Direct correlation function asymptotic behavior

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

Here, w r) is the attractive (or repulsive) tail of the potential, a is the diameter of spheres, and we have assumed that the fluid is uniform, therefore translational invariance is implied. The first equality in the above equation embodies the physical requirement that the center of a sphere can not penetrate the excluded volume of other spheres. The second equality is just obtained from (1.25) by linearizing the entire exponential factor. Actually, it is the asymptote of the direct correlation function at the infinite separation. The approximation is known to be superior for describing the critical phenomena. The radial distribution function, however, shows an ill-behavior for a Coulombic system, similar to those from the PY closure. 

The study of the asymptotic behavior for long distances of pair correlations in classical fluids is a step beyond the application of Eq. (115). In classical fluids this study can be traced back to the works by Kirkwood [198] and by Fisher and Widom [199]. This is a well-established topic, which has benefited from the advances in the field of direct correlation functions and density functional theories. As was shown by Tago and Smith [187] and, independently, by Evans et al. [185,188,200] the cfR) function plays a central role in this important issue. As a consequence, its formulation is made in terms of the total correlation function hfR), rather than in terms of gfR). This study of asymptotics has become indispensable to the understanding of a wide range of phenomena. In addition to the fundamental features of hfR) (e.g., monotonic or oscillatory decay), one can mention the stability of colloid dispersions, the properties of ionic fluids, or the plethora of phenomena at fluid interfaces [201-206]. 

RhJiR) in terms of the complex zeros of l-p Cj(fe) = 0 (i.e., the poles kj. In general, an infinite number of poles contribute to this expansion, and some convergence problans related to the theoretical long tail of the direct correlation functions are encountered [185, 188, 200], However, the asymptotic behavior can be extracted by keeping only the pole =iy and the pair of lowest y -lying poles that are denoted conventionally by = x, +iy. Pure monotonic exponential decay or exponentially damped oscillatory decay can be identified from these poles. 

Closely related to the QHS fluid is the QHSY fluid. In this regard, one notes that while QHS state points can be characterized with two parameters, that is, (Ag, pI ), QHSY state points need two additional parameters, which are the de Boer quantumness /C=hl(me(T and the inverse range of the attraction k = kdifferent ranges of conditions within (0.2 < A < 0.6 0.27 pi 0.5). Use of direct correlation functions (BDH) was also made, and its reliability to identify the onset of critical behavior was clearly stated [108]. These QHSY studies covered the following issues mechanical and pair structural properties [108] the asymptotic behavior of the pair radial correlations, with a view to the existence of FW lines [159] and the features of triplet correlations in Fourier space [161]. 

The proof of this result is based on exact bounds on the asymptotic behavior of the interfacial correlation functions, obtainable from the Bogoliubov inequality, and uses a reductio ad absurdum a self-maintained interface is assumed to exist and it is then shown that this assumption leads to a contradiction. The key step in the demonstration consists in deriving and making use of the asymptotic behavior at large separations of the direct correlation function of Ornstein-Zernike, c(r, r ), defined in terms of the more familiar pair correlation function ft(r, r ) which measures the probability of having a molecule at point r given that there is one at point r ... 

The correlation length E, along the imaginary time direction is the other quantity we can use to determine the critical region. The correlation length is defined as the asymptotic behavior of the correlation function... 

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