Grand Canonical Description of Solutions at Finite Concentration

The statistical mechanics of solutions at nonvamshing concentration is best-formulated within the grand canonical ensemble, where the number of solute molecules contained in volume Si is allowed to fluctuate. We only control the average concentration cp via the chemical potential fi.p This has great technical advantages, allowing for a very simple, analysis of the thermodynamic limit. In contrast, in the canonical ensemble, where the particle number M = Qc.p is taken as basic variable, an analysis of the thermodynamic limit is more tricky. A short discussion is given in the appendix. 

However, these formal expansions are not adapted to an actual evaluation valid outside the dilute region. A simple consideration reveals the problem. Consider some chain of length n in a solution of segment concentration c. Following Chap. 2, Eq. (2,10ii) we estimate the number of two-body contacts of that chain with all the other chains as 

This characteristic energy naturally shows up in the cluster expansion. Note that W can become large even for /3e l, dc C 1, just because the chains 

Grand Canonical Description of Solutions at Finite Concentration 

This consideration alone, though physically most reasonable, does not guarantee that the loop expansion, i.e. the expansion in fluctuations about the average density, really eliminates all problems connected to W 1. We have to add one further essential observation in some sense the densitv 

We should immediately note that the picture described here in the context of scaling or renormalization theory Icnids to the important concept of concentration blobs. (See Sect. 9.1.) 

The organization of this riiapter is as follows. In Sect. 5.1 we present the brisie formalism and work out the Feynman niles for the grand canonical (ensemble. Diagrammatic representations valid in the thermodynamic limit u e derived for both thermod3uiamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2 

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