arXiv subjects

Patrick van Meurs

Publications and source records attributed to Patrick van Meurs.

At least 19 recordsLinked to original sources

arXiv2026

Simulations of dislocation dynamics on an atomic lattice: the effect of collision rules

The stochastic dynamics of dislocations on a one-dimensional periodic lattice domain are considered. Two models are studied: one without a collision rule, and one which annihilates colliding dislocations if they have opposite orientation. The behaviour of both models is investigated by means of a series of numerical simulations exploring the asymptotic behaviour of these models as the number of dislocations increases. From these simulations, evidence is obtained that the discrete model with annihilation tends to a PDE for the dislocation density that accounts for annihilation. However, the discrete model without a collision rule does not appear to exhibit consistent convergence behaviour; instead, it appears that the expected PDE with conserved dislocation density appears in the limit for some parameters, but that for other parameters the density appears to follow to the evolution of the PDE with annihilation. These findings provide evidence that a careful treatment of dislocation collisions is important in discrete dislocation dynamics models.

cond-mat.mtrl-sci↗

arXiv2026

Strong well-posedness of a singular SDE for signed Coulomb particles

We consider an SDE system for signed Coulomb particles moving in $\mathbb R^2$. Due to the singular Coulomb interaction force, collisions between particles of opposite sign will happen in finite time. Upon collision, the colliding particles are removed from the system. Our main results are the existence and uniqueness of strong global solutions and the characterization of all possible collisions. The challenge of the proofs is to deal with the singularity of the interactions. We overcome this by using scaling invariance of the process and by putting together several tools from [FT25] developed for the similar Keller--Segel particle system.

math.AP↗

arXiv2026

Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle creation and annihilation with birth-death rates depending locally on the particle configuration. In the macroscopic limit, the particle density evolves according to a nonlinear reaction--diffusion equation, where the reaction term is explicitly determined by the microscopic rates. The key new ingredient is a replacement lemma adapted to the fractal geometry of the Sierpiński gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpiński gasket graph, which require new arguments due to the absence of classical lattice structures.

math.PR↗

arXiv2026

Non-equilibrium functional inequalities for finite Markov chains

Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.

math.PR↗

arXiv2024

Global existence and mean-field limit for a stochastic interacting particle system of signed Coulomb charges

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.

math.PR↗

arXiv2024

Discrete-to-continuum limits of interacting particle systems in one dimension with collisions

We study a class of interacting particle systems in which $n$ signed particles move on the real line. At close range particles with the same sign repel and particles with opposite sign attract each other. The repulsion and attraction are described by the same singular interaction force $f$. Particles of opposite sign may collide in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Požár and the author, one particular particle system of this type was studied; that in which $f(x) = \frac1x$ is the Coulomb force. Global well-posedness of this particle system was shown and a discrete-to-continuum limit (i.e. $n \to \infty$) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. In the present paper we extend these results to a large class of particle systems in order to cover many new applications. Motivated by these applications, we consider the presence of an external force $g$, consider interaction forces $f$ with a large range of singularities and allow $f$ to scale with $n$. To handle this class of $f$ we develop several new proof techniques in addition to those used for the Coulomb force.

math.AP↗

arXiv2024

Incompressible limit for weakly asymmetric simple exclusion processes coupled through collision

We establish the incompressible limit of weakly asymmetric simple exclusion processes coupled through particle collisions. The incompressible limit depends on various parameters in the particle system and is linked to fluid dynamics equations. Our main contributions to previous results are the extension of the parameter space and the focus on local particle jumps. Our proof uses the relative entropy method. The main novelties in the proof are a Boltzmann-Gibbs principle (a replacement lemma) and a spectral gap estimate.

math.PR↗

arXiv2023

Hölder regularity for the trajectories of generalized charged particles in 1D

We prove Hölder regularity for the trajectories of an interacting particle system. The particle velocities are given by the nonlocal and singular interactions with the other particles. Particle collisions occur in finite time. Prior to collisions the particle velocities become unbounded, and thus the trajectories fail to be of class $C^1$. Our Hölder-regularity result supplements earlier studies on the well-posedness of the particle system which imply only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relies on standard ODE techniques: we transform the system into different variables to expose and exploit the hidden monotonicity properties.

math.DS↗

arXiv2022

Motion by mean curvature from Glauber-Kawasaki dynamics with speed change

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a `Boltzmann-Gibbs' principle which covers a class of local particle interactions.

math.PR↗

arXiv2022

Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics

We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.

math.PR↗

arXiv2022

Discrete Dislocations Dynamics with annihilation as the limit of the Peierls-Nabarro model in one dimension

Plasticity of metals is the emergent phenomenon of many crystal defects (dislocations) which interact and move on microscopic time and length scales. Two of the commonly used models to describe such dislocation dynamics are the Peierls-Nabarro model and the so-called discrete dislocation dynamics model. However, the consistency between these two models is known only for a few number of dislocations or up to the first time at which two dislocations collide. In this paper we resolve these restrictions, and establish the consistency for any number of dislocations and without any restriction on their initial position or orientation. In more detail, the evolutive Peierls-Nabarro model which we consider describes the evolution of a phase-field function $v_\e(t,x)$ which represents the atom deformation in a crystal. The model is a reaction-diffusion equation of Allen-Cahn type with the half Laplacian. The small parameter $\ep$ is the ratio between the atomic distance and the typical distance between phase transitions in $v_\e$. The position of a phase transition determines the position of a dislocation, and the sign of the transition (up or down) determines the orientation. The goal of this paper is to derive the asymptotic behavior of the function $v_\e$ as $\ep\to0$ up to arbitrary end time $T$; in particular beyond collisions. We prove that $v_\e$ converges to a piecewise constant function $v$, whose jump points in the spatial variable satisfy the ODE system which represents discrete dislocation dynamics with annihilation. Our proof method is to explicitly construct and patch together several sub- and supersolutions of $v_\e$, and to show that they converge to the same limit $v$.

math.AP↗

arXiv2022

Convergence rates for energies of interacting particles whose distribution spreads out as their number increases

We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.

math.AP↗

arXiv2022

Discrete-to-continuum convergence of charged particles in 1D with annihilation

We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are effectively removed from the system (annihilation). The two applications we have in mind are vortices and dislocations in metals. In this paper we reach two goals. First, we develop a rigorous solution concept for the interacting particle system with annihilation. The main innovation here is to provide a careful management of the annihilation of groups of more than two particles, and we show that the definition is consistent by proving existence, uniqueness, and continuous dependence on initial data. The proof relies on a detailed analysis of ODE trajectories close to collision, and a reparametrization of vectors in terms of the moments of their elements. Secondly, we pass to the many-particle limit (discrete-to-continuum), and recover the expected limiting equation for the particle density. Due to the singular interactions and the annihilation rule, standard proof techniques of discrete-to-continuum limits do not apply. In particular, the framework of measures seems unfit. Instead, we use the one-dimensional feature that both the particle system and the limiting PDE can be characterized in terms of Hamilton--Jacobi equations. While our proof follows a standard limit procedure for such equations, the novelty with respect to existing results lies in allowing for stronger singularities in the particle system by exploiting the freedom of choice in the definition of viscosity solutions.

math.AP↗

arXiv2021

Expansions for the linear-elastic contribution to the self-interaction force of dislocation curves

The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the nonlocal interaction between dislocation curve segments. While these nonlocal segment-segment interactions can be accurately described by linear elasticity when the segments are further apart than the atomic scale of size $\varepsilon$, this model breaks down and blows up when the segments are $O(\varepsilon)$ apart. To separate the nonlocal interactions from the local contribution, various models depending on $\varepsilon$ have been constructed to account for the nonlocal term. However, there are no quantitative comparisons available between these models. This paper makes such comparisons possible by expanding the self-interaction force in these models in $\varepsilon$ beyond the $O(1)$-term. Our derivation of these expansions relies on asymptotic analysis. The practical use of these expansions is demonstrated by developing numerical schemes for them, and by -- for the first time -- bounding the corresponding discretization error.

math.AP↗

arXiv2021

Uniqueness of local, analytic solutions to singular ODEs

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.

math.DS↗

arXiv2021

Boundary-layer analysis of repelling particles pushed to an impenetrable barrier

This paper considers the equilibrium positions of $n$ particles in one dimension. Two forces act on the particles; a nonlocal repulsive particle-interaction force and an external force which pushes them to an impenetrable barrier. While the continuum limit as $n \to \infty$ is known for a certain class of potentials, numerical simulations show that a discrete boundary layer appears at the impenetrable barrier, i.e. the positions of $o(n)$ particles do not fit to the particle density predicted by the continuum limit. In this paper we establish a first-order $Γ$-convergence result which guarantees that these $o(n)$ particles converge to a specific continuum boundary-layer profile.

math.AP↗

arXiv2020

Particle dynamics with elastic collision at the boundary: existence and partial uniqueness of solutions

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $Ω$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.

math.AP↗

arXiv2020

Quantitative estimate of the continuum approximations of interacting particle systems in one dimension

We consider a large class of interacting particle systems in 1D described by an energy whose interaction potential is singular and non-local. This class covers Riesz gases (in particular, log gases) and applications to plasticity and approximation theory of functions. While it is well established that the minimisers of such interaction energies converge to a certain particle density profile as the number of particles tends to infinity, any bound on the rate of this convergence is only known in special cases by means of quantitative estimates. The main result of this paper extends these quantitative estimates to a large class of interaction energies by a different proof. The proof relies on one-dimensional features such as the convexity of the interaction potential and the ordering of the particles. The main novelty of the proof is the treatment of the singularity of the interaction potential by means of a carefully chosen renormalisation.

math.AP↗