Mean square displacement, single molecule

The Monte Carlo simulation comprises three distinct moves (i) Canonical Monte Carlo moves update the molecular conformations in the Mr repUca. In this specific application, we employ a Smart Monte Carlo algorithm [119] that utilizes strong bonded forces to propose a trial displacement [43, 87]. The amplitude of the trial displacement has been optimized in order to maximize the mean-square displacement of molecules [91], and the single-chain dynamics closely resembles the Rouse-dynamics of unentangled macromolecules [120]. (ii) Since each replica is an... 

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. 

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. 

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 ... 

Anomalous diffusion is also possible in microporous solids. For instance, it is possible for molecules to be confined in a channel system in which they cannot pass each other, and this will obviously affect molecular displacement in a time interval. This case is termed single-file diffusion , and the mean square displacement in a time t is then given... 

There has recently been much interest in the phenomenon of single file diffusion, which occurs in a unidimensional pane system when the diffusing molecules are too large to pass one another. In this situation the mean square displacement increases in proportion to the square root of elapsed... 

The mean square displacement in single-file systems may quite generally be shown to be related to the movement of a sole molecule by the expression [8,10]... 

Molecular dynamics (MD) simulations in single-file systems are additionally comphcated by the requirement that in the absence of external forces the center of mass must be preserved. This comphcation results from the fact that, as a consequence of the correlated motion in a single-file system, the shift of a particular molecule must be accompanied by shifts of other molecules in the same direction. Depending on the total amoimt of molecules under consideration, the conservation of the center of mass therefore prohibits arbitrarily large molecular shifts. The maximum mean square displacement may be shown to obey the relation [22]... 

The concept of single-file diffusion has most successfully been applied for MD simulations in carbon nanotubes [36-39], yielding both the square-root time dependence of the molecular mean square displacement and a remarkably high mobility of the individual, isolated diffusants. In [40-42], the astonishingly high single-particle mobilities in single-file systems have been attributed by MD simulations to a concerted motion of clusters of the adsorbed molecules. 

Concerning PCS, the most interesting observables from the simulations are the trajectories of single diffusion molecules or particles, respectively. From these trajectories, the mean square displacements and the autocorrelation functions can be calculated. This way, it can be analyzed how heterogeneity expresses itself in the PCS results, i.e., how anomalous diffusion is averaged over the relevant PCS time and length scales. Also, the question how interactions between dye and polymer chains influence PCS results has been recently addressed using a combination of PCS experiments with simulations [46] (Fig. 14). 

Berg (1983) shows very clearly how the equations for the macroscopic diffusion of an ensemble of molecules can be derived starting with the random motion of a single particle in one dimension. Just as in other statistical problems when the mean is zero, the important parameter for the distribution is the mean square displacement of the particle, , which is... 

The rapid diffusibUity of NO has critically important imphcations for its chemistry in the biological setting. The speed with which NO moves by random diffusion can be illustrated by consideration of its root mean square distance of displacement, which describes the distance a single NO molecule will move in any time interval based on its diffusion constant D (which is similar for aqueous solution and also tissue (brain) ) ... 

The INS intensity, 5 (g,fo), as calculated from the Scattering Law, Eq. (2.41), is related to the mean square atomic displacements, weighted by the incoherent scattering cross sections. What is required to calculate this quantity is the mean square atomic displacement tensor, Bi, and this can be obtained from the crystalline equivalent of L/ " ( A2.3), the normalised atomic displacements in a single molecule Eq. (4.20). This is and was introduced above, in Eq. (4.55). We have seen how... 

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