THE GRAND CANONICAL FORMALISM

Whenever the Hamiltonian of a system obeys a global conservation law, like the conservation of energy or of the number of particles, one might be tempted to describe the statistics of the system by giving a well-defined value to the conserved quantity. However, such an approach would be clumsy because, in this case, any local fluctuation of the conserved quantity must be compensated elsewhere by fluctuations with the opposite sign, and thus the conservation law would entail correlations devoid of interest and difficult to take into account. 

In order to break free from these correlations, larger ensembles are used in which a chemical potential or fugacity is associated with each conserved quantity so as to fix only averages of these conserved quantities. Thus, when the energy is conserved in the system, canonical ensembles are used when, moreover, the system consists of particles whose number is also conserved, we use grand canonical ensembles. 

This is the reason why the grand canonical formalism is used to study polymer solutions, and this fact has important practical consequences. Actually, it is in the grand canonical framework that the simplest perturbation expansions can be obtained, and we note that these expansions are not only useful by themselves they also serve as a basis for all the extensions that are derived from prescriptions of the renormalization theories. 

to study polymer solutions, the grand canonical formalism plays a role which, from a theoretical point of view, is quite essential. 

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The grand canonical formalism is useful only provided n[iJ,p] exists in the thermodynamic limit Q — oo. This property is not completely obvious, as becomes dear from an analysis of the canonical partition function Z M Since the positions of all M chains are integrated independently over all the volume, we find... 

The argument is easily extended to the grand canonical formalism. We simply absorb the prefactor into the chemical potential, defining... 

The grand canonical formalism can be developed for various polymer models. 

In the framework of the grand canonical formalism, we define the correlations corresponding to by means of the probability (or the probability distribution)... 

Pressure and osmotic pressure in the grand canonical formalism... 

For reasons expounded at the beginning of this chapter, it is convenient to express these mean values in the grand canonical formalism. Thus, for instance, the quantity C C j (r) Ca (r) is just one of these functions p 38, / ) defined... 

We also note that the grand canonical formalism deals with symmetry factors in a very elegant way. In fact, with the help of (9.4.2) and of (9.4.3), it is possible to rewrite (9.3.43) in the form... 

The grand canonical formalism provides expansions of the osmotic pressure and of the concentrations in powers of the fugacities but, in principle, it is possible to eliminate these fugacities so as to obtain an expansion of the osmotic pressure in powers of the concentrations moreover, in practice, the calculations are nearly always performed in the limiting case V - go. ... 

For example, let us consider simple molecules in solution. Using the grand canonical formalism, we can find a parametric representation of the dependence of the osmotic pressure with respect to the concentration... 

The grand canonical formalism, presented in Section 5 allows us to express the osmotic pressure and the concentrations in parametric form as functions of the fugacities f(S). Moreover, perturbation theory, expounded in Section 4, enables us to construct connected diagrams representing the expansions of these physical quantities in powers of interactions and fugacities. 

The grand canonical formalism is used to calculate the osmotic pressure of chains as a function of the chain concentrations. Then, the physical quantities are given in parametric form as functions of the fugacities. Moreover, it is convenient to express these expansions in reciprocal space. A number Q of independent internal wave vectors (number of loops) corresponds to each diagram. The contribution of a diagram is obtained by summations over all these wave vectors and by summations over all the positions of the interaction points on the chains. 

Let us note that we can also calculate, without difficulty, the first term (in ) of the expansion of PF with respect to . All these elements enable us to compare theoretical predictions and experimental results in a satisfactory manner (des Cloizeaux and Noda 1982)44. It is also possible to calculate F(x) for all values of x. L. Schafer took a special interest in this problem and, of course, used the grand canonical formalism which gives the dependence of /7 with respect to C in parametric form. In a first article (1981)45 written in collaboration with Knoll and Witten, Schafer derived expansions to order 2 then, later (1982),46 he summarized the complicated results he had obtained, with the help of a semi-phenomenological formula, and he showed that they were in good agreement with experimental results. His formula which takes polydispersity into account can be written in the following way (here d = 3)... 

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