Computer simulation trajectories

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

Classical ion trajectory computer simulations based on the BCA are a series of evaluations of two-body collisions. The parameters involved in each collision are tire type of atoms of the projectile and the target atom, the kinetic energy of the projectile and the impact parameter. The general procedure for implementation of such computer simulations is as follows. All of the parameters involved in tlie calculation are defined the surface structure in tenns of the types of the constituent atoms, their positions in the surface and their themial vibration amplitude the projectile in tenns of the type of ion to be used, the incident beam direction and the initial kinetic energy the detector in tenns of the position, size and detection efficiency the type of potential fiinctions for possible collision pairs. 

Simulations. In addition to analytical approaches to describe ion—soHd interactions two different types of computer simulations are used Monte Cado (MC) and molecular dynamics (MD). The Monte Cado method rehes on a binary coUision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Cado suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—soUd interactions has been provided (43). 

An important issue, the significance of which is sometime underestimated, is the analysis of the resulting molecular dynamics trajectories. Clearly, the value of any computer simulation lies in the quality of the information extracted from it. In fact, it is good practice to plan the analysis procedure before starting the simulation, as the goals of the analysis will often detennine the character of the simulation to be performed. 

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. 

But a computer simulation is more than a few clever data structures. We need algorithms to manipulate our system. In some way, we have to invent ways to let the big computer in our hands do things with the model that is useful for our needs. There are a number of ways for such a time evolution of the system the most prominent is the Monte Carlo procedure that follows an appropriate random path through configuration space in order to investigate equilibrium properties. Then there is molecular dynamics, which follows classical mechanical trajectories. There is a variety of dissipative dynamical methods, such as Brownian dynamics. All these techniques operate on the fundamental degrees of freedom of what we define to be our model. This is the common feature of computer simulations as opposed to other numerical approaches. 

There are two important consequences of this equality for computer simulations of many-body systems. First, it means that statistically averaged properties of these systems are accessible from simulations that are aimed at generating trajectories -e.g., molecular dynamics, or ensemble averages such as Monte Carlo. Furthermore, for sufficiently long trajectories, the time-averaged properties become independent of the initial conditions. Stated differently, it means that for almost all values of qo, Po, the system will pass arbitrarily close to any point x, p, in phase space at some later time. 

The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. 

Which model provides the best representation for local mobility in a particular group remains unclear, as a detailed picture of protein dynamics is yet to be painted. This information is not directly available from NMR measurements that are necessarily limited by the number of experimentally available parameters. Additional knowledge is required in order to translate these experimental data into a reliable motional picture of a protein. At this stage, molecular dynamic simulations could prove extremely valuable, because they can provide complete characterization of atomic motions for all atoms in a molecule and at all instants of the simulated trajectory. This direction becomes particularly promising with the current progress in computational resources, when the length of a simulated trajectory approaches the NMR-relevant time scales [23, 63, 64]. 

Fig. 1. Computer simulations of four selective excitation pulses. (Top) Pulse shapes. From left to right 90° rectangular pulse, 270° Gaussian truncated at 2.5%, Quaternion cascade Q, and E-BURP-1. The vertical axis shows the relative rf amplitudes, whereas the horizontal axis shows the time. (Middle) Trajectories of Cartesian operators in the rotating frame...

Since proteins are essentially polymers, they may adopt many, almost iso-energetic and very similar structures called protein substates (Austin et al. 1975 Frauenfelder et al. 1988). At ambient temperatures, proteins switch between their substates hence, they are constantly changing their structure (Parak et al. 1982 Parak 2003). By knowing the coordinates of each of the atoms and their velocity vectors at any instant of time, the dynamics of the structural changes, i.e., the trajectories of the molecule s atoms, can be followed exactly. However, this has been achieved only in computer simulations so far. 

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

A theoretical analysis of the possible conformations of polylp-phenylene terephthalate) (PPTA) and polylp-phenylene isophthalate) (PPIA) is performed on the basis of molecular mechanics and molecular dynamics trajectories. The dependence of the persistence length on the fluctuations of the torsional angle around the ester bond is discussed for PPTA in the frame of the RIS model. Realistic parameters like bond length and bond angles are provided by computer simulations using MD. 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

Figure 12. Computer simulation of drop trajectory and deposit.

P 62] A Lagrangian particle tracking technique, i.e. the computation of trajectories of massless tracer particles, which allows the computation of interfacial stretching factors, was coupled to CFD simulation [47]. Some calculations concerning the residence time distribution were also performed. A constant, uniform velocity and pressure were applied at the inlet and outlet, respectively. The existence of a fully developed flow without any noticeable effect of the inlet and outlet boundaries was assured by inspection of the computed flow fields obtained in the third mixer segment for all Reynolds numbers under study. 

The need for computer simulations introduces some constraints in the description of solvent-solvent interactions. A simulation performed with due care requires millions of moves in the Monte Carlo method or an equivalent number of time steps of elementary trajectories in Molecular Dynamics, and each move or step requires a new calculation of the solvent-solvent interactions. Considerations of computer time are necessary, because methodological efforts on the calculation of solvation energies are motivated by the need to have reliable information on this property for a very large number of molecules of different sizes, and the application of methods cannot be limited to a few benchmark examples. There are essentially two different strategies. 

When S(t) is calculated from computer simulation on a system containing a single solute molecule, the overbar is interpreted as an average over statistically independent nonequilibrium trajectories. The total Stokes shift can also be obtained via equilibrium statistical mechanical theory or simulation,... 

It was assumed that the motion of the fictitious particle within the above time steps is rectilinear. This simplification, which accelerates the computer calculation of the trajectories of the fictitious particle, has been show n to be justified (9). The Philips slip correction factor for an accommodation coefficient of unity (Eq. [4]) was used in the calculation of the diffusion coefficients of particles. The values of the dimensionless coagulation coefficients % obtained by the computer simulation for different particle sizes, are given in Table I. The statistical errors of the Monte Carlo simulation were estimated by the standard 3 a method (corresponding to a probability of 0.997) (13). The number of particle pairs that must be generated in order to lower the error to a reasonable level depends both on the initial distance of separation between... 

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

The previous analysis may be extended to spatially periodic suspensions whose basic unit cell contains not one, but many particles. Such models would parallel those employed in liquid-state theories, which are widely used in computer simulations of molecular behavior (Hansen and McDonald, 1976). This subsection briefly addresses this extension, showing how the trajectories of each of the particles (modulo the unit cell) can be calculated and time-average particle stresses derived subsequently therefrom. This provides a natural entree into recent dynamic simulations of suspensions, which are reviewed later in Section VIII. 

In all of these computations, there is a dense manifold of excited states present [83], Thus the computations are sensitive to dynamic electron correlation and the details of the reaction coordinates involved. In the cytosine-guanine base pair simulations, trajectory calculations proved to be necessary to determine the extent of the conical intersection that is actually accessible. Subsequent improvements in the level of theory used for the static calculation of single molecules will be possible, but these should be balanced against a more realistic treatment of vibrational kinetic energy and environmental effects (solvent/protein). 

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