Partition function particle insertion

The usual view of the PDT Eq. (3.3) is that it is a particle insertion formula (Valleau and Torrie, 1977 Frenkel and Smit, 2002). This is a view for implementing the indicated averaging solution configurations are sampled without the distinguished molecule, and then the distinguished molecule is imposed upon the solution. The natural statistical estimate of the Boltzmann factor then provides the approximate evaluation of this partition function. This procedure has been extensively used, and naturally it will fail as a practical matter if the variance of the estimator is too large. 

In the context of van der Waals theory, a and b are positive parameters characterizing, respectively, the magnitude of the attractive and repulsive (excluded volume) intermolecular interactions. Use this partition function to derive an expression for the excess chemical potential of a distinguished molecule (the solute) in its pure fluid. Note that specific terms in this expression can be related to contributions from either the attractive or excluded-volume interactions. Use the Tpp data given in Table 3.3 for liquid n-heptane along its saturation curve to evaluate the influence of these separate contributions on test-particle insertions of a single n-heptane molecule in liquid n-heptane as a function of density. In light of your results, comment on the statement made in the discussion above that the use of the potential distribution theorem to evaluate pff depends on primarily local interactions between the solute and the solvent. 

The Rory lattice model [88] for polymeric LCs has received the most attention, although it lay dormant for more than a decade after it had been introduced in 1956. It is ideally suited for lyotropic LCs consisting of solvent and rigid rods, although it has been considered (with modifications) in the context of semillexible linear polymers and thermotropic monomer LCs more recently. The crux of this model is the derivation of the partition function Z corresponding to the inserfion of p rod-like solute particles (each comprised of x segments) into a lattice with no sites all sites are filled in the solution by inserting s = o — xn solvent particles. Zis the product of two components a combinational part,... 

The power 3A is justified easily, because for a particle in a three-dimensional box we have from quantum mechanics k = k + kx + k and for each -component i we have v = L/jt)kvi (note V = L ) with v — 1,2, — The factor results from the proper normalization of the N-particle state v > to be used in Eq.(5.48) instead of the single particle state. However, there is alternative classical motivation for this factor in the context of the so called Gibbs paradox. But let us first proceed with the three dimensional ideal gas. Inserting the partition function Eq. (5.55) into the free energy Eq. (5.10) immediately yields for the pressure, P = —dF/dV T,... 

---