Single mean square displacement

The simulations to investigate electro-osmosis were carried out using the molecular dynamics method of Murad and Powles [22] described earher. For nonionic polar fluids the solvent molecule was modeled as a rigid homo-nuclear diatomic with charges q and —q on the two active LJ sites. The solute molecules were modeled as spherical LJ particles [26], as were the molecules that constituted the single molecular layer membrane. The effect of uniform external fields with directions either perpendicular to the membrane or along the diagonal direction (i.e. Ex = Ey = E ) was monitored. The simulation system is shown in Fig. 2. The density profiles, mean squared displacement, and movement of the solvent molecules across the membrane were examined, with and without an external held, to establish whether electro-osmosis can take place in polar systems. The results clearly estab-hshed that electro-osmosis can indeed take place in such solutions. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

The mean-square displacement is again proportional to T, as it is for the harmonic oscillator [Eq. (2.52)], but the slope of the versus T curve is quite different from the value of kB/(hv), predicted for the single harmonic oscillator [Expression (2.52)]. Expression (2.65) represents the temperature dependence of an assembly of harmonic oscillators, rather than that of a single oscillator. 

Fig. 2 The mean square displacement of a single harmonic oscillator as a function of temperature. Units are arbitrary because (uj) and T values depend on the frequency of the oscillator and the mass of the particle. The plot shows that at low temperature the displacement is almost constant, whereas at high temperature it varies linearly with T. The change of regime occurs approximately at 0e/2...

The growth of (Z2) (the particle accumulation) is accompanied by the growth of clusters of similar particles (cf. [91]). The characteristic spatial dimension of the cluster grows according to the same law as the mean square displacement of a single particle,... 

A basic characteristic of a single polymer is its spatial dimensions, such as the radius of gyration. The average size of the ideal chain is identical to the mean square displacement of the random walker J(R fd) — IN1/2 (the bracket... 

When a particle moves in brownian motion, the chance that it will ever return to its initial position is negligibly small. Thus, there will be a net displacement with time of any single particle, even though the average displacement for all particles is zero. For example, during a short time interval one particle may move a distance sls another a distance s2 and so on. Some of these displacements will be positive, others negative some up, others down but with equilibrium conditions the sum of the displacements will be zero. It is possible to estimate the displacement of any particle in terms of its root-mean-square displacement. 

Another deviation from the pattern of ordinary diffusion must be expected if the reactant and product molecules are subjected to single-file conditions, i.e. if (i) the zeolite pore system consists of an array of parallel channels and if (ii) the molecules are too big to pass each other. In this case, the molecular mean-square displacement z t)) is found to be proportional to the square root of the observation time, rather than to the observation time itself. First PFG NMR studies of such systems are in agreement with this prediction [8]. By introducing a mobility factor F, in analogy to the Einstein relation for ordinary diffusion. 

We consider the first derivative of a) instead of Og itself, because the former requires only single integration while the latter requires double integrations [Eq. (9)]. We first omit the dependence on x of the parameter C )( 0 x) and fix it to a constant value to observe how it affects the anomaly in diffusion. We then fix the other two parameters fCOrr(x) and ( (xj to determine their effect on the mean-square displacement. 

In order to understand the effect of temperature on the water dynamics and how it leads to the glass transition of the protein, we have performed a study of a model protein-water system. The model is quite similar to the DEM, which deals with the collective dynamics within and outside the hydration layer. However, since we want to calculate the mean square displacement and diffusion coefficients, we are primarily interested in the single particle properties. The single particle dynamics is essentially the motion of a particle in an effective potential described by its neighbors and thus coupled to the collective dynamics. A schematic representation of the d)mamics of a water molecule within the hydration layer can be given by ... 

In this case and under these conditions (no cell is fllled to capacity) the movement of any particle between the cells is independent of the presence of the other particles. The coefficient of diffusion for the migration of a single component [(case (a)] must then be the same as the diffusion coefficient of this component in the presence of another sorbate [binary or self-diffusion, case (j8)]. In both cases the mean square displacement for i > 1/r is... 

At the moment the noise in the mean squared displacement versus time curves does not permit easy observation of this type of time-dependent behavior, and it remains a matter for future investigation. Also of interest is the question of whether, under the highly diffusive conditions of computer simulations, the systems very near equilibrium relax exponentially ( single relaxation time ) or otherwise, as for laboratory glasses. ... 

This gives 29% population of the n (NO) orbitals, in good agreement with other estimates. Since the 3i/-electron population is effectively only 4-5, and there is a 45-population of about 0-5, the electron density at the nucleus is much higher than usual and confers on sodium nitroprusside its unusually low chemical isomer shift. The electric field gradient and mean square displacement tensors have been completely determined in sodium nitroprusside single-crystal absorbers from the polarisation dependence of the absorption cross-section [37, 38]. The principal axes of these tensors do not coincide. 

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