Statistical mechanics weight factor

Given an approximation to V( 0s ) acceptable for the purposes at hand, one can proceed to compute equilibrium, i.e., statistically mechanically averaged, values

for properties P( 0s ) of interest using standard procedures which weight each conformation of the carbohydrate molecule by the Boltzmann factor of V( 0s ) normalized by the configuration integral given in eqn. (9). 

The statistical mechanical theory of the helix-coil transition of DNA is improved by introducing approximate normalization factors for the unnormalized statistical weights of finding a given molecule of the assembly in a given microscopic state. 

According to the rules of statistical mechanics, in order to calculate the mean value of a quantity depending on p and q the quantity must be multiplied by the weighting factor er w, where /3=1/ T, and then averaged over the whole of tho phase space (p, q) corresponding to possible motions. Thus the mean energy becomes... 

Molecular architectures can be structurally classified as being more comb-like or Cayley tree-like. Structure has impact on the radius of gyration, which is larger for linear molecules than for branched molecules of the same weight (number of monomer units), since the latter are more compact. The ratio between branched and linear radius is usually described by a contraction factor . Furthermore, Cayley tree-like structures are more compact than comb-like structures [33, 56]. We will show here how to obtain the contraction factor from the architectural information. The squared radius of gyration is expressed in monomer sizes. According to a statistical-mechanical model [55] it follows from the architecture as represented in graph theoretical terms, the KirchhofF matrix, K, which is derived from the incidence matrix, C [33] ... 

For a gas containing N molecules of the same chemical species, the molecules would all be indistinguishable from one another. The factor W has to be divided by Nl in this case. The proper explanation can only be understood through a detailed discussion of quantum mechanics and Bose-Einstein statistics. This explanation is beyond the realm of interest here, and we simply state the proper weighting for a collection of N indistinguishable molecules as... 

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