Statistical mechanical configurational partition function

Both Flory and Huggins derived statistical mechanical expressions for aS . Their expressions are still among the best available. For this reason, Prigogine and his co-workers concentrated their efforts on revising the statistical mechanical configurational partition function which leads, among other things, toAH. ... 

In many cases in statistical mechanics, we are not interested in the configurational part of the partition function itself, but in averages of the type in Eq. (1.1), where the ratio between integrals is involved. Metropolis et a1. [1] showed that it is possible to devise an efficient Monte Carlo scheme to sample such a ratio even when we do not know the probability density P(q) in configuration space ... 

We also have an interest in solution processing, that is in ternary phase diagrams of the type PLC + engineering polymer + solvent. A statistical mechanical theory of rigid-rod systems has been developed by Flory he started the work in 1956 (44.) when most of the present applications of PLCs were unknown and continued it more than two decades later (45). The Flory configurational partition function is... 

Here we review well-known principles of quantum statistical mechanics as necessary to develop a path-integral representation of the partition function. The equations of quantum statistical mechanics are, like so many equations, easy to write down and difficult to implement (at least, for interesting systems). Our purpose here is not to solve these equations but rather to write them down as integrals over configuration space. These integrals can be seen to have a form that is isomorphic to the discretized path-integral representation of the kernel developed in the previous section. 

This is one of the most important parameters in statistical mechanics since it is directly related to the thermodynamic properties of a system. The summation in (6.10) is made over all energy states, so z is a function of the partitioning of all particles among all energies for the equilibrium configuration. 

The entropy theory is the result of a statistical mechanical calculation based on a quasi-lattice model. The configurational entropy (S, ) of a polymeric material was calculated as a function of temperature by a direct evaluation of the partition function (Gibbs and Di Marzio (1958). The results of this calculation are that, (1) there is a thermodynamically second order liquid to glass transformation at a temperature T2, and 2), the configurational entropy in the glass is zero i.e. for T > T2, => 0 as T... 

Consider a statistical mechanical description of the system where F represents a point in phase space (or just in configuration space if only the configurational statistics are being discussed). Let C(F) be a mathematical description of the constraint. For a system with a particular value C of C(F), the inicrocanonical ensemble partition function is... 

Sanchez and Lacombe developed the lattice fluid EOS theory using statistical mechanics. Gibbs free energy can be expressed in terms of configurational partition function Z in the pressure ensemble. In the lattice fluid theory development the problem is to determine the number of configurations available for a system of N molecules each of which occupies r sites and vacant sites or holes. Mean field approximation was used to evaluate the partition function. The SL EOS has the capability to account for molecular weight effects, unlike other EOS theories. Characteristic lattice fluid EOS parameters were tabulated for 16 commonly used polymers. 

Models for Stiff-Chain Polymers.— Flory briefly reviewed some of the consequences of separating the configurational partition function for long chain molecules and their solutions into inter- and intra-molecular parts. In particular, he pointed out that this separation, and hence the partition function derived therefrom, are valid only for sufficiently flexible chains, or when the polymer concentration is sufiiciently low. He indicates how this can be rectified in statistical mechanical models for semi-rigid and rod-like polymer molecules. For the latter case this is pursued in considerable detail in a very recent series of papers by Flory and co-workers. ... 

The statistical distribution function (or Slatersum) expresses the relative probability of a spatial configuration of a quantum mechanical system of N molecules in equilibrium with its surroundings. For a system in thermal equilibrium Pjv, the probability that a given state of energy E, is occupied, is proportional to exp (—E,), where = IjkT. The constant of proportionality is Zm, where Zjv is partition function given by... 

The main difficulty of statistical mechanics consists in the evaluation of the configurational partition function. 

The evaluation of the configurational partition function (6.2.5) is now reduced to the simple mathematical problem of picking out the maximum term taking account of the conditions (6.2.2)-(6.2.4) by the Lagrange multipliers X, Y, Z. This procedure is exactly the same as for perfect gases and is explained in every text-book on statistical mechanics. 

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