Periodic boundary condition equations

This text of the two-volume treatment contains most of the theoretical background necessary to understand experiments in the field of phonons. This background is presented in four basic chapters. Chapter 2 starts with the diatomic linear chain. In the classical theory we discuss the periodic boundary conditions, equation of motion, dynamical matrix, eigenvalues and eigenvectors, acoustic and optic branches and normal coordinates. The transition to quantum mechanics is achieved by introducing the Sohpddingev equation of the vibrating chain. This is followed by the occupation number representation and a detailed discussion of the concept of phonons. The chapter ends with a discussion of the specific heat and the density of states. 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

The original particle mesh (P3M) approach of Hockney and Eastwood [42] treats the reciprocal space problem from the standpoint of numerically solving the Poisson equation under periodic boundary conditions with the Gaussian co-ion densities as the source density p on the right-hand side of Eq. (10). Although a straightforward approach is to... 

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions,... 

Consider a more general eigenvalue equation without imposing the periodic boundary condition,... 

In order to write down the microscopic equations of motion more formally, we consider a size N x N 4-neighbor lattice with periodic boundary conditions. At each site (i, j) there are four cells, each of which is associated with one of the four neighbors of site (i,j). Each cell at time t can be in one of two states defined by a Boolean variable where d = 1,..., 4 labels, respectively, the east, north,... 

Turbulent inlet conditions for LES are difficult to obtain since a time-resolved flow description is required. The best solution is to use periodic boundary conditions when it is possible. For the remaining cases, there are algorithms for simulation of turbulent eddies that fit the theoretical turbulent energy distribution. These simulated eddies are not a solution of the Navier-Stokes equations, and the inlet boundary must be located outside the region of interest to allow the flow to adjust to the correct physical properties. 

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

For homogeneous turbulent flows (no walls, periodic boundary conditions, zero mean velocity), pseudo-spectral methods are usually employed due to their relatively high accuracy. In order to simulate the Navier-Stokes equation,... 

During the time intervals between random eddy events, (4.37) is solved numerically using the scalar fields that result from the random rearrangement process as initial conditions. A standard one-dimensional parabolic equation solver with periodic boundary conditions (BCs) is employed for this step. The computational domain is illustrated in Fig. 4.3. For a homogeneous scalar field, the evolution of t) will depend on the characteristic length... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

With this simplification of the two dimensional step flow problem, we can study the long time behavior of the step train well beyond the initial onset of instability. We start with an array of 40 steps with small perturbations from an initial uniform configuration. We discretize the y coordinate so that each step has 2000 segments. Periodic boundary conditions are used in x and y direction. The time evolution problem of Eqs. (15) using (16) is converted into a set of difference equations. We control the time step so... 

In practice the equation of motion is solved first without considering the constraint force and in the next step the constraint forces are obtained by correcting the positions such that the molecule conserves its minimum structure, i.e., such that the constraints are fiilfilled. For small molecules direct inversion is possible, for large molecules iterative procedures are applied (60). This means that each constraint is corrected after the other until a certain convergence is reached. This algorithm is called Shake (65). Another important aspect of simulations concerns periodic boundary conditions. A virtual replication of the central box at each of its planes is carried out in order to avoid surface effects. A detailed description can be found in the excellent textbooks of Allen and Tildesley (60) and Frenkel and Smit (61). 

Fig. 6.9. Decay of the reactant concentrations n(t) for d = 1 obtained in the computer simulations. Realizations of the decay process were considered with 104 particles of each kind, initially distributed on a chain of 106 lattice sites, periodic boundary conditions are imposed. Full curves give the simulation data for m = 6 and m = 10, as indicated the dashed line gives the slope —d/(2m — d), the dotted line gives the slope -d/2m, to be compared with equations (6.1.41) and (6.1.43).

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

The computer simulations employed the molecular dynamics technique, in which particles are moved deterministically by integrating their equations of motion. The system size was 864 Lennard-Jones atoms, of which one was the solute (see Table II for potential parameters). There were no solute-solute interactions. Periodic boundary conditions and the minimum image criterion were used (76). The cutoff radius for binary interactions was 3.5 G (see Table II). Potentials were truncated beyond the cutoff. 

Equation (18) is a statement of the Bom and von Karman periodic boundary conditions. 

The direction of the displacement comes from the periodic boundary condition of the solution of the wave equation. It is normal to the flat of the crystal... 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

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