Particle-in-cell

This example shows the round particle in cell B,B with two possible nonbonded cutoffs. With the outer cutoff, the round particle interacts with both the rectangle and its periodic image. By reducing the nonbonded cutoff to an appropriate radius (the inner circle), the round particle can interact with only one rectangle—in this case, the rectangle also in cell B,B. ... 

Approaches used to model ozone formation include box, gradient transfer, and trajectoty methods. Another method, the particle-in-cell method, advects centers of mass (that have a specific mass assigned) with an effective velocity that includes both transport and dispersion over each time step. Chemistry is calculated using the total mass within each grid cell at the end of each time step. This method has the advantage of avoiding both the numerical diffusion of some gradient transfer methods and the distortion due to wind shear of some trajectory methods. 

Particle-in-Cell Discharge Models 1.4.3.1. Particle-in-Cell Model Principles... 

A complete model for the description of plasma deposition of a-Si H should include the kinetic properties of ion, electron, and neutral fluxes towards the substrate and walls. The particle-in-cell/Monte Carlo (PIC/MC) model is known to provide a suitable way to study the electron and ion kinetics. Essentially, the method consists in the simulation of a (limited) number of computer particles, each of which represents a large number of physical particles (ions and electrons). The movement of the particles is simply calculated from Newton s laws of motion. Within the PIC method the movement of the particles and the evolution of the electric field are followed in finite time steps. In each calculation cycle, first the forces on each particle due to the electric field are determined. Then the... 

At the start of each modulation pulse, a sharp peak in optical emission is seen. Similar SiH emission peaks in pulsed plasmas have been found by Scarsbrook et al. [516] and Howling et al. [321]. The sharp peak was claimed to be caused by a pulse of high-energy electrons. Overzet and Verdeyen [517] measured electron densities at a 2.9-MHz excitation frequency and modulation frequencies up to 20 kHz. The optical emission of a SQWM argon plasma was measured by Booth et al. [518], who also performed particle-in-cell modeling. 

Kinetic detection provides an opportunity to assess the concentration of active particles in cell s volume through monitoring the electric pa-... 

The collisions that take place at the times x represent the effects of many real collisions in the system.1 These effective collisions are carried out as follows.2 The volume V is divided into Nc cells labeled by cell indices Each cell is assigned at random a rotation operator 6v chosen from a set Q of rotation operators. The center of mass velocity of the particles in cell , is Vj = AT1 JTJj v where is the instantaneous number of particles in the cell. The postcollision velocities of the particles in the cell are then given by... 

Here V = (N Vj + N2S2) / N + N2) is the mean velocity of the pair of cells. By summing over the particles in cell 1 it is easy to verify that velocities normal to collision rule treats the particles in the two cells as groups that undergo elasticlike collisions. 

Three dimensional particle-in-cell simulation performed with the numerical code CALDER [77] reveals that the unprecedented efficiency of this accelerator was due to the achievement of a physical regime in which multiple electron bunches are accelerated in the gas-jet plasma during the action of each laser shot. This effect is shown in Fig. 8.7 by a snap-shot from the simulation sequence. 

Particle-in-cell simulation, 154 Phonon stiffening, 36 Phonon-magnon coupled mode, 39 Photo-absorption cross section, 156 Photo-induced phase transitions, 42 Photo-nuclear activation, 173 PIC, 135... 

With A/(x), the right-hand side will just be the sum over all notional particles in cell / divided by Wp/. 

Sklarew et al. used a particle-in-cell K theory (pick) approach to... 

Sklarew et al. evaluated their particle-in-cell A -theory approach for atmospheric diffusion of carbon monoxide and for photochemical smog. All-day averages of carbon monoxide concentration were predicted to be within 20% of the measured averages at 12 monitoring stations, and the correlation coefficient of measured with observed concentrations was 0.73. 

Rept 2139(Nov 1957) [Numerical method "Particle-in Cell , used later by A. Vidart et al (Ref 21) 2b) F.H. Harlow, LASL Rept... 

Ibid, pp 527-37 [A brief description of the following numerical methods for calculation a) "Finite Difference Scheme in Lagrangian Coordinates , previously described by Goad (Ref 5) b) Particle-in Cell Method, previously described by Evans Harlow (Ref 1)... 

Subdivide the total volume Q into cells A and call nk the number of particles in cell X. The cells must be so small that inside each of them the above mentioned condition of homogeneity prevails. Let P( nk, t) be the joint probability distribution of all nk. At t + dt it will have changed because of two kinds of possible processes. Firstly, the nk inside each separate cell X may change by an event that creates or annihilates a particle. In the master equation for P( nk, t) this gives a corresponding term for each separate cell. 

Secondly, P changes because during the time dt a particle may move from a cell k into a cell /a (which need not be an adjacent one). The probability for a particle in cell k to arrive in p is proportional to A and dt and will therefore be denoted by w A dt. The corresponding contribution to the M-equation is... 

We demonstrate the method on the following concrete - if somewhat trivial - example. A swarm of particles is moving freely in space, but each particle has a probability a per unit time to disappear, through spontaneous decay or through a reactive collision. To cover the latter possibility we allow a to depend on v. The (r, u)-space is decomposed in cells A and nx is the number of particles in cell X. The joint probability distribution P( nx, t) varies through decay and through the motion of the particles. The decay is described by... 

Exercise. Consider an ideal Fermi gas whose particles can jump to another level because of the interaction with some heat bath. Subdivide the levels in cells labelled p, v,..., each containing N levels, and call their occupation numbers nfl(= 0, 1,..., N). A particle in cell p has a probability wVfl per unit time to jump to v when cell v is... 

Following the approach discussed in Section 2.2.2, let us divide the whole reaction volume V of the spatially extended system into N equivalent cells (domains) [81]. However, there is an essential difference with the mesoscopic level of treatment in Section 2.2.2 a number of particles in cells were expected to be much greater than unity. Note that this restriction is not imposed on the microscopic level of system s treatment. Their volumes are chosen to be so small that each cell can be occupied by a single particle only. (There is an analogy with the lattice gas model in the theory of phase transitions [76].) Despite the finiteness of vq coming from atomistic reasons or lattice discreteness, at the very end we make the limiting transition vo - 0, iV - oo, v0N = V, to the continuous pattern of point dimensionless particles. 

Kantor, A. B., Gibbons, I., Miltenyi, S., and Schmitz, J. (1998) Magnetic cell sorting with colloidal superparamagnetic particles, in Cell Separation Methods and Applications (Recktenwald, D., and Radbruch, A., eds.), Marcel Dekker, Inc., New York, pp. 153. 

Sherrill ME, Abdallah Jr. J, Csanak G, Dodd ES, Fukuda Y, Akahane Y, Aoyama M, Inoue N, Ueda H, Yamakawa K, Faenov AYa, Magunov AI, Pikuz TA, Skobelev IYu (2006) Spectroscopic characterization of an ultrashort laser driven Ar cluster target incorporating both Boltzmann and particle-in-cell models. Phys. Rev. E 73 0664041-0664046... 

In principle, this mode of operation creates the possibility of producing high-information-content displays due to the short frame times associated with bistable displays, since they are basically a memory effect and only new information must be changed. Unfortunately, metastable twist states of intermediate twist, which degrade the optical performance of the device, form around dust particles in cells with a cell gap below a certain value (d < 20 m). Therefore, since the response time is proportional to d, very long response times are observed ( 1 s) for LCDs with a cell gap above this critical value. These optically disruptive metastable twist states also form at the interface with spacers used to generate a uniform cell gap. 

Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

Particle in Cell Methods. An alternative to the direct finite-diflFerence solution of (7) is the so-called particle in cell (PIC) technique. The distinguishing feature of the PIC technique is that the continuous concentration field is treated as a collection of mass points, each representing a given amount of pollutant and each located at the center of mass of the volume of material it represents. The mass points, or particles, are moved by advection and diffusion. It is convenient but not necessary, to have each of the particles of a given contaminant represent the same mass of material. The application of the PIC technique in hydrodynamic calculations is discussed by Harlow (32). Here we consider the use of the PIC technique in the numerical solution of (7). 

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