Classical fluids and the superposition approximation

Abstract

Contents 1. Introduction . 2. Microscopic specification of a simple fluid 2.1. Three states of matter . 2.2. Empirical determination . 2.3. The radial distribution and total correlation functions 2.4. Certain physical associations . 3. Formulae from statistical mechanics . 3.1. The energy function . . 3.2. Correlation functions . 3.3. Formal link with thermodynamic functions . 3.4. Thermodynamics and the pair correlation . 3.5. The use of the equations . 3.6. The virial coefficients . 4. Equations for the pair correlation , 4.1. Method of approach . 4.2. First-order equations.4.3. The superposition approximation . 4.4. Fisher's application of the approximation 4.5. A second-order equation . 4.6. Dependence on density and temperature . 5. Attempts to improve the closure procedure . 5.1. Comments on functional differentiation . 5.2. Rederivation of established equations . 5.3. The triplet indirect correlation . 5.4. Alternative closure approximations . . 6.1. The interparticle force potential . . 6.2. The virial coefficients . 6.3. Liquid densities . 6.4. Further deductions from the theory 6.5. The use of laboratory models..

Entropy and Ordering of Hard Rods in One Dimension

“…The reader is obviously aware that the above expression for the TDF does in fact coincide with the celebrated Kirkwood superposition approximation (KSA) [58,59], which has been widely used in the past as an approximate "closure" of the Born-Green-Yvon (BGY) integro-differential equation for the calculation of the PDF in liquids, both in two and three dimensions [60]. However, here we intend to exploit the "augmented" form of the TDF -which, as an approximation, is obviously unneeded in one dimension aside from being manifestly wrong -in a different speculative context: we suggest to reread Eq.…”

Multi-particle critical correlations

“…, g k−1 and not only of g 2 [16,20]. For instance, a GSA at the level of g 4 is [16,21] g 4 (1, 2, 3, 4) = g 3 (1, 2, 3)g 3 (1, 2, 4)g 3 (1, 3, 4)g 3 (2, 3, 4) g 2 (1, 2)g 2 (1, 3)g 2 (1, 4)g 2 (2, 3)g 2 (2, 4)g 2 (3,4) .…”

Continuum Approximations to Systems of Correlated Interacting Particles

“…It is clear from (10)-(11) that to solve (8) one needs f 3 , the three particle pdf. One can write the equation for f 3 similar to (5) and (8), and this equation will depend on f 4 . We can continue in this manner to obtain a system of N coupled partial differential equations for f 1 , f 2 , ..., f N .…”

“…Following the general idea of closure approximations of the BBGKY hierarchy described above, in KSA a single ansatz for f 3 in terms of f 1 and f 2 is substituted in the equation for f 2 . This ansatz is presented in Section 2 and may be formulated in words as follows: the probability of finding the particle triple in a given configuration equals to the probability of finding each pair independently from the third particle [8]. Though KSA is a phenomenological ansatz, formal justification and further improvements are available [7,26].…”