Pair interactions statistics

It has long been recognized that the validity of the BKW EOS is questionable.12 This is particularly important when designing new materials that may have unusual elemental compositions. Efforts to develop better EOSs have been based largely on the concept of model potentials. With model potentials, molecules interact via idealized spherical pair potentials. Statistical mechanics is then employed to calculate the EOS of the interacting mixture of effective spherical particles. Most often, the exponential-6 (exp-6) potential is used for the pair interactions ... 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

One says that the above results are valid for a chain with non-interacting particles. However, the monomers in a real macromolecule interact with each another, and this ensures, above all, that parts of the molecule cannot occupy the place already occupied by other parts i.e. the probabilities of successive steps are no longer statistically independent, as was assumed in the derivation of the above probability distribution functions and mean end-to-end distance (Flory 1953). So, considering the coarse-grained model, one has to introduce lateral forces of attractive and repulsive interactions. The potential energy of lateral interactions U depends on the differences of the position vectors of all particles of the chain and, in the simplest case, can be written as a sum of pair interactions... 

Abstract. The statistical calculation of the temperature dependence of heat capacity of ordering two-component fullerite has been fulfilled in the approximation of pair interaction between fullerenes by the method of average energies in the model of spherically symmetrical stiff balls. 

At least in lattice statistics, pair interaction energies are usually accounted for by a parameter w, which is positive for repulsion and negative for attraction for m = 0 the situation reduces to the Langmuir case. 

Atom-Pair Interaction Potentials. Affinities can be calculated based on ligand-receptor atom-pair interaction potentials that are statistical in nature rather than empirical. Muegge and Martin (320) derived these potentials from crystallographic data in the Protein... 

Again harking back to our use of the Cu-Au system to serve as a case study, many of the ideas associated with the approximation schemes given above can be elucidated pictorially. For the present discussion, we consider a model Hamiltonian in which only pair interactions between neighboring atoms are considered. That is, the cluster expansion of the total energy (as given in eqn (6.22)) is carried only to the level of terms of the form CTiO). Even within the confines of this trimmed down effective Hamiltonian, the treatment of the statistical mechanics can be carried out... 

Knowledge-based functions are based on the derivation of statistical preferences in the form of potentials for protein ligand atom pair interactions. Similar to potentials derived for protein folding and protein structure evaluation (e.g., Ref. 148), pair potentials akin to potentials of mean force (PMFs) are derived for various protein and ligand atom types using the PDB as a knowledge base. The PMF scoring function [118]... 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

Two sets of methods for computer simulations of molecular fluids have been developed Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. 

The classic rigid lattice model still proves to be a valuable tool for predictive calculations of phase stability in polymer blends. It depends on the required accuracy whether the simple model needs improvement. Miscibility behaviour of blends containing statistical copolymers do not seem to require such Improvements since values of pair-interaction parameters obtained on one system appear to be transferable to another, fig.6 bears witness of this welcome feature. A similar transferability was observed previously in an analysis of critically demixing poly(styrene) solutions in n-alkanes and n-alcohols varying in number of carbon atoms [31]. The data allowed extraction of the end/middle group interaction function for n-alkanes that correctly predicted the lower critical miscibility behavior of n-alkane /poly(ethylene) systems. 

Interlude 4.2 Getting Microscopic Information from Macroscopic Observations The Inverse Problem When one thinks of equilibrium statistical mechanics, what usually comes to mind is its classical mission, namely, predicting macroscopic structure and properties from microscopic, pairwise interaction forces. However, more often than not, we are faced with a need to deduce information on the (conservative) pair interaction forces [equivalently, the pair potential observed macroscopic properties. The information thus obtained can then be used to predict other properties of interest. The problem of extracting microscopic information from macroscopic observations is known as the inverse problem. 

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