Fluctuations local density

In chapters 1 and 4, we have discussed fluctuations in the number of particles in the entire system. Here, we shall be interested in the fluctuations of the density in a given region S within the system. 

Consider, for example, a system in the T, V, N ensemble. We select a region S within the system and inquire as to the number of particles that fall within S for a given configuration RN  

Each term in the sum over i is unity whenever R, is in S, and is zero otherwise. Therefore, the sum over i counts all the particles that are within S at a given configuration RN. The average number of particles in S is 

we have used the definition of the singlet molecular distribution function. The last relation holds for a homogenous fluid, where V(S) is the volume of the region S, and p is the the bulk density p = N/V. 

In the second step on the rhs of (F.4), we have split the double sum over i and j into two sums the first over all the terms with i=j, and the second over the terms with i j. We have also used the identity of the product of two Dirac functions  

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This is the result for monatomic fluids and is well approximated by a sum of tliree Lorentzians, as given by the first tliree temis on the right-hand side. The physics of these tliree Lorentzians can be understood by thinking about a local density fluctuation as made up of tliemiodynamically independent entropy and pressure fluctuations p = p s,p). The first temi is a consequence of the themial processes quantified by the entropy... 

Light scattering from a solution is due both to the scattering from local density fluctuations and to the scattering from the solvent [9,18], This scattering may be described by the Rayleigh scattering ratio [9,18] ... 

DISCUSSION INFLUENCE OF LOCAL DENSITY FLUCTUATIONS ON SOLVATION PROPERTIES... 

In order to investigate the effects of local density fluctuations on solvation properties, we decided to study two supercritical thermodynamic state points of the same density (5.7 at/nm3) but at different temperatures (295 and 153 K). The low temperature state point, close to the Ar critical point (Tc= 150.8 K, pc= 8.1 at/nm3), is expected to involve significant local density enhancements [5]. 

Smoothing out of local density fluctuations 2.1.2.1 Balance equations... 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

The density cumulant therefore measures the correlation among local density fluctuations p(r) — (p(r). It is these fluctuations which lead to scattering. 

Local density fluctuations occur in penetrant polymer systems both above and below Tg. It is then reasonable to expect that a free-volume diffusion model should also provide an adequate description of the diffusion of small penetrants in glassy polymers. To reach this goal the free-volume model for diffusion of small penetrants in rubbery polymers, second part of Section 5.1.1, was modified to include transport below Tg (64,65,72,91-93). 

Keywords Field theory, ionic systems, local density fluctuations, Dyson-like equation,... 

To illustrate a FT, we use the Dyson equation that is characteristic of helds and we focus on the local density fluctuations that is the most natural variable in this approach. Two systems are considered a homogeneous Yukawa fluid and an ionic soluhon near a hard wall. 

A very similar situation holds for the system with a = 3.1. The NVT results are completely featureless. This can be explained by the fact that the interactions are so long ranged that local density fluctuations within the simulation cell do not give rise to any appreciable energy fluctuations. Since the overall number density is held constant, energy fluctuations are almost completely suppressed. The pVT results, on the other hand, show a very sharp... 

The original derivation of the Ornstein-Zernike relation (Ornstein and Zernike 1914) employs arguments on local density fluctuations in the fluid. We present here a different derivation based on the method of functional derivatives (Appendix B). A very thorough discussion of this topic is given by Munster (1969), and by Gray and Gubbins (1984). 

The scaled particle theory of fluids developed by Reiss, Lebowitz, Helfand and Frisch > " need concern itself [in the case of hard spheres by virtue of Eq. (26)] only with calculating g a). To accomplish this we focus our attention on a spherical cavity of radius at least r centered about a fixed point in the fluid. A cavity is defined as a region of space devoid of molecular (hard sphere) centers (see Fig. 8). Such a cavity can be formed spontaneously in our fluid as a result of a local density fluctuation. 

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