Triplet and higher correlations for fluid mixtures in the long-wavelength limit

Particle and Energy Pair and Triplet Correlations in Liquids and Liquid Mixtures from Experiment and Simulation

“…The above relationships also provide the equations for the compressibility, thermal expansion, and heat capacity of a protein through eq 18 (see the Appendix). The key term for each derivative is the fluctuating quantity 13) where ⟨δXδY⟩ N = ⟨XY⟩ N − ⟨X⟩ N ⟨Y⟩ N , etc. The other terms in eq 12 are usually small in magnitude and/or cancel when considering the difference between two conformations (see below).…”

“…The theory used to obtain the fluctuation densities in terms of experimental thermodynamic data is quite involved. ,, Hence, only the final isothermal equations for pure liquids and a binary mixture of components 1 and 2 are provided here. The equations are simplified by defining the quantities b αβ = B αβ /ρ and c αβγ = C αβγ /ρ, where ρ = ρ 1 + ρ 2 is the total number density.…”

“…However, little subsequent application of these triplet integrals has appeared, despite the comparatively prevalent use of the KB pair integrals. A more general approach for pure liquids was also provided by Egelstaff in terms of scattering data, but the applications were limited to very simple systems. , Finally, the ability to obtain triplet correlations for liquid mixtures was essentially achieved in the early 1970s by Bhatia and Ratti and Parrinello and Tosi. , However, these studies focused on the use of a (long wavelength) structure factor formulation of the triplet fluctuations, with applications to liquid metal alloys. Hence, the use of the above approaches for common liquids and liquid mixtures has been very limited.…”

Experimental triplet and quadruplet fluctuation densities and spatial distribution function integrals for liquid mixtures

“…They are also closely related to structure factors of various order. 34,35,46 The B's were already defined by Kirkwood and Buff. 27 For a multivariate symmetric (e.g., Gaussian) distribution, the C's and D's would be zero.…”

“…The first expression summarizes the relationships derived by Bhatia and Ratti for the third order structure factors at long wavelengths in binary mixtures. 34 The expressions for the b α β quantities depend on the number of components, although a general expression can be written in terms of matrix cofactors as indicated by Eq. (7).…”