Newtons Equations of Motion

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

MD calculations integrate Newton equations of motion forward in time, so that the dynamical behaviour of the system can be predicted. The calculations... 

The Vlasov-Newton equation of motion of/is the closed equation ... 

Electrokinetic phenomena can be understood with the help of two equations The known Poisson equation and the Navier3-Stokes4 equation. The Navier-Stokes equation describes the movement of a Newtonian liquid, i.e., a liquid whose viscosity does not change when it flows and when it is sheared. In order to make the equation plausible we consider an infinitesimal quantity of the liquid having a volume dV = dx dy dz and a mass dm. If we want to write Newtons equation of motion for this volume element we have to consider three forces ... 

In the latter method we simply specify the initial conditions of the system, the coordinates of the atoms of the biomolecule in its initial conformation, as well as those of the water molecules constituting its environment, along with a set of initial velocities for these atoms. Having specified the initial conditions, Newtons equation of motion... 

A MD simulation, on the other hand, is based on the following Newton equations of motion ... 

The algorithm chosen to integrate the Newton equations of motion is a slight variation of the popular Verlet algorithm. If / , + , r, , and indicate the relative position of the particle i at the time steps n +1, n, and n — 1, respectively, then... 

Another useful way to express the Newton equations of motion is in the Hamiltonian representation. One starts with the generalized momenta... 

Another important outcome of these considerations is the following. The uniqueness of solutions of the Newton equations of motion implies that phase point trajectories do not cross. If we follow the motions of phase points that started at a given volume element in phase space we will therefore see all these points evolving in time into an equivalent volume element, not necessarily of the same geometrical shape. The number of points in this new volume is the same as the original one, and Eq. (1.107) implies that also their density is the same. Therefore, the new volume (again, not necessarily the shape) is the same as the original one. If we think of this set of points as molecules of some multidimensional fluid, the nature of the time evolution implies that this fluid is totally incompressible. Equation (1.107) is the mathematical expression of this incompressibility property. 

Consider an equilibrium thennodynamic ensemble, say a set of atomic systems characterized by the macroscopic variables T (temperature), Q (volume), andTV (number of particles). Each system in this ensemble contains N atoms whose positions and momenta are assigned according to the distribution function (5.2) subjected to the volume restriction. At some given time each system in this ensemble is in a particular microscopic state that coiTesponds to a point (r, p- ) in phase space. As the system evolves in time such a point moves according to the Newton equations of motion, defining what we call a phase space trajectory (see Section 1.2.2). The ensemble coiTesponds to a set of such trajectories, defined by their starting point and by the Newton equations. Due to the uniqueness of solutions of the Newton s equations, these trajectories do not intersect with themselves or with each other. 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, zi and z, which satisfy the Markovian equations of motion... 

Once the initial coordinates, momenta, and weights are generated, a swarm of classical trajectories is propagated by numerically solving the Newton equations of motion [26] in a standard way [2]. Typically, several hundreds trajectories are required for converged results. 

Molecular mechanics does not deal with nuclear motion as a function of time, as well as with the kinetic energy of the system (related to its temperature). This is the subject of molecular dynamics, which means solving the Newton equation of motion for all the nuclei of the system interacting through potential energy V R). Various approaches to this question (of general importance) will be presented at the end of the chapter. 

The Newton equations of motion for aU the atoms of the system can be written in matrix form as (jc means the second derivative with respect to time t)... 

Region I is treated as a quantum meehanieal objeet and deseribed by the proper time-dependent Sehrddinger equation, while region II is treated elassieaUy by the foree-field deseription and the eorresponding Newton equations of motion and region IE is simulated by a eontinuous medium (no atomic representation) with a certain dielectric permittivity. 

In the MD, we solve Newton equations of motion for all atoms of the system. Imagine we have a large moleeule in an aqueous solution (biology offers us important examples). We have no chance to solve Newton equations because there are too many of them (a lot of water molecules). What do we do then Let us recall that we are interested in the macromolecule the water molecules are interesting only as a medium that changes the conformation of the macromolecule. 

Molecular mechanics does not involve atomic kinetic energy, molecular dynamics (MD) does. MD represents a method of solving the Newton equations of motion for all the atoms of the system. The forces acting on each atom at a given configuration of the nuclei are computed (from the potential energy assumed to be known )... 

MD Moleeular Dynamics The solution of the Newton equation of motion for the... 

The ion trajectory within such a potential is described by the Newton equations of motion (Equation [6.2], Appendix 6.1), one for motion in the z-direction and one for each of the x- and y-directions. The component of force exerted on an ion with charge magnitude ze in the x-direction, for example, is ze.E where E, is the electric field component (Equation [6.9]) ... 

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