1D Particle Research Articles

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 achieve 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. Second, 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.

A novel algorithm for generating minimum energy points from identically charged particles in 1D, 2D and 3D unit hypercubes

Generating minimum energy points (MEPs) is an optimal solution of many real-world problems, such as the selection of best locations for hospitals inside a city that reduce the overcrowding and competition and avoid the less-populated regions. The key idea is considering these locations as charged particles with the same sign (i.e., repel each other) inside a box and distribute these points by minimizing the total electric potential energy (TEPE) among them. The practice demonstrated that most of the existing techniques for generating MEPs are complex, especially for non-mathematicians. Therefore, the greedy algorithm (GreA) is the classical widely used algorithm for its simplicity even though a satisfactory result is not guaranteed. This paper gives a novel algorithm for generating MEPs from identically charged particles in 1D, 2D and 3D unit hypercubes. The results show that the new algorithm distributes the points far away from each other to reduce the TEPE of the generated MEPs more effectively than the GreA. The new algorithm is a significant improvement of the GreA to overcome its unsatisfactory results. Therefore, the new algorithm in its current form or after some improvements is highly recommended to be used instead of the GreA for many different applications.

Prediction of mineral liberation characteristics of comminuted particles of high grade ores

In mineral processing, the liberation of valuable mineral is of key importance in achieving high recoveries from downstream separation processes such as froth flotation and gravity concentration. To quantify mineral liberation, information on ore texture of the parent rock as well as properties of comminuted particles is essential. These properties have been quantified by statistical measures such as the proximity function and covariance function, which were extracted from SEM images of parent rock and particle polished sections, using convenient and efficient image analysis techniques based on Labview™ software. To quantify fully liberated particles, a phase specific line segment function has been introduced and evaluated by placing random line segments on the image. It was also found that the ore texture assumptions made by Barbery are not valid for the high grade sulphide ore tested and the general applicability of these assumptions is therefore questionable. Using the measured information above, predictive liberation models to quantify volumetric grade distribution of particles in 1D and 3D have been developed based on Barbery’s work. Results show that the grade distributions of composite particles predicted from the proposed 1D model is closest to measured data than those of Barbery’s 1D model. The predictions using the proposed 3D model are similar to those predicted from Barbery’s model and are considered more realistic as the model does not rely on assumed ore texture but on measurements made on the parent rock and particle sections.

Modeling selective local interactions with memory

Recently we developed a stochastic particle system describing local interactions between cyanobacteria. We focused on the common freshwater cyanobacteria Synechocystis sp., which are coccoidal bacteria that utilize group dynamics to move toward a light source, a motion referred to as phototaxis. We were particularly interested in the local interactions between cells that were located in low to medium density areas away from the front. The simulations of our stochastic particle system in 2D replicated many experimentally observed phenomena, such as the formation of aggregations and the quasi-random motion of cells. In this paper, we seek to develop a better understanding of group dynamics produced by this model. To facilitate this study, we replace the stochastic model with a system of ordinary differential equations describing the evolution of particles in 1D. Unlike many other models, our emphasis is on particles that selectively choose one of their neighbors as the preferred direction of motion. Furthermore, we incorporate memory by allowing persistence in the motion. We conduct numerical simulations which allow us to efficiently explore the space of parameters, in order to study the stability, size, and merging of aggregations.

Dynamical properties of spatially anisotropic frustrated Heisenberg models in a magnetic field

We investigate spectral features of spatially anisotropic spin-1/2 frustrated antiferromagnets in a magnetic field by a weak-coupling approach from the one-dimensional (1D) limit. Using exact results of the Heisenberg chain, we calculate dynamical structure factors of spatially anisotropic frustrated antiferromagnets in two dimensions. We obtain rich peak structures depending on the momentum and the strength of magnetic fields. We identify the sharp peaks as bound and antibound states between particles carrying fractional Sz in 1D in a magnetic field, psinons and antipsinons, and those originating from 2-string solutions. We compare these spectral features with available experimental results on Cs2CuCl4, and suggest that the dynamical properties of Cs2CuCl4 in a magnetic field can be interpreted in terms of particles in 1D in a magnetic field.