear Envelope Junctions Reveals Efficient Transport via Simple Diffusion

The endoplasmic reticulum (ER) is the largest continuous membrane-bound organelle in the cell and plays a central role in the synthesis and turnover of many lipids and proteins. It connects directly to the nucleus through specialized contact points known as ER-nuclear envelope (NE) junctions. In our recent study, we found that these ER-NE junctions are both narrow and infrequent, measuring less than 20 nanometers in diameter and occurring at a frequency of approximately 0.1 per square micrometer. However, it remains unclear whether such limited and narrow connections are sufficient to support efficient transport between the ER and NE. Here, we built a mathematical model of ER-to-NE protein diffusion, incorporating ultrastructural parameters, the frequency of ER-NE junctions, and the diffusion coefficient of proteins within the ER lumen. To validate the model, we experimentally quantified the transport rate of ER luminal proteins to the NE using fluorescence recovery after photobleaching (FRAP). Our model and experimental data demonstrate that simple diffusion is sufficient to account for the rapid transport of proteins from the ER to the NE, despite the limited and narrow nature of the connecting junctions. Together, these findings offer mechanistic insight into how ER-NE connectivity enables rapid protein transport and lay the groundwork for future studies on ER-nucleus communication.

math-ph↗

arXiv2025

Macroscopic effects of an anisotropic Gaussian-type repulsive potential: nematic alignment and spatial effects

Elongated particles in dense systems often exhibit alignment due to volume exclusion interactions, leading to packing configurations. Traditional models of collective dynamics typically impose this alignment phenomenologically, neglecting the influence of volume exclusion on particle positions. In this paper, we derive nematic alignment from an anisotropic repulsive potential, focusing on a Gaussian-type potential and first-order dynamics for the particles. By analyzing larger particle systems and performing a hydrodynamic limit, we uncover the effects of anisotropy on both particle density and direction. Our findings reveal that while particle density evolves independently of direction, anisotropy slows down nonlinear diffusion. The direction dynamics are affected by the particles' position and involve complex transport and diffusion processes, with different behaviors for oblate and prolate particles. The key to obtaining these results lies in recent advancements in Generalized Collision Invariants offered by Degond, Frouvelle and Liu (KRM 2022).

math.AP↗

arXiv2024

Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation

The Vicsek-BGK equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace-Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data smooth solutions might blow up in finite time whereas weak solutions with bounded Boltzmann entropy are shown to exist globally.

math.AP↗

arXiv2023

Convergence proof for first-order position-based dynamics: An efficient scheme for inequality constrained ODEs

NVIDIA researchers have pioneered an explicit method, position-based dynamics (PBD), for simulating systems with contact forces, gaining widespread use in computer graphics and animation. While the method yields visually compelling real-time simulations with surprising numerical stability, its scientific validity has been questioned due to a lack of rigorous analysis. In this paper, we introduce a new mathematical convergence analysis specifically tailored for PBD applied to first-order dynamics. Utilizing newly derived bounds for projections onto uniformly prox-regular sets, our proof extends classical compactness arguments. Our work paves the way for the reliable application of PBD in various scientific and engineering fields, including particle simulations with volume exclusion, agent-based models in mathematical biology or inequality-constrained gradient-flow models.

math.NA↗

arXiv2022

Hyperbolicity and non-conservativity of a hydrodynamic model of swarming rigid bodies

In this paper, we study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird flocks, fish schools or fleets of drones. We show that the system is hyperbolic and can be approximated by a conservative system through relaxation. We also derive viscous corrections to the model from the hydrodynamic limit of a kinetic model. This analysis prepares the future development of numerical approximations of this system.

math.AP↗

arXiv2022

How environment affects active particle swarms: a case study

We investigate the collective motion of self-propelled agents in an environment filled with obstacles that are tethered to fixed positions via springs. The active particles are able to modify the environment by moving the obstacles through repulsion forces. This creates feedback interactions between the particles and the obstacles from which a breadth of patterns emerges (trails, band, clusters, honey-comb structures,...). We will focus on a discrete model first introduced in [Aceves2020] and derived into a continuum PDE model. As a first major novelty, we perform an in-depth investigation of pattern formation of the discrete and continuum models in 2D: we provide phase-diagrams and determine the key mechanisms for bifurcations to happen using linear stability analysis. As a result, we discover that the agent-agent repulsion, the agent-obstacle repulsion and the obstacle's spring stiffness are the key forces in the appearance of patterns, while alignment forces between the particles play a secondary role. The second major novelty lies in the development of an innovative methodology to compare discrete and continuum models that we apply here to perform an in-depth analysis of the agreement between the discrete and continuum models.

math-ph↗

arXiv2021

Cauchy theory for general Vicsek models in collective dynamics and mean-field limit approximations

In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity and nonlinear friction. Particularly, we include normalised kernels which display a singularity when the flux of particles vanishes. Thus, in terms of the Cauchy theory for the kinetic equation, we extend to more general interactions and complete the program initiated in Gamba et. al. (2016) (where the authors assume that the singularity does not take place) and in Figalli et. al. (2017) (where the authors prove that the singularity does not happen in the space homogeneous case). Moreover, we derive an explicit lower time of existence as well as a global existence criterion that is applicable, among other cases, to obtain a long time theory for non-renormalised kernels and for the original Vicsek problem without any a priori assumptions. On the second part of the paper, we also establish the mean-field limit in the large particle limit for an approximated (regularized) system that coincides with the original one whenever the flux does not vanish. Based on the results proved for the limit kinetic equation, we prove that for short times, the probability that the dynamics of this approximated particle system coincides with the original singular dynamics tends to one in the many particle limit.

math.AP↗

arXiv2020

Large-scale dynamics of self-propelled particles moving through obstacles: model derivation and pattern formation

We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of non-linear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.

math.AP↗

arXiv2020

Continuum and thermodynamic limits for a simple random-exchange model

We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann's program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.

math.PR↗

arXiv2019

Nematic alignment of self-propelled particles in the macroscopic regime

Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross diffusion system poses many new challenging questions.

math-ph↗

arXiv2019

Phase transitions and macroscopic limits in a BGK model of body-attitude coordination

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body-attitude of an agent is modelled by a rotation matrix in $\mathbb{R}^3$ as in [Degond, Frouvelle, Merino-Aceituno, 2017]. The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher dimensional space from which we deduce the complete description of the possible equilibria Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated to the stable equilibria in the spirit of [Degond, Frouvelle, Merino-Aceituno, 2017] and [Degond, Frouvelle, Liu, 2015].

math.AP↗

arXiv2018

Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations

The goal of these lecture notes is to present in a unified way various models for the dynamics of aligning self-propelled rigid bodies at different scales and the links between them. The models and methods are inspired from [12,13], but, in addition, we introduce a new model and apply on it the same methods. While the new model has its own interest, our aim is also to emphasize the methods by demonstrating their adaptability and by presenting them in a unified and simplified way. Furthermore, from the various microscopic models we derive the same macroscopic model, which is a good indicator of its universality.

math.AP↗

arXiv2017

A new continuum theory for incompressible swelling materials

Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.

math.AP↗

arXiv2017

Quaternions in collective dynamics

We introduce a model of multi-agent dynamics for self-organised motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbours. The body attitudes are represented through unitary quaternions. We prove the correspondance with the model presented in a previous work by the three first authors where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual based model (IBM) introduced here is based on nematic (rather than polar) alignment. From the IBM, the kinetic and macroscopic equations are derived. The benefit of this approach is twofold: firstly, it allows for a better understanding of the macroscopic equations obtained and, secondly, these equations are prone to numerical studies, which is key for applications.

math-ph↗

arXiv2017

Coupled Self-Organized Hydrodynamics and Stokes models for suspensions of active particles

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek-Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery's equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics (SOH)-Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek-Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.

physics.flu-dyn↗