Long-range correlation distance

Hence, from the slope of vs or R vs sin d/2, one can determine the long-range correlation distance (Stanley, 1971 Skripov, 1972 Vuks, 1977). 

In the region of critical opalescxince, the parameters of the fluctuation correlation (the long-range correlation distance the radius of intermolecular interaction forces, i.e. 

However, near the spinodal, the approach based on Onsager s hypothesis and on the solution of the hydrodynamic equations loses its validity. With allowance for the long-range correlation distance of the concentration fluctuations ft, the validity condition of... 

Hence, away from the critical point (see Equation 15), the applicability range of Landau s approximation extends with increasing the long-range correlation distance. Increasing away from Tc is equivalent to an admission of a longer range of particle interactions in the system, which is the premise of applicability of the mean field theories (see section 1.5). 

The correlation of density fluctuations in the critical region is determined by the long-range correlation distance (Equation 2.3 129) characterizing the efficiency of the correlation falls of density fluctuations with distance. 

We now recognize that correlation between densities at different locations can arise from two sources. One is due to direct intermolecular interactions, and the other is due to the closure condition. We assume that the first is operative only at short distances say 0 < R < Rcor> where Rqor is the correlation distance beyond which a particle placed at fixed position does not have any influence on the density at any other position. This is the distance RCor> beyond which g(R) is nearly equal to 1. The second, is normally referred to as long-range correlation. We shall refer to it as the closure correlation (CC). Since this part has the 1ST 1 dependence on N, it has a negligible effect on g(R) at R < Rqok- It becomes important when integration extends to R —> oo. 

For a theoretical ideal gas, U(R) = 0, equation (G.24) is also the total correlation as in (G.3). But for real gas as p — 0, we have an additional contribution due to the intermolecular interaction as in (G.9). The latter, is in general of short range, hence the former is referred to as the long-range correlation. As we have noted before, this part of the correlation for an ideal gas holds true at any distance, not necessarily for R —> oo. Therefore, we feel that the term closure correlation is more appropriate for gc(CC). 

Shortcomings in the MNDO model as described in the previous section led to a reexamination of the model, leading to Austin Model 1, AM1. ° In this model a term was added to MNDO to correct for the excessive repulsions at van der Waals distances. Toward this end, each atom was assigned a number of spherical gaussians, which were intended to mimic long range correlation effects. The core-core repulsion term was modified and became... 

Fig. (8). H- H, Long-range correlation, results of NOE experiments and distances...

However, the sensitivity for direct detection is required to be improved before long-range correlations to obtain distance constraints can be used in... 

Consequently the range of the correlation mediated by P(x) and the domain of the coherence will extend over a distance Finite (but long) range correlation forces are then expected in realistic (finite volume) systems. A similar conclusion can be drawn from Eq. (3.58) when an appropriate cut-off function /,(x) is introduced. In conclusion a relation between the size of the system and the not exactly zero mass of the P-field quantum is obtained. As announced in Section 2 a more correct treatment, which considers the electric field as a gauge field, would lead to a dynamically constructed (nonzero) mass of the quanta, producing thus a finite volume for the biological system. 

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