Number density fluctuation correlation

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]. 

A wide variety of density- and temperature-dependent input parameters are required. These include p, the number density of the solvent, kj, the isothermal compressibility, f, the correlation length of density fluctuations, y = Cp/Cv, the ratio of specific heats, and 37, the viscosity. Very accurate equations of state for ethane (74,75,99) and CO2 (74) are available that provide the necessary input information. The necessary input parameters for fluoroform were obtained by combining information from a variety of sources (76,100,101). There is somewhat greater uncertainty in the fluoroform parameters. 

In fluorescence correlation spectroscopy (FCS) a small volume element or a small area) of a sample is illuminated by a laser beam and the autocorrelation function of fluctuations in the fluorescence is determined by photon counting. From this autocorrelation function the mean number densities of the fluorophores and their diffusion coefficients can be extracted. Measurement and analysis of higher order correlation functions of the fluorescence has been shown to yield information concerning aggregation states of fluorophores ). 

There are several similar correlation functions related to G r) by multiplicative and additive constants. They contain the same structural information but are subtly different in some detail. Their interrelationship has been discussed in detail in the literature. G r) is the function obtained directly from the Fourier transform of the scattered data. The function oscillates around zero and asymptotes to zero at high-r. It also tends to zero at r = 0 with the slope - npo, where pq is the average number density of the material. From a practical point of view G r) is an attractive function because the random uncertainties on the data (propagated from the measurement) are constant in r. This means that fluctuations in the difference between a calculated and measured G(r) curve have the same significance at all values of r. Thus, for example, if the observed fluctuations in the... 

An important consequence of the above assumption is the presence of density fluctuations with a non-zero correlation length. That is because a molecule with a larger than average number of HBs is more likely to be surrounded by other molecules also with a larger than average number of HBs. In this way, it is possible to justify the anomalous increase of compressibility with decreasing temperature. At low temperatures, the number of bonds increases and the density fluctuations increase as well. These correlated fluctuations are superimposed on the normal thermally driven density fluctuations present in other non-associated liquids. The combination of the two competing behaviors yields the compressibflity minimum of the temperature dependence of isothermal compressibility. 

In the framework of the scaling theory, the corona of a spherical micelle can be envisioned [53-56] as an array of concentric spherical shells of closely packed blobs. The blob size, (r) = r/grows as a function of the radial distance r from the center of the core. Each blob comprises a segment of the chain within the local correlation length of the monomer density fluctuations [57], and corresponds to a contribution to the free energy of steric repulsion between the coronal chains. After calculating the total number of blobs in the micellar corona, one finds fhe free energy (per coronal chain) as ... 

The earliest work was done for hard disks and spheres, with no longer range forces. In that case there is no direct effect due to interactions with distant (but correlated) parts of the periodic system. There is nevertheless some dependence of the results on the size and shape of the periodic sample. This arises in various ways, which of course also persist for systems with other forces. For example, the number of particles within any of the periodic boxes is fixed at N. This is a serious constraint on the density fluctuations for small N, and will lead to errors in the resulting thermodynamic averages, in particular to a diminished entropy. (For mixtures, the concentration fluctuations are similarly constrained, and this could be a still more severe problem.) With small samples the range of structural fluctuations may be similarly constrained. This will clearly be the case if significant interparticle configurational correlations have a... 

The tendency for a hydrophobic material to adsorb onto a hydrophobic surface is frequently described in terms of a "hydrophobic interaction (attraction)." While entropic arguments for the presence of such a phenomenon are easy to make, the fact that long-range attractive forces between hydrophobic surfaces have been observed cannot be accounted for by typical DLVO theory [20]. A number of explanations of this phenomenon have been advanced, such as enhanced hydrogen bonding, electrostatic correlation forces, film stability, dissolved gases, and density fluctuations, but none has been successful in explaining all of the features of this... 

The density fluctuation of the total sample is the superposition of local fluctuations in a dynamic, time dependent, equilibrium. We now consider, as a subsystem of the total liquid sample, a fixed (but arbitrary) volume AV which contains a variable number of molecules. The local fluctuations are considered random with no correlation between the various AV s. This subsystem can be considered as a grand canonical ensemble in equilibrium with its surroundings (the balance of the sample). Then,... 

If there is a dispersion of sizes, or shapes, or if the number density is large enough that there is scattering interference between particles, then interpretation of the data is more difficult (see reviews cited). In the general case, the interpretation is based on a theoretical development of Debye and Bueche (1949) (also see Debye (I960)) in which the scattered intensity is written in terms of a correlation function (c(r)) for the electron density fluctuations (5pg) ... 

On the microscopic level, both spatial and temporal fluctuations in number density are to be expected. The decay of these should be rapid compared to the time and distance scales appropriate to the macroscopic self-diflusion phenomenon. Thus to study the presence of a steady state, the M observations of JVi and /i are pooled AM at a time so that the resulting observations appear to be uncorrelated. Typically (AM)fi 200to seems to be satisfactory for this purpose at a reduced volume t = 3 for hard disks (but pooling over even longer intervals is frequently used to assure the independence of the observations). While there are rather long-time correlation effects, there seems little doubt that on any macroscopic time scale the flow is steady. 

This result can be deduced from a number of perspectives such as renormalized perturbation theory [93], polymer density fimctional theory [95,96], and Per-cus functional expansion methods [97]. The medium-induced pair potential is determined by the direct correlation function and collective density fluctuations which are both functionally related to the intramolecular pair correlations via the PRISM equation. Hence, a coupled intramolecular/intermolecular theory is obtained. 

The ri t-hand side of this equation is called Ae density nsity correlation function, smce it is a measure of the oonelation of fluctuations in numbers of molecules at f, and fa cf. (4.16). The last two terms can be expressed in terms of the total correlation function oi (4.23),... 

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