Stochastic particle method

The first maj or extension of the stochastic particle method was made by O Rourke 5501 who developed a new method for calculating droplet collisions and coalescences. Consistent with the stochastic particle method, collisions are calculated by a statistical, rather than a deterministic, approach. The probability distributions governing the number and nature of the collisions between two droplets are sampled stochastically. This method was initially applied to diesel sprays13171... 

Particle method for turbulent flows Integration of stochastic model equations. 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

Another method of controlling the temperature that can be used in CP MD is the stochastic thermostat of Andersen.27 In this approach the velocity of randomly selected nucleus is rescaled this corresponds in a way to the stochastic collisions with other particles in the system. Therefore, this approach is often called a stochastic collision method. The Andersen thermostat has recently been shown28 to perform very well in the Car-Parinello molecular dynamic simulations of bimolecular chemical reactions. 

An alternative to stochastic reconstruction of multiphase media is the reconstruction based on the direct simulation of processes by which the medium is physically formed, e.g., phase separation or agglomeration and sintering of particles to form a porous matrix. An advantage of this approach is that apart from generating a medium for the purpose of further computational experiments, the reconstruction procedure also yields information about the sequence of transformation steps and the processing conditions required in order to form the medium physically. It is thereby ensured that only physically realizable structures are generated, which is not necessarily the case when a stochastic reconstruction method such as simulated annealing is employed. 

As a result of the mechanical action of mixing tools in high intensity mixers (see Section 7.4.2) an aerated, turbulent particulate matter system with stochastic particle movement develops. Similar conditions exist if the particles are suspended in a fluidized bed. The main difference between the two methods is that in the mixers particle movement is caused by mechanical forces while in fluidized beds drag forces, that are induced by a flow of gas, are the main reason for the movement of the particulate matter. Therefore, fluidized beds are not only used as excellent environments in which gas efficiently and intimately contacts particles but also for dry mixing of particulate solids and coalescence of particles which, in the presence of binding mechanisms, causes agglomeration. 

Instead of transforming a MOO problem into a SOO problem and solving it repeatedly, researchers have modified stochastic SOO methods for solving MOO problems. MOO methods such as NSGA-n, I-MODE and Multi-objective Particle Swarm optimization (MOPSO) can generate many Pareto-optimal solutions in a single run even for problems with many... 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

In recent years, new discrete-particle methods have been developed for modeling physical and chemical phenomena occurring in the mesoscale. The most popular are grid-type techniques such as cellular automata (CA), LG, LBG, diffusion- and reaction-limited aggregation [37] and stochastic gridless methods, e.g., DSMC used for modeling systems characterized by a large Knudsen number [41], and SRD [39]. Unlike DSMC, in SRD collisions are modeled by simultaneous stochastic rotation of the relative velocities of every particle in each cell. 

Atomistic MD becomes very ineffident for low-density problems, for example, for long, single chains. For this method to be useful, we need to compute long enough for the chain to evolve on the maaoscopic time and space scales while using a specified miaoscopic particle interaaion law. The time step, however, is limited. If the chain units are too far apart, it could take many millions of time steps to simulate a few collisions. The expense becomes unacceptably high. Thus, MD can be too ineffeaive for dilute systems. Stochastic MC methods, event-driven molecular dynamics, or a hybrid pivot MC/MD generation procedure can be much more successful. 

The integration of the single-particle joint-PDF transport equation (12.4.1-11) is tedious. Computer requirements for standard CFD techniques rise exponentially with the dimensionality of the joint-PDF. Therefore, micro-PDF methods commonly use a Monte-Carlo approach [Spielman and Levenspiel, 1965 Kattan and Adler, 1967, 1972 Pope, 1981]. A deterministic system is constructed with stochastic particles whose joint-PDF evolves in the same way as the joint-PDF of fluid particles. The trajectories of the so-called conditional particles define a formal solution of the joint-PDF transport equation (12.4.1-11). Ramkrishna [2000] presents details on the computational methods. 

During the last decade two novel stochastic optimisation methods came into use which, like evolutionary algorithms, are based on heuristics inspired by nature. These methods are particle swarm optimisation (Kennedy and Eberhart, 2001), and ant colony optimisation (Dorigo and... 

Particle Swarm Optimization (PSO) is a stochastic optimization method evolved from Swarm Theory and Evolutionary Computation [4]. It is instigated by animals natural swarming behavior [5]. PSO has been proven to be a suitable technique for solving various optimization problems [6, 7]. Among the advantages of PSO are that it allows efficient and rapid optimization of the problem, due to its parallel nature, it requires only basic mathematical operators for optimization and it provides low computational and memory costs for each iteration [8]. Many variants of the PSO algorithm exist, such as PSO with inertia weight [9], PSO with constriction factor [10], and mutative PSO [11]. 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

To carry out this method, values are chosen for Tq, the desired temperature, and v, the mean frequency with which each particle experiences a stochastic collision. If successive collisions are uncorrected, then the distribution of time intervals between two successive stochastic collisions, P(v, t), is of the Poisson fonn. 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

Steadiness of vacuum and one-particle states, 657 Steck B.,5Z8 Steck operator, 538 Steepest descents, method of, 62 Stochastic processes, 102,269 Strangeness quantum number, 516 Strategic saddle point, 309 Strategy, 308 mixed, 309... 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

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