Energy-momentum schemes

J. C. Simo, N. Tarnow, and K. K. Wang. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100 63-116, 1994. 

The most important category of dimensionless groups is that of the numerics connected with transport (of mass, energy and angular momentum). Scheme 3.1 shows the three fundamental equations of conservation, written in their simplest form (i.e., one-dimensional). A complete system of numerics can be derived by forming "ratios" of the different terms of these three equations, as was suggested by Klinkenberg and Mooy (1943). This system is reproduced in Scheme 3.2. 

An interaction that is never directly seen in liquid spectra but that, if present, always dominates solid-state spectra is the quadrupole interaction. In Figure 15.16 we see the energy level scheme for a spin I - 1 nucleus, such as deuterium, 2H. Recall that a system with an angular momentum I has 21 + 1... 

The advantage of the energy-based schemes is that a well-defined Hamiltonian exists. The PAP, SAP, and DAS schemes conserve energy and momentum in the propagation of trajectories in NVE simulations. It is recognized that conservations... 

Most schemes that have been proposed to propel starships involve plasmas. Schemes differ both in the selection of matter for propulsion and the way it is energi2ed for ejection. Some proposals involve onboard storage of mass to be ejected, as in modem rockets, and others consider acquisition of matter from space or the picking up of pellets, and their momentum, which are accelerated from within the solar system (184,185). Energy acquisition from earth-based lasers also has been considered, but most interstellar propulsion ideas involve nuclear fusion energy both magnetic, ie, mirror and toroidal, and inertial, ie, laser and ion-beam, fusion schemes have been considered (186—190). 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

In principle, the calculation of bonding in two or three dimensions follows the same scheme as outlined for the chain extended in one dimension. Instead of one lattice constant a, two or three lattice constants a, b and c have to be considered, and instead of one sequential number k, two or three numbers kx, ky and k- are needed. The triplet of numbers k = (kx, ky, kz) is called wave vector. This term expresses the relation with the momentum of the electron. The momentum has vectorial character, its direction coincides with the direction of k the magnitudes of both are related by the de Broglie relation [equation (10.5)]. In the directions a, b and c the components of k run from 0 to nja, njb and n/c, respectively. As the direction of motion and the momentum of an electron can be reversed, we also allow for negative values of kx, ky and kz, with values that run from 0 to —nja etc. However, for the calculation of the energy states the positive values are sufficient, since according to equation (10.4) the energy of a wave function is E(k) = E(—k). 

---