Displacement random

The nature of dispersion. The effect which the solid packing has on the flow pattern within a tubular reactor can sometimes be of sufficient magnitude to cause significant departures from plug flow conditions. The presence of solid particles in a tube causes elements of flowing gas to become displaced randomly and therefore produces a mixing effect. An eddy diffusion coefficient can be ascribed to this mixing effect and becomes superimposed on the transport processes which normally occur in unpacked tubes—either a molecular diffusion process at fairly low Reynolds... 

One by one, each of the N atoms is displaced randomly, and the closest local minima is determined. If the new structure has a lower total energy than the original one, this new one is kept, and the old one discarded. This is repeated approximately 500 1000 times (depending on cluster size). 

The simplest sorts of defect are those based on points Cations and anions are removed, or displaced, randomly in two sorts of defect. 

The relationship between mean squared phase shift and mean squared displacement can be modelled in a simple way as follows This motion is mediated by small, random jumps in position occurring with a mean interval ij. If the jump size in the gradient direction is e, then after n jumps at time the displacement of a spin is... 

Selecting trial moves in an unbiased way typically means (a) choose an atom randomly , with equal probability from the complete set (b) displace it by random amounts in the v, y and z directions, chosen... 

This ensemble is a weighted superposition of NVT ensembles with different values, of N. As a rule of tiuiinb, a typical MC sweep consists of N attempted moves, each of which is chosen randomly to be (i) a displacement (liandled exactly as in constant-AfTMC) (ii) the creation of a new particle at a randomly selected position (iii) the destruction of a randomly selected particle from the system. The probabilities for attempting creation and destniction must be equal (for consistency with what follows), but they need not be equal to the probability for attempting displacement (although they often are). 

As an alternative to the random selection of particles it is possible to move the atom sequentially (this requires one fewer call to the random number generator per iteration) Alternatively, several atoms can be moved at once if an appropriate value for the maximun displacement is chosen then this may enable phase space to he covered more efficiently. 

The probability calculated so far is too low because it describes one specific sequence of heads and tails. From the point of view of net displacement, the sequence does not matter. Hence the above results must be multiplied by the number of different ways this outcome can arise. Instead of tossing one coin n times, we could toss n coins drawn at random from a piggy bank. For the first, we have a choice of n to draw from for the second, n - 1 for the third, n - 2, and so on. The total possible ways the toss could be carried out is given by the product of these different choices, that is by n ... 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

Successful measurement of machine vibration requires more than a transducer randomly selected, installed, and a piece of wire to carry the signal to the analyzer. When the decision to monitor vibration is made, three choices of measurement are available (1) displacement, (2) velocity, and... 

The concentration profiles of the solute in both the mobile and stationary phases are depicted as Gaussian in form. In due course, this assumption will be shown to be the ideal elution curve as predicted by the Plate Theory. Equilibrium occurs between the mobile phase and the stationary phase, when the probability of a solute molecule striking the boundary and entering the stationary phase is the same as the probability of a solute molecule randomly acquiring sufficient kinetic energy to leave the stationary phase and enter the mobile phase. The distribution system is continuously thermodynamically driven toward equilibrium. However, the moving phase will continuously displace the concentration profile of the solute in the mobile phase forward, relative to that in the stationary phase. This displacement, in a grossly... 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

The MC method can be implemented by a modification of the classic Metropolis scheme [25,67]. The Markov chain is generated by a three-step sequence. The first step is identical to the classic Metropolis algorithm a randomly selected molecule i is displaced within a small cube of side length 26r centered on its original position... 

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