Lattice dynamics basic equation

A formal analc y is evident between Eq. (2.5), describing electron propagation, and Eq. (2.9), which is the basic equation of lattice dynamics. 

Let us briefly define the elementary parameters and review the basic equations for the treatment of molecular and lattice dynamics. The concepts presented here fonn the background for a further discussion later in the chapter for the case of large molecular systems. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

This monograph deals with kinetics, not with dynamics. Dynamics, the local (coupled) motion of lattice constituents (or structure elements) due to their thermal energy is the prerequisite of solid state kinetics. Dynamics can explain the nature and magnitude of rate constants and transport coefficients from a fundamental point of view. Kinetics, on the other hand, deal with the course of processes, expressed in terms of concentration and structure, in space and time. The formal treatment of kinetics is basically phenomenological, but it often needs detailed atomistic modeling in order to construct an appropriate formal frame (e.g., the partial differential equations in space and time). 

If one is interested in properties that vary on very long distance and time scales it is possible that a drastic simplification of the molecular dynamics will still provide a faithful representation of these properties. Hydrodynamic flows are a good example. As long as the dynamics preserves the basic conservation laws of mass, momentum and energy, on sufficiently long scales the system will be described by the Navier-Stokes equations. This observation is the basis for the construction of a variety of particle-based methods for simulating hydrodynamic flows and reaction-diffusion dynamics. (There are other phase space methods that are widely used to simulate hydrodynamic flows which are not particle-based, e.g. the lattice Boltzmann method [125], which fall outside the scope of this account of MD simulation.)... 

An even more drastic simplification of the dynamics is made in lattice-gas automaton models for fluid flow [127,128]. Here particles are placed on a suitable regular lattice so that particle positions are discrete variables. Particle velocities are also made discrete. Simple rules move particles from site to site and change discrete velocities in a manner that satisfies the basic conservation laws. Because the lattice geometry destroys isotropy, artifacts appear in the hydrodynamics equations that have limited the utility of this method. Lattice-gas automaton models have been extended to treat reaction-diffusion systems [129]. 

The Lattice-Boltzmann method is a numerical scheme for fluid simulations which originated from molecular dynamics models such as the lattice gas automata. In contrast to the prediction of macroscopic properties such as mass, momentum and energy by solving conservation equations, e.g. the Navier-Stokes equations, the LBM describes the fluid behaviour on a so-called mesoscopic scale [7, 19]. The basic parameter in the Boltzmann statistics is the distribution function f = f(x,, 0, which represents the number of fictitious fluid elements having the velocity at the location x and the time t. The temporal and spatial development of the distribution function is described by the Boltzmann equation in consideration of collisions between fluid elements. 

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