Grand canonical ensemble partition function potentials

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

Grand-canonical ensemble GCE (each system has constant V,T, and p the walls between systems are rigid, but permeable and diathermal each system keeps its volume, temperature and chemical potential, but can trade both energy and particles with neighboring systems). The relevant partition function is the grand canonical partition function E ( V, T, fi) ... 

To transform from the canonical to the grandcanonical ensemble with respect to guest molecules using the chemical potential of the guest species, the grand partition function, E, is written as... 

An important partition function can be derived by starting from Q (T, V, N) and replacing the constant variable AT by fi. To do that, we start with the canonical ensemble and replace the impermeable boundaries by permeable boundaries. The new ensemble is referred to as the grand ensemble or the T, V, fi ensemble. Note that the volume of each system is still constant. However, by removing the constraint on constant N, we permit fluctuations in the number of particles. We know from thermodynamics that a pair of systems between which there exists a free exchange of particles at equilibrium with respect to material flow is characterized by a constant chemical potential fi. The variable N can now attain any value with the probability distribution... 

The first of these difficulties can be avoided for symmetrical polymer mixtures (Na = Nb = N) by working in the semigrandcanonical ensemble of the polymer mixture [107] rather than keeping the volume fractions < )A, B and hence the numbers of chains nA, nB individually fixed, as one would do in experiment and in the canonical ensemble of statistical thermodynamics, we keep the chemical potential difference Ap = pA — pB between the two types of monomers fixed as the given independent variable. While the total volume fraction 1 — < )v taken by monomers is held constant, the volume fractions < )A, B of each species fluctuate and are not known beforehand, but rather are an output of the simulation. Thus in addition to the moves necessary to equilibrate the coil configuration (Fig. 16, upper part), one allows for moves where an A-chain is taken out of the system and replaced by a B-chain or vice versa. Note that for the symmetrical polymer mixture the term representing the contributions of the chemical potentials pA, pB to the grand-canonical partition function Z... 

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