Discrete-particles boundary conditions

The probability density for a particle at a location is proportional to the square of the wavefunction at that point the wavefunction is found by solving the Schrodinger equation for the particle. When the equation is solved subject to the appropriate boundary conditions, it is found that the particle can possess only certain discrete energies. 

An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus. We can therefore expect the electron s wavefunctions to obey certain boundary conditions, like the constraints we encountered when fitting a wave between the walls of a container. As we saw for a particle in a box, these constraints result in the quantization of energy and the existence of discrete energy levels. Even at this early stage, we can expect the electron to be confined to certain energies, just as spectroscopy requires. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

Fig. 7. Schematic illustrating the coupling of a fluid-phase mass transfer model with a discrete, particle model, such as KMC, through the boundary condition. The continuum model passes the external field and the KMC simulation computes spatial and temporal rates that are needed in the boundary condition of the continuum model.

The solver used for this study is the same as in Chapter 9 a parallel fully compressible code for turbulent reacting two-phase flows, on both structured and unstructured grids. The fully explicit finite volume solver uses a cell-vertex discretization with a Lax-Wendroff centered numerical scheme [296] or a third order in space and time scheme named TTGC [268]. Characteristic boundary conditions NSCBC [339 329] are used for the gas phase. Boundary conditions are easier for the dispersed phase, except for solid walls where particles may bounce off. In the present study it is simply supposed that the particles stick to the wall, with either a slip or zero velocity. 

In the first instance, this applies only to a particle with constant kinetic energy. Nevertheless, it shows with unexpected clarity the origin of quanti sation—when a particle is bound within a certain region of space it can exist only in certain discrete energy states. For if the particle is now confined to a certain region, by means of barriers a distance I apart, p must vanish at and outside the end points and these boundary conditions require that... 

Almost all QMC calculations in periodic boundary conditions have assumed that the phase of the wave function returns to the same value if a particle goes around the periodic boundaries and returns to its original position. However, with these boundary conditions, delocalized fermion systems converge slowly to the thermodynamic limit because of shell effects in the filling of single particle states. Indeed, with periodic boundary conditions the Fermi surface of a metal will be reduced to a discrete set of points in k-space. The number of k-points is equal to the number of electrons of same spin and therefore it is quite limited. 

LBM was originally proposed by McNamara and Zanetti [3] to circumvent the limitations of statistical noise that plagued lattice gas automata (EGA). LBM is a simplified kinetic (mesoscopic) and discretized approximation of the continuous Boltzmarui equation. LBM is mesoscopic in nature because the particles are not directly related to the number of molecules like in DSMC or MD but representative of a collection of molecules. Hence, the computational cost is less demanding compared with DSMC and MD. Typical LBM consists of the lattice Boltzmann equation (LBE), lattice stmcture, transformation of lattice units to physical units, and boundary conditions. 

Abstract A novel lattice-gas approach has been developed to model the effect of molecular interactions on dynamic interfacial structure and flows of liquid-vapor and liquid-liquid systems in microcapillaries, Within a mean-field approximation, discrete time evolution of species and momentum densities consists of alternating convective and diffusive steps subject to local conservation laws. Stick boundary conditions imposed during the convective step cause momentum transfer to lattice particles in contact... 

This example is selected with a view to show how discrete particle states can arise rather than develop a very realistic model of a yeast population. Also, it gives us an opportunity to discuss differences in the boundary condition from that used in the previous example. 

---