Stochastic particle

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

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... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Although the iGLE with the nonstationarity of Eq. (15) is formally correct, it is nonetheless too strict. The imderlying assumption is that the environment is homogeneous at a given time t, and hence the solvation of the environment to each stochastic particle is exactly the same and characterized by g(t). However, in many cases, each of the particles will be in a unique environment, and can each be characterized by its own iGLE,... 

Figure 4 The average position of a stochastic particle in a double well is displayed as a function of time for the two different environments of Fig. 3 and for two different nonequilibrated initial conditions corresponding to localization at each of the wells.

The classical motion of a stochastic particle in the potential defined by Eq. (25) may be described by the following set of equations ... 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

Braumann, a., Kraft, M. Wagner, W. 2010 Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation. Journal of Computational Physics 229, 7672-7691. 

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. 

Quasi-elastic neutron scattering (QENS) is related to stochastic particle motions. Because the displacements are random, the diffusive motion of particles in liquids caimot be quantized, and the energies are continuously distributed. Unlike the case of cooperative motions like phonons in solids or molecular vibrational excitations, in the dynamic scattering function S(Q,(o), there are no 5-functions at finite momentum and energy transfers. Instead, the dynamic scattering function is centered at zero-energy transfer with a characteristic quasielastic line width proportional to the diffusivity of the particles. 

The basic idea of the MC approach lies in the discrete representation of the joint PDF by an ensemble of stochastic particles. Each particle carries an array of properties denoting position, velocity and scalar composition. During a fractional time stepping procedure [6] the particles are submitted to certain deterministic and stochastic processes changing each particle s set of properties in accordance with the different terms in the PDF evolution equation. Afterwards the statistical moments may be derived in the simplest case by averaging from the ensemble of particles. 

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. 

Other experiments revealed an increased rate of transformation or mutation compared to the expectation value based on the number of traversed cells [94,95]. Furthermore, induction of gene expression has been studied using directed as well as stochastic particle Irradiation. These experiments also indicate a more pronounced increase of the response than expected on the basis of the number of tra-... 

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