arXiv subjects

Matthew Dobson

Publications and source records attributed to Matthew Dobson.

At least 19 recordsLinked to original sources

arXiv2025

What Do Bouncing Balls Tell Us About the Universe? A Journey into Billiard Systems

Have you ever played or watched a game of pool? If so, you have already seen a billiard system in action. In mathematics and physics, a billiard system describes a ball that moves in straight lines and bounces off walls. Despite these simple rules, billiard systems can produce remarkably rich behaviors: some table shapes generate regular, periodic patterns, while others give rise to complete chaos. Scientists also study what happens when we shrink the ball down to the size of an electron to a world where quantum effects take over and the familiar reflection rules no longer apply. In this article, we discuss billiard systems in their many forms and show how such a simple setup can reveal fundamental insights into the behavior of nature at both classical and quantum scales.

math.DS↗

arXiv2025

Mesoscale Modeling of an Active Colloid's Motion

This paper uses Cahn-Hilliard equations as a mesoscale model of the motion of active colloids. The model attempts to capture the driving mechanisms and qualitative behavior of the isotropic colloids originally proposed by J. Decayeaux in 2021. We compare our model against the single colloid behavior presented in that work, as well as against multi-colloid systems.

cond-mat.soft↗

arXiv2022

Convergence of Nonequilibrium Langevin Dynamics for Planar Flows

We prove that incompressible two dimensional nonequilibrium Langevin dynamics (NELD) converges exponentially fast to a steady-state limit cycle. We use automorphism remapping periodic boundary conditions (PBCs) techniques such as Lees-Edwards PBCs and Kraynik-Reinelt PBCs to treat respectively shear flow and planar elongational flow. After rewriting NELD in Lagrangian coordinates, the convergence is shown using a technique similar to [R. Joubaud, G. A. Pavliotis, and G. Stoltz,2014].

math.PR↗

arXiv2021

Simple Periodic Boundary Conditions for Molecular Simulation of Uniaxial Flow

We present rotating periodic boundary conditions (PBCs) for the simulation of nonequilibrium molecular dynamics (NEMD) under uniaxial stretching flow (USF) or biaxial stretching flow (BSF). Such nonequilibrium flows need specialized PBCs since the simulation box deforms with the background flow. The technique builds on previous models using one or lattice remappings, and is simpler than the PBCs developed for the general three dimensional flow. For general three dimensional flows, Dobson \cite{Dobson} and Hunt \cite{Hunt} proposed schemes which are not time-periodic since they use more than one automorphism remapping. This paper presents a single automorphism remapping PBCs for USF and BSF which is time periodic up to a rotation matrix and has a better minimum lattice spacing properties.

physics.comp-ph↗

arXiv2020

A deep learning method for solving Fokker-Planck equations

The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by our early study in \cite{li2018data}, we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the the demand of data in the training process. Several high dimensional numerical examples are demonstrated.

math.NA↗

arXiv2020

SN 2019muj -- a well-observed Type Iax supernova that bridges the luminosity gap of the class

We present early-time ($t < +50$ days) observations of SN 2019muj (= ASASSN-19tr), one of the best-observed members of the peculiar SN Iax class. Ultraviolet and optical photometric and optical and near-infrared spectroscopic follow-up started from $\sim$5 days before maximum light ($t_{max}(B)$ on $58707.8$ MJD) and covers the photospheric phase. The early observations allow us to estimate the physical properties of the ejecta and characterize the possible divergence from a uniform chemical abundance structure. The estimated bolometric light curve peaks at 1.05 $\times$ 10$^{42}$ erg s$^{-1}$ and indicates that only 0.031 $M_\odot$ of $^{56}$Ni was produced, making SN 2019muj a moderate luminosity object in the Iax class with peak absolute magnitude of $M_{V}$ = -16.4 mag. The estimated date of explosion is $t_0 = 58698.2$ MJD and implies a short rise time of $t_{rise}$ = 9.6 days in $B$-band. We fit of the spectroscopic data by synthetic spectra, calculated via the radiative transfer code TARDIS. Adopting the partially stratified abundance template based on brighter SNe Iax provides a good match with SN 2019muj. However, without earlier spectra, the need for stratification cannot be stated in most of the elements, except carbon, which is allowed to appear in the outer layers only. SN 2019muj provides a unique opportunity to link extremely low-luminosity SNe Iax to well-studied, brighter SNe Iax.

astro-ph.HE↗

arXiv2020

SN 2019ehk: A Double-Peaked Ca-rich Transient with Luminous X-ray Emission and Shock-Ionized Spectral Features

We present panchromatic observations and modeling of the Calcium-rich supernova 2019ehk in the star-forming galaxy M100 (d$\approx$16.2 Mpc) starting 10 hours after explosion and continuing for ~300 days. SN 2019ehk shows a double-peaked optical light curve peaking at $t = 3$ and $15$ days. The first peak is coincident with luminous, rapidly decaying $\textit{Swift}$-XRT discovered X-ray emission ($L_x\approx10^{41}~\rm{erg~s^{-1}}$ at 3 days; $L_x \propto t^{-3}$), and a Shane/Kast spectral detection of narrow H$\alpha$ and He II emission lines ($v \approx 500$ km/s) originating from pre-existent circumstellar material. We attribute this phenomenology to radiation from shock interaction with extended, dense material surrounding the progenitor star at $r<10^{15}$ cm and the resulting cooling emission. We calculate a total CSM mass of $\sim$ $7\times10^{-3}$ $\rm{M_{\odot}}$ with particle density $n\approx10^{9}\,\rm{cm^{-3}}$. Radio observations indicate a significantly lower density $n < 10^{4}\,\rm{cm^{-3}}$ at larger radii. The photometric and spectroscopic properties during the second light curve peak are consistent with those of Ca-rich transients (rise-time of $t_r =13.4\pm0.210$ days and a peak B-band magnitude of $M_B =-15.1\pm0.200$ mag). We find that SN 2019ehk synthesized $(3.1\pm0.11)\times10^{-2} ~ \rm{M_{\odot}}$ of ${}^{56}\textrm{Ni}$ and ejected $M_{\rm ej} = (0.72\pm 0.040)~\rm{M_{\odot}}$ total with a kinetic energy $E_{\rm k}=(1.8\pm0.10)\times10^{50}~\rm{erg}$. Finally, deep $\textit{HST}$ pre-explosion imaging at the SN site constrains the parameter space of viable stellar progenitors to massive stars in the lowest mass bin (~10 $\rm{M_{\odot}}$) in binaries that lost most of their He envelope or white dwarfs. The explosion and environment properties of SN 2019ehk further restrict the potential WD progenitor systems to low-mass hybrid HeCO WD + CO WD binaries.

astro-ph.HE↗

arXiv2019

Using coupling methods to estimate sample quality for stochastic differential equations

A probabilistic approach for estimating sample qualities for stochastic differential equations is introduced in this paper. The aim is to provide a quantitative upper bound of the distance between the invariant probability measure of a stochastic differential equation and that of its numerical approximation. In order to extend estimates of finite time truncation error to infinite time, it is crucial to know the rate of contraction of the transition kernel of the SDE. We find that suitable numerical coupling methods can effectively estimate such rate of contraction, which gives the distance between two invariant probability measures. Our algorithms are tested with several low and high dimensional numerical examples.

math.NA↗

arXiv2019

An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker-Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker-Planck equations proposed in [15]. Based on these results, we propose a new `block solver' for the stationary Fokker-Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

math.NA↗

arXiv2017

Strong Convergence of Integrators for Nonequilibrium Langevin Dynamics`

Several numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the rate of convergence is analyzed. Due to the special deforming boundary conditions used, care must be taken when using standard stochastic integration schemes, and we demonstrate a loss of convergence for a naive implementation. We then present several first and second order schemes, in the sense of strong convergence.

math.NA↗

arXiv2016

On assessing the accuracy of defect free energy computations

We develop a rigorous error analysis for coarse-graining of defect-formation free energy. For a one-dimensional constrained atomistic system, we establish the thermodynamic limit of the defect-formation free energy and obtain explicitly the rate of convergence. We then construct a sequence of coarse-grained energies with the same rate but significantly reduced computational cost. We illustrate our analytical results through explicit computations for the case of harmonic potentials and through numerical simulations.

math.NA↗

arXiv2014

Cell List Algorithms for Nonequilibrium Molecular Dynamics

We present two modifications of the standard cell list algorithm for nonequilibrium molecular dynamics simulations of homogeneous, linear flows. When such a flow is modeled with periodic boundary conditions, the simulation box deforms with the flow, and recent progress has been made developing boundary conditions suitable for general 3D flows of this type. For the typical case of short-ranged, pairwise interactions, the cell list algorithm reduces computational complexity of the force computation from O($N^2$) to O($N$), where $N$ is the total number of particles in the simulation box. The new versions of the cell list algorithm handle the dynamic, deforming simulation geometry. We include a comparison of the complexity and efficiency of the two proposed modifications of the standard algorithm.

math.NA↗

arXiv2014

Periodic Boundary Conditions for Long-time Nonequilibrium Molecular Dynamics Simulations of Incompressible Flows

This work presents a generalization of the Kraynik-Reinelt (KR) boundary conditions for nonequilibrium molecular dynamics simulations. In the simulation of steady, homogeneous flows with periodic boundary conditions, the simulation box moves with the flow, and it is possible for particle replicas to become arbitrarily close, causing a breakdown in the simulation. The KR boundary conditions avoid this problem for planar elongational flow and general planar mixed flow [J. Chem. Phys 133, 14116 (2010)] through careful choice of the initial simulation box and by periodically remapping the simulation box in a way that conserves replica locations. In this work, the ideas are extended to a large class of three dimensional flows by using multiple remappings for the simulation box. The simulation box geometry is no longer time-periodic (which was shown to be impossible for uniaxial and biaxial stretching flows in the original work by Kraynik and Reinelt [Int. J. Multiphase Flow 18, 1045 (1992)]). The presented algorithm applies to all flows with nondefective flow matrices, and in particular, to uniaxial and biaxial flows.

math.NA↗

arXiv2012

Derivation of Langevin Dynamics in a Nonzero Background Flow Field

We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of [D. Dürr, S. Goldstein, and J. L. Lebowitz, 1981 and 1983; P. Calderoni, D. Dürr, and S. Kusuoka, 1989] and provides extensions to handle the nonzero background flow. The derived nonequilibrium Langevin dynamics is similar to the dynamics in [M. McPhie, et al., 2001]. Some numerical experiments illustrate the use of the obtained dynamic to simulate homogeneous liquid materials under flow.

math-ph↗

arXiv2011

There is no pointwise consistent quasicontinuum energy

Much work has gone into the construction of quasicontinuum energies that reduce the coupling error along the interface between atomistic and continuum regions. The largest consistency errors are typically pointwise $O(\frac{1}{\eps})$ errors, and in some cases this has been reduced to pointwise O(1) errors. In this paper we show that one cannot create a coupling method using a finite-range coupling interface that has o(1)-consistency in the interface, and we use this to give an upper bound on the order of convergence in discrete $w^{1,p}$-norms in 1D.

math.NA↗

arXiv2010

Symplectic schemes for highly oscillatory Hamiltonian systems: the homogenization approach beyond the constant frequency case

We follow up on our previous works which presented a possible approach for deriving symplectic schemes for a certain class of highly oscillatory Hamiltonian systems. The approach considers the Hamilton-Jacobi form of the equations of motion, formally homogenizes it and infers an appropriate symplectic integrator for the original system. In our previous work, the case of a system exhibiting a single constant fast frequency was considered. The present work successfully extends the approach to systems that have either one varying fast frequency or several constant frequencies. Some related issues are also examined.

math.NA↗

arXiv2010

The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a positive spectrum, which is identical to the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbour interactions. Moreover, we establish a bound on the condition number of a matrix of eigenvectors that is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent conjectures ([arXiv:0907.3861, Conjecture 2] and [arXiv:0910.2013, Conjecture 8]) which were based on numerical experiments. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms, as well as a new stability estimate for the QCF method.

math.NA↗

arXiv2009

Iterative Methods for the Force-based Quasicontinuum Approximation

Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular class of atomistic-continuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the force-based approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method.

math.NA↗