Collision dynamics numerical solution

The harmonic approximation is unrealistic in a dynamical description of the dissociation dynamics, because anharmonic potential energy terms will play an important role in the large amplitude motion associated with dissociation. An accurate potential energy surface must be used in order to obtain a realistic dynamical description of the dissociation process and, as in the quasi-classical approach for bimolecular collisions, a numerical solution of the classical equations of motion is required [2]. 

A more complete treatment of the classical dynamics of a system containing an ion and a rotating polar molecule involves the numerical solution of the equation of motion (trajectory calculation) by Dugan et al. [64—68]. In their treatment, a capture collision is defined by an ion trajectory that penetrates to within a certain value of r. Their results also show that the locking in of the dipole is not likely to occur because of the conservation of angular momentum. 

At the intermediate wavelengths no useful analytic forms of solution are known. On the other hand, (138) yields readily to numerical solutions. Such results show a smooth interpolation between characteristic hydrodynamic and free-particle behavior. Notice, however, that in this region it is essential to treat the collisions and molecular flow on equal footing for this reason it would be inappropriate to apply either (154) or (155). Since the intermediate k range is particularly relevant to neutron inelastic scattering studies of liquids and related computer molecular dynamics simulations, the validity of our kinetic model solutions is of interest. 

The beauty of LGA is their simplicity. A lattice gas is an attempt to define the simplest system of interacting paitides on a lattice, whose collisions conserve mass and momentum, and thus whose behavior can be expected to obey the Navie-Stokes equations. The intent here is not to start with the continuum equations and discretize them, but rather to start with a fundamentally discrete system of partides, and make them interact in such a way that the Navier-Stokes equations are emergent, just as they are for natural fluids. That is, LGA, like continuous EDMD, are a particulate simulation of the physical system under study, rather than the numerical solution of hydrodynamic PDEs. These modds range from purdy discrete lattice molecular dynamics to the highly dabo-... 

Classical mechanics provides a direct route from the potential energy surface to the dynamics of the collision, namely, the (numerical) solution of the classical equations of motion for the atoms. The solution uses Newton s law of motion to determine the position of each atom as a function of time. This output is known as a trajectory. It allows us to visualize how each atom moves as the reaction is taking place. Trajectory computations are carried out for two purposes. First, as a diagnostic of trends, i.e., features of the dynamics arising from different featnres of the surface or from changes in reactants energies, masses, and so... 

For the numerical simulation of flowing polymers, several mesoscopic models have been proposed in the last few years that describe polymer (hydro-)dynamics on a mesoscopic scale of several micrometers, typically. Among these methods, we like to mention dissipative particle dynamics (DPD) [168], stochastic rotation dynamics (sometimes also called multipartide collision dynamics) [33], and lattice Boltzmann algorithms [30]. Hybrid simulation schemes for polymer solutions have been developed recenfly, combining these methods for solvent dynamics with standard particle simulations of polymer beads (see Refs [32, 169, 170]). Extending the mesoscopic fluid models to nonideal fluids including polymer melts is currently in progress [30, 159,160,171]. 

These authors use the lattice-Boltzmann discretization method for solving the Navier-Stokes equations. In lattice-Boltzmann discretization, the motion of the fluid is represented by the motion and collision of discrete particles moving between the nodes of a fixed grid (the lattice). This discretization method, which has recently become more frequently used, but is still not standard, is thus completely different from the ones briefly discussed in Sect. 7.1.1. If the rules for the particle collisions and the morphology of the lattice upon which they move are set correctly, the dynamics of these particles will represent a numerical solution to the Navier-Stokes equations (Wolf-Gladrow, 2000 Chen and Doolen, 1998). 

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

The dynamics is obtained hy numerical solving a set of the coupled Boltzmann-BGK transport equations (d. eqn [35]) on a spatial lattice in discrete time steps with a discrete set of microscopic vdodties. At each time step, the prohahility density evolved hy each LB equation is adverted to nearest neighhoting lattice sites and modified by molecular collisions, which are local and conserve mass and momentum. As a result, a LB fluid is shown to obey the Navier-Stokes equation (in the limit of a small lattice spacing and small time step). For dilute polymer solutions, the method typically involves phenomenological coupling between the polymer chain and the flowing fluid in the form of a linear friction term based on an effective viscosity. 

MD simulations involve numerical determinations of individual trajectories, that is, solutions to Newton s equation of motion for particular initial conditions. In chemistry, this technique began with the study of individual gas-phase collisions, where large numbers of trajectories were run to explicitly average over initial conditions. For macromolecules or liquid simulations, however, the frequency of atomic collisions becomes so great that simulations often appear to be ergodic, such that a single trajectory samples phase space with the same distribution as do multiple simulations with randomized starting points. This implies that a dynamics simulation can be used to explore phase space and make connections to classical thermodynamics and kinetics. 

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