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… Show more
“…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.…”
Section: Three-body Correlations Kirkwood's Coupling and Periodic Stmentioning
We revisit the equilibrium properties of a classical one-dimensional system of hardcore particles in the framework provided by the multiparticle correlation expansion of the configurational entropy. The vanishing of the cumulative contribution of more-than-two-particle correlations to the excess entropy is put in relation with the onset of a solidlike behavior at high densities
“…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.…”
Section: Three-body Correlations Kirkwood's Coupling and Periodic Stmentioning
We revisit the equilibrium properties of a classical one-dimensional system of hardcore particles in the framework provided by the multiparticle correlation expansion of the configurational entropy. The vanishing of the cumulative contribution of more-than-two-particle correlations to the excess entropy is put in relation with the onset of a solidlike behavior at high densities
“…, 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) .…”
We study the role of multi-particle spatial correlations in the appearance of
a liquid-vapour critical point. Our analysis is based on the exact infinite
hierarchy of equations relating spatial integrals of $(k+1)$-particle
correlations to the $k$-particle ones and their derivatives with respect to the
density. Critical exponents corresponding to generalized compressibility
equations resulting from the hierarchy are shown to grow linearly with the
order of correlations. We prove that the critical behaviour requires taking
into account correlation functions of arbitrary order. It is however only a
necessary condition. Indeed, approximate closures of the hierarchy obtained by
expressing higher order correlations in terms of lower order ones (Kirkwood's
superposition approximation and its generalizations) turn out to be
inconsistent with the critical behaviour.Comment: 14 pages, 1 figure; v2: minor change
“…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 .…”
Section: Description Of Continuum Approximationsmentioning
confidence: 99%
“…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].…”
We consider a system of interacting particles with random initial conditions. Continuum approximations of the system, based on truncations of the BBGKY hierarchy, are described and simulated for various initial distributions and types of interaction. Specifically, we compare the Mean Field Approximation (MFA), the Kirkwood Superposition Approximation (KSA), and a recently developed truncation of the BBGKY hierarchy (the Truncation Approximation -TA). We show that KSA and TA perform more accurately than MFA in capturing approximate distributions (histograms) obtained from Monte Carlo simulations. Furthermore, TA is more numerically stable and less computationally expensive than KSA.Keywords Many Particle System · Mean Field Approximation · Closure of BBGKY hierarchy Here X i (t) is the position of the ith particle at time t, S is either self-propulsion, internal frequency (as in Kuramoto model, see Subsection 3.3), or a conservative force field (e.g., gravity), W i (t) denotes the Weiner process, and u(x, y) is an interaction Robert Creese
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