Manipulation of closed form expressions

1 Manipulation of closed form expressions. We consider the canonical partition function Z M of a system of M chains in volume 17. The number of chains of length n is denoted by Mn. and dearly the relations 

Grand Canonical Description of Solutionis at Finite Concentration 

The Gaussian integral over is easily carried out. Using Stirling s formula to evaluate Mnl after a little calculation we find an expression for the density of the free energy in the large volume limit  

We now turn to the evaluation of Z M. The argument of the logarithm in Eq. (A 5.36) can be expanded in 0. closely following Eq. (A 5.29). No linear term occurs, since p contains no k = 0 mode. In Fourier representation the result can be written as 

Here the 5 s are Kronecker symbols and the primed summations extend only over nonvanishing k-vectors. In deriving this result we used the orthonormality of the plane waves Q-l/2e-ifcr 

Clearly /o is the mean field result, corresponding to Flory-Huggins theory. 

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