Attractive Repulsive Potentials Research Articles

Enabling attractive-repulsive potentials in binary-collision-approximation monte-carlo codes for ion-surface interactions

Binary Collision Approximation (BCA) codes for ion-material interactions, such as SRIM, Tridyn, F-TRIDYN, and SDtrimSP, have historically been limited to screened Coulomb potentials even at low energies due to the difficulty in numerically solving the Distance of Closest Approach (DOCA) problem for attractive-repulsive potentials. Techniques such as direct n-body simulation or modifications to Newton’s method are either prohibitively costly or not guaranteed to work for all potentials. Advanced rootfinding techniques, such as companion matrix solvers, offer a solution. For many attractive-repulsive potentials, however, a companion matrix cannot be used directly, because there is no way to put the associated functions into a monomial basis form. A complementary technique is proxy rootfinding—by finding the best-fit polynomial approximant of a function, the zeros of the approximant can be guaranteed to be close to the zeros of the function. Using the Chebyshev basis and grid offers additional guarantees with regards to the quality of the approximation, the speed of convergence, and the avoidance of Runge’s phenomenon. By finding Chebyshev interpolants and using the Chebyshev-Frobenius companion matrix, the zeros of any real function on a bounded domain can be found. Here we show that using an Adaptive Chebyshev Proxy Rootfinder with Automatic Subdivision (ACPRAS) with appropriate scaling functions, numerical issues presented by attractive-repulsive potentials, including those of scale, can be handled. Using these techniques, we show that it is possible to include any physically reasonable interatomic potential in a BCA code, and to guarantee correctness of the resulting scattering angle calculations.

Lie-Group Type Quadcopter Control Design by Dynamics Replacement and the Virtual Attractive-Repulsive Potentials Theory

The aim of the present research work is to design a control law for a quadcopter drone based on the Virtual Attractive-Repulsive Potentials (VARP) theory. VARP theory, originally designed to enable path following by a small wheeled robot, will be tailored to control a quadcopter drone, hence allowing such device to learn flight planning. The proposed strategy combines an instance of VARP method to control a drone’s attitude (SO(3)-VARP) and an instance of VARP method to control a drone’s spatial location (R3-VARP). The resulting control strategy will be referred to as double-VARP method, which aims at making a drone follow a predefined path in space. Since the model of the drone as well as the devised control theory are formulated on a Lie group, their simulation on a computing platform is performed through a numerical analysis method specifically designed for these kinds of numerical simulations. A numerical simulation analysis is used to assess the salient features of the proposed regulation theory. In particular, resilience against shock-type disturbances are assessed numerically.

Virtual Attractive-Repulsive Potentials Control Theory: A Review and an Extension to Riemannian Manifolds

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds.

Anticipation Breeds Alignment

We study the large-time behavior of systems driven by radial potentials, which react to anticipated positions, \(\mathbf{x}^\tau (t)=\mathbf{x}(t)+\tau \mathbf{v}(t)\), with anticipation increment \(\tau >0\). As a special case, such systems yield the celebrated Cucker–Smale model for alignment, coupled with pairwise interactions. Viewed from this perspective, such anticipation-driven systems are expected to emerge into flocking due to alignment of velocities, and spatial concentration due to confining potentials. We treat both the discrete dynamics and large crowd hydrodynamics, proving the decisive role of anticipation in driving such systems with attractive potentials into velocity alignment and spatial concentration. We also study the concentration effect near equilibrium for anticipated-based dynamics of pair of agents governed by attractive–repulsive potentials.

Uniform in Time L ∞ $L^{\infty }$ -Estimates for Nonlinear Aggregation-Diffusion Equations

We derive uniform in time $L^{\infty }$ -bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the repulsive part of the potential has a weaker singularity ( $2-n\leq B< A\leq 2$ ), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time $L^{\infty }$ -bound. When the repulsive part of the potential has a stronger singularity ( $-n< B<2-n \leq A\leq 2$ ), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case $2-n\leq A\leq 2$ within the diffusion dominated regime.

A reaction model for cement solidification: Evolving the C–S–H packing density at the micrometer-scale

Cement paste is a multiphase material of complex chemistry, of which 60% by volume is typically composed of calcium–silicate–hydrates (C–S–H), the phase that lends the material its strength and stiffness. Moreover, it has been shown that the C–S–H phase is a dispersion of nanometer-sized particles that are characterized by an attractive–repulsive potential and densify in course of the hydration reaction. Herein, we model the nucleation and growth of the nanoparticles as a continuous density field subject to a reaction equation. Using this phase-field approach, we aim to reduce the parameter space present in similar hydration models and create a vehicle to upscale mechanical information from the nanometer-scale to the micrometer-scale. Despite the apparent simplification of the physics at play, we readily reproduce the cement paste reaction kinetics and microtexture—functions of temperature, coarseness of the calcium–silicate source particles, and initial water-to-cement ratio—, and vet them against experimental observations. Presenting results for two-dimensional simulations, we achieve excellent agreement with measurements of hydration heat curves, pore-chord-length and solid-chord-length density functions, distributions of low- and high-density C–S–H products, and elasticity.

Discrete minimisers are close to continuum minimisers for the interaction energy

Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $\Gamma$-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.

Spatial Formation Control with Volume Information: Application to Quadcopter UAV’s

Abstract: This paper extends the distance-based formation control for the case of holonomic robots moving in 3D space. The approach is addressed for agents modeled as double-integrators with any undirected communication graphs. The control strategy uses a combined distance-based attractive-repulsive potentials to ensure convergence to the formation pattern avoiding possible inter-robot collisions. In order to avoid unwanted formation patterns that verify the distance constraints, each robot control law includes a volume condition which provides information about the unique desired position of each robot in the formation pattern. The proposed algorithm is tested by numerical simulations and extended to the case of quadcopters UAV’s by an input-output linearization showing good behavior.

Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

Conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles. We address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities. Currently, no existence theory for such potentials is available. We develop and compare two complementary solution methods, a continuous pseudoinverse method and a discrete stochastic lattice approach, and formally show a connection between the two. Interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive-repulsive potential are determined. For a swarming Morse potential, characteristic slow-fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches. The discrete approach is found to be remarkably robust to modifications in movement rules, related to the potential function. The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations.

Role of the electrical double layer in particle deposition by convective diffusion

A theoretical analysis is presented to describe the deposition of Brownian particles which interact with collectors through an attractive-repulsive potential. When the interactions are confined sufficiently near the collector surface, it is shown that the process is equivalent to ordinary convective diffusion in the bulk with a first-order surface reaction at the collector; a formula for the surface reaction coefficient is derived in terms of the net interaction potential. Large Peclet solutions appropriate to hydrocolloid collection by spheres and cylinders with first-order surface reaction are obtained and available solutions are discussed. The ratio of overall deposition to deposition under diffusion control proves to be a function of a single dimensionless parameter. At extreme values of this parameter collection becomes surface reaction controlled or diffusion controlled.

Estimate of Bounds on the Minimum Position of an Attractive-Repulsive Potential from Elastic Scattering Data

Estimate of Bounds on the Minimum Position of an Attractive-Repulsive Potential from Elastic Scattering Data