Single-particle properties

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. 

From Single Particle Properties to Collective Charge Transport... 

The single particle properties, which have been discussed in the previous section showed the meaning of the large charging energy of metal nanoclusters as a prerequisite for... 

Water-soluble polymers in general, and especially polyelectrolytes, are often difficult due to their specific and long range electrostatic interactions, which complicate all analytical techniques that rely on single particle properties that are usually realized by high dilution. In most cases the ionic strength of the solution must be increased by the addition of salt in order to screen electrostatic forces. Ideally, SEC separation is predominantly governed by entropic interactions,... 

Let us assume that the single particle properties Am and Bm have definite symmetry under coordinate inversion, P, so that... 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

Spectral functions, described before, determine single-particle properties of the system, such as quasiparticle energy, broadening of the levels (life-time), and density of states. These functions can be modified in nonequilibrium state, but most important kinetic properties, such as distribution function, charge, and current, are determined by lesser Green function... 

Conclusion We have used several, partially new methods to study single particle properties in a mixture of metal clusters. We are now moving to more refractory metals where we expect to see rigid structure at small particle size and a large range of chemical properties depending strongly on size. 

Collective dynamical properties, on the other hand, cannot be averaged over the N particles of the system. Hence, a much longer simulation, an order of magnitude say, is required to achieve a statistical uncertainty comparable to that affecting single-particle properties [55]. 

We reviewed some aspect of passive transport in fluid flows. As far as inert substances are concerned, we described the problem of transport from a Lagrangian point of view focusing on single-particle properties. In particular, the conditions for having asymptotic standard or anomalous diffusion have been discussed in details. As for the problem of reacting substances, we study the... 

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 ... 

Given the structure in Fig. 2.3 as a prototype of the monolayer arrangement of adsorbed water on kaolinite group minerals, there remains only the question of its single-particle properties. Proton and deuteron NMR measurements are instructive in this respect.As indicated in Table 2.1, NMR line shapes show whether a preferred orientation of water molecules exists on the NMR time scale, and NMR relaxation data can yield a value for the rotational correlation time, In the case of 10-A halloysite, the NMR signal from water protons or deuterons in the... 

The increase in the value of to over the rotational correlation time can be understood quantitatively in terms of the micro-macro relation, which provides a relationship between the orientational correlation time (a microscopic, single-particle property) and the DR (a collective phenomenon). Simple continuum model arguments give the following relation between the two relaxation times [11],... 

If C t) is a time correlation function of a single-particle property, then in addition to time averaging, C t) may be averaged over the N particle labels in the molecular dynamics system. This additional averaging decreases the error by the factor (1/N). Even if C(t) is the time correlation function of a... 

When the property calculated is a single particle property, such as the velocity autocorrelation function, or the particle s mean squared displacement, averaging over the particles does help a lot. Averaging over even a moderate number of particles of about 1000 decreases the error by more than an order of magnitude. Unfortunately, this is not possible for all properties. For instance, the calculation of viscosity calls for the stress correlation function [Eq. (67)], which is not a single particle property. This is why self-diffusion coefficients are usually estimated with much better accuracy than viscosity. 

The third region, the high momentum transfer, is the one where the single particle properties of the electron are prominent. This region is investigated mainly by Y-ray Compton scattering experiments and equation (1) is reduced within the limits of Impulse Approximation to ... 

The VH spectrum thus tracks single-particle properties. Under the approximation that sphere rotations and translations must by symmetry be decoupled, one obtains for identical particles... 

The particle number average of a generalized single particle property V i it) is defined as [89] ... 

It is also important to notice that self-diffusion is a single particle property but all the others are collective properties of the whole system. This means, in practice, that at every timestep there are A - 1 independent contributions to the velocity autocorrelation function average. In the case of the other properties, there is only one contribution per timestep which makes the calculation of these transport coefficients much more demanding. (It is worth mentioning that the isotropic character of the equilibrium fluid provides a limited time-saving opportunity during the computation. In the... 

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