Velocities, random distribution

Hqii ilibraliori corrects the velocities of atom s. Velocities rcsii Itin g from heating dt) not simulate the type of motion found in a real molecular system. Instead, these velticities depend on a random distribution of values corresponding to a given temperature and on the forces in a partially minimized structure. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

Initializing the initial kinetic energy and temperature of the system it is necessary to start the motion at some level, eg, assume a Boltzmann (random) distribution of atomic velocities, at 300 K. 

When a fluid is flowing under streamline conditions over a surface, a forward component of velocity is superimposed on the random distribution of velocities of the molecules, and movement at right angles to the surface occurs solely as a result of the random motion... 

Initial velocities needed to kickstart the simulation is taken as a random distribution. Temperature is calculated from atomic velocities using the ideal-gas relationship... 

We generate the start-up configuration - all particles in the box are assigned positions r, and velocities v . Velocities are randomly distributed according to a Maxwellian distribution for some given temperature. 

Cook (Ref 15) also reported that the shock transmitted thru a barrier into a transparent liq expl, appeared (from the partial opacity of the liq behind the shock front) to initiate some reaction at once. The high-velocity deton appeared, on the other hand, to start as a much more intense luminosity at one or more centers randomly distributed within the reacting liq. Often an intense flame is observed to flash across the region just traversed by the shock wave, at a velocity far above the normal deton vel, and upon reaching the shock front to start a high-order deton (See also Ref 17, p 13b)... 

M 3] [P 3] Fluorescence images at various times were taken in the main channel, i.e. along the direction of the electric field, of the first-generation micro mixer [25, 93]. After a period of 2 s, the flow becomes unstable and transverse velocities stretch and fold material lines in the flow. The initial seeded/unseeded interface becomes rapidly deformed. Finally after about 13 s, a random distribution of the tracer transverse to the applied AC field is achieved. EKI action is visible throughout the whole channel length of 7 mm. Thus, feasibility of EKI action for micro mixing has been demonstrated. 

P. Fevrier, O. Simonin, and K. Squires. Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution Theoretical formalism and numerical study. J. Fluid Mech., 533 1-46, 2005. 

Figure 15 shows the CI2 bond distance for two different trajectories for which the initial conditions of the Ari25Cl2 cluster are the same, i.e. the configuration and the center of mass velocity of the clusters at the beginning of each trajectory (before the collision with the surface) are identical. The only differences between the two trajectories are the velocities (randomly chosen from a one-dimensional thermal distribution at 30 K) of the hard cubes that mimic the surface. Despite the rather low temperature of the surface, one of the trajectories results in the dissociation of the diatomic molecule while the other one ends with a vibrationally excited reactant molecule. The effect of the hard cube velocity on the energy of the atom scattering from the surface is negligible but the history of a single trajectory is extremely sensitive to the details of the collisions with the surface, as shown in Fig. 15. This is a characteristic of so called chaotic systems. In...

These phenomena can be interpreted in terms of molecular orientation by the velocity gradient in the flowing liquid, opposed by the rotary Brownian movement which produces disorientation and a tendency toward a purely random distribution. The intensity of this Brownian movement is charaterized by the rotary diffusion constants, 0, discussed in the preceding section. The fundamental treatment of this problem, for very thin rod-shaped particles, was given by Boeder (5) the treatment has been generalized, and extended to rigid ellipsoids of revolution of any axial ratio, by Peterlin and STUARTi 56), [98), (99) and by Snell-MAN and Bj5knstAhl (J9J). The main features of their treatment are as follows 1 ... 

We assume molecular chaos. This means that in binary collisions both sets of molecules are randomly distributed so that the molecular velocity is uncorrelated with their position. 

At any given instance, a number of chain molecules of equal length will have a random distribution of chain end-to-end distances. This information can be obtained from a derivation analogous to that used to derive the Maxwellian velocity distribution of molecules in an ideal gas. 

Simulations using BOMD or CPMD give as result a set of snapshots of the system, as coordinates, velocities, and forces. Exploitation of this information allows to know statistical quantities as well as dynamic quantities. As an example, the radial distribution function gives the probability to find a pair of atoms a distance r apart, relative to the probability for a random distribution at the same density [27]... 

Once new droplets are created, the product droplet velocity is computed by adding a factor Wbu to the primary drop velocity. This additional velocity is randomly distributed in a plane normal to the relative velocity vector between the gas phase and parent drop, and the magnitude is determined by the radius of the parent drop and the breakup frequency, wbu = ru. This modification of newly formed droplets follows the physical picture of parent droplets being tom apart by aerodynamic forces giving momentum to the newly formed droplets in the direction normal to the relative velocity between the gas phase and parent drops [17]. As new droplets are formed, parent droplets are destroyed and Lagrangian tracking in the physical space is continued till further breakup events. 

The clustering of these channels generates various flow structures at the bed scale between which the liquid velocities are randomly distributed. ... 

Based on their previous work on molecular dynamics using noisy forces, Kiihne et al. [58] demonstrated a novel AIMD method that combines Langevin dynamics for the nuclei with time reversible dynamics for the electronic degrees of freedom and incomplete SCF convergence. The method is based on the observations [59] that the error of the nuclear forces has a Gaussian distribution, that the autocorrelation function of the force errors decays rapidly, and that the force errors show a random distribution with respect to the velocities. From these observations it was... 

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