Cluster distribution function

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

Scaling theory assumes (Stauffer 1979, Stauffer and Aharony 1992) that the cluster distribution function ns p) is a homogeneous one near p — p. Thus ns p) is basically a function of the single scaled variable s/S p), where the denote the typical cluster size ... 

When condensation occurs, the equilibrium relation +A Ag no longer holds. With the nonequilibrium cluster distribution function now given by /i. the dilTerence... 

This expression is different for p>p and for p

p contribute to the communal entropy Sq(p) and are therefore favored. This difference in the dependence of 5i,(p) and the resulting discontinuity at I = is the only difference from the usual percolation theory. The desired cluster distribution function C (p) is porportional to q T)/V, obtained by substituting (5.18) into (5.12) and treating p as a function of temperature T. Thus C Xp) f e same as that already given by (4.4) with T f-a (p) for Fotp

r, so that clusters with... 

A realistic range would be 10 — 10 . A more precise estimate must await a study of the cluster distribution function... 

There has been no direct verification of the conceptual structure of the theory. That is, a microscopic determination of the cluster distribution function has not been made, and the effects of percolation have not been seen. Assuming that the structure of the glass is well-defined liquidlike clusters in a denser solidlike background, one might expect to be able to see these clusters by either neutron or X-ray scattering. Since v is probably between 100 and 400 A and r c SO at Tscattered wave vectors on the order of 0.1 A could be used. 

In H2S decomposition, A represents the sulfur clusters, which diffuse and drift in the centrifugal field. Consider the cluster size n as a continuous coordinate (varying from 1 to 00). Another coordinate is the radial one, x, varying from 0 to the radius of the discharge tube, R. Clusterization in the centrifugal field can be considered an evolution of the distribution function f n, x, t) in the space n, x) of cluster sizes. Dissociation of H2S takes into account a source A (sulfur atoms) that is located at the point x = 0. Thus, evolution of the cluster distribution function can be described by the continuity equation in the space (n, x) of cluster sizes (Macheret, Rusanov, Fridman, 1985) ... 

The coupled equations (6.81) and (6.82) should be solved for x and y to find the cluster distribution function in terms of the polymer volume fraction and the temperature. Upon eliminating x, the second equation is transformed to... 

In the critical region, the cluster distribution function obeys the scaling law... 

Fig. 8 Normalized cluster distribution functions f(m) = ntnj(mn ), for ctmstant (j, = 2), sum (kij = i + j), and product fty = 2ij) kernels at different dimensionless times t = tlz ...

On the other hand, whenever AV exceeds the value of AVq the formation of a dense monolayer film appears to be the continuous process. It has been demonstrated that the observed crossover between those two regimes is due to the changes in the mechanism of the adsorbate nucleation, as determined by the calculation of the nucleated cluster size distribution functions. For... 

Fig. 8.20. Metallicity distribution function of globular clusters (crosses indicating error bars and bin widths) and halo field stars (boxes), after Pagel (1991). Copyright by Springer-Verlag.

The distribution function for field stars in the halo is reasonably well fitted by the Simple model equation (8.20) with a small remaining gas fraction, but with a very low effective yield p 10-11Z for oxygen (see earlier comments on dwarf galaxies). This was first noted (actually for globular clusters) by Hartwick (1976), who pointed out that it could be readily explained by continuous loss of gas from the halo in the form of a homogeneous wind with a mass loss rate from the system proportional to the rate of star formation. In this case,... 

The distribution function for globular clusters is somewhat more complicated, as there appear to be two (probably overlapping) distributions corresponding to the halo and the thick disk, respectively. These have been tentatively fitted in Fig. 8.20 with a Simple model truncated at [Fe/H] = —1.1 for the halo and a model for the thick disk clusters with an initial abundance [Fe/H] = —1.6 (the mean metallicity of the halo) and truncated at [Fe/H] = —0.35. The disk-like character of the more metal-rich clusters is supported by their spatial distribution (Zinn 1985). Furthermore, there is a marginally significant shortage of globular clusters in the lowest... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

The discussion of the defect distribution functions and potentials of average force follows along rather similar fines to that for the activity coefficient. The formal cluster expansions, Eqs. (90)-(91), individual terms of which diverge, must be transformed into another series of closed terms. This can clearly be done by... 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

The point of departure of this method is the "cluster expansion of the non-equilibrium distribution functions ... 

These expressions are analogous to the series expansions of the equilibrium distribution functions in terms of the activity in which appear, in the coefficients, the integrals of the Ursell cluster functions Us (see, for example, ref. 30). 

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