arXiv subjects

Florian Theil

Publications and source records attributed to Florian Theil.

13 recordsLinked to original sources

arXiv2025

Leading order asymptotics for non-local energies and the Read-Shockley law

We study an energy minimization problem $\sum_{i \neq j} W(z_i - z_j)$ for $N$ points $\left\{z_1, \dots, z_N\right\}$ with applications in dislocation theory. The $N$ points lie in the two-dimensional domain $\mathbb{R} \times [-\pi, \pi]$, %who are trying to minimize their interaction energy where where the kernel $W$ is derived from the Volterra potential $V(x,y) = \frac{x^2}{x^2+y^2}-\frac12\log(x^2+y^2)$. We prove that the minimum energy is given by $- N \log{N} +\mathcal{O}(N)$. This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals.

math.CA↗

arXiv2022

Coupling Temperature Distribution with the Single Particle Model

The DFN (Doyle-Fuller-Newman) model is well know for being accurate and computationally expensive. In situations where temperature gradients are important (eg fast charging) it is desirable to couple the temperature dynamics within a battery into the DFN model. This leads to even greater computational complexity. Inspired by the work of Marquis et al [1] we present the derivation of a reduced-order model based on the DFN model with temperature in the macroscale. The complexity of the reduced-order model is characterised by the local temperature plus one internal electro-chemical dimension and the electrolyte dynamics is accounted for by a simple correction term.

physics.chem-ph↗

arXiv2021

Long range order in atomistic models for solids

The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read-Shockley law for grain boundaries, which we rigorously derive in a simple, explicit, geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza-Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques.

math.AP↗

arXiv2020

Distribution of cracks in a chain of atoms at low temperature

We consider a one-dimensional classical many-body system with interaction potential of Lennard-Jones type in the thermodynamic limit at low temperature $1/β\in(0,\infty)$. The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with $N$ particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of $N\exp(- βe_\mathrm{surf}/2)$ with $e_\mathrm{surf}>0$ a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.

math-ph↗

arXiv2020

Derivation of an Effective Thermal Electrochemical Model for Porous Electrode Batteries using Asymptotic Homogenisation

Thermal electrochemical models for porous electrode batteries (such as lithium ion batteries) are widely used. Due to the multiple scales involved, solving the model accounting for the porous microstructure is computationally expensive, therefore effective models at the macroscale are preferable. However, these effective models are usually postulated ad hoc rather than systematically upscaled from the microscale equations. We present an effective thermal electrochemical model obtained using asymptotic homogenisation, which includes the electrochemical model at the cell level coupled with a thermal model that can be defined either at the cell or the battery level. The main aspects of the model are the consideration of thermal effects, the diffusion effects in the electrode particles, and the anisotropy of the material based on the microstructure, all of them incorporated in a systematic manner. We also compare the homogenised model with the standard electrochemical Doyle, Fuller & Newman model.

physics.chem-ph↗

arXiv2019

Surface energy and boundary layers for a chain of atoms at low temperature

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $β^{-1}$ goes to zero. Our main results are: (1) As $β\to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $μ_β$ and $ν_β$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathrm{surf}$. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $\overline{\mathcal{E}}_\mathrm{surf}$. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in $β$.

math.PR↗

arXiv2018

Rescaled Objective Solutions of Fokker-Planck and Boltzmann equations

We study the long-time behavior of symmetric solutions of the nonlinear Boltzmann equation and a closely related nonlinear Fokker-Planck equation. If the symmetry of the solutions corresponds to shear flows, the existence of stationary solutions can be ruled out because the energy is not conserved. After anisotropic rescaling both equations conserve the energy. We show that the rescaled Boltzmann equation does not admit stationary densities of Maxwellian type (exponentially decaying). For the rescaled Fokker-Planck equation we demonstrate that all solutions converge to a Maxwellian in the long-time limit, however the convergence rate is only algebraic, not exponential.

math.AP↗

arXiv2017

Long Range Particle Dynamics and the Linear Boltzmann Equation

This paper gives the first full proof of the justification of the linear Boltzmann equation from an underlying long range particle evolution. We suppose that a tagged particle is interacting with a background via a two body potential that is decaying faster than $ C\exp\left(-C|x|^{\frac{3}{2}}\right) $, and that the background is initially distributed according to a function in $L^1((\mathbb{R}^3,(1+|v|^2)\mathrm{d}v)$ in velocity and uniformly in space. Under finite mass and energy assumptions on the initial density, the tagged particle density converges weak-$\star$ in $L^\infty$ to a solution of the linear Boltzmann equation. The proof uses estimates on two body scattering and on the relationship between long range dynamics and dynamics with a truncated interaction potential to explicitly estimate the error between densities for long and short range dynamics. To compare the difference between the short range dynamics and the linear Boltzmann equation, we use a tree based structure to encode the collisional history of the tagged particle.

math.AP↗

arXiv2017

The Derivation of the Linear Boltzmann Equation from a Rayleigh Gas Particle Model

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.

math.AP↗

arXiv2016

Asymptotic Analysis of the Ginzburg-Landau Functional on Point Clouds

The Ginzburg-Landau functional is a phase transition model which is suitable for clustering or classification type problems. We study the asymptotics of a sequence of Ginzburg-Landau functionals with anisotropic interaction potentials on point clouds $Ψ_n$ where $n$ denotes the number data points. In particular we show the limiting problem, in the sense of $Γ$-convergence, is related to the total variation norm restricted to functions taking binary values; which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

math.AP↗

arXiv2015

Convergence of the $k$-Means Minimization Problem using $Γ$-Convergence

The $k$-means method is an iterative clustering algorithm which associates each observation with one of $k$ clusters. It traditionally employs cluster centers in the same space as the observed data. By relaxing this requirement, it is possible to apply the $k$-means method to infinite dimensional problems, for example multiple target tracking and smoothing problems in the presence of unknown data association. Via a $Γ$-convergence argument, the associated optimization problem is shown to converge in the sense that both the $k$-means minimum and minimizers converge in the large data limit to quantities which depend upon the observed data only through its distribution. The theory is supplemented with two examples to demonstrate the range of problems now accessible by the $k$-means method. The first example combines a non-parametric smoothing problem with unknown data association. The second addresses tracking using sparse data from a network of passive sensors.

math.ST↗

arXiv2014

Face-centered cubic crystallization of atomistic configurations

We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance as it presents a significant milestone towards understanding, on the atomistic level, phenomena such as melting or plastic behavior. We characterize a set of rotation- and translation-invariant two- and three-body energies such that the energy minimum per particle over all n-particle configurations converges to the energy per particle in the face-centered cubic (fcc) lattice as tends to infinity. The proof involves a careful analysis of the symmetry properties of the fcc lattice.

math.AP↗

arXiv2006

Energy transport by acoustic modes of harmonic lattices

We study the large scale evolution of a scalar lattice excitation which satisfies a discrete wave-equation in three dimensions. We assume that the dispersion relation associated to the elastic coupling constants of the wave-equation is acoustic, i.e., it has a singularity of the type |k| near the vanishing wave vector, k=0. To derive equations that describe the macroscopic energy transport we introduce the Wigner transform and change variables so that the spatial and temporal scales are of the order of epsilon. In the continuum limit, which is achieved by sending the parameter epsilon to 0, the Wigner transform disintegrates into three different limit objects: the transform of the weak limit, the H-measure and the Wigner-measure. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave-equation for the weak limit of the rescaled initial data, a dispersive transport equation for the regular limiting Wigner measure, and a geometric optics transport equation for the H-measure limit of the initial data concentrating to k=0. A simple consequence of our result is the complete characterization of energy transport in harmonic lattices with acoustic dispersion relations.

math-ph↗