Partition function grand-canonical

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... 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

B Grand canonical partition function for the guest-host ensemble (5.5)... 

As a convenient starting point for the model, the grand canonical partition function is developed from the canonical partition function, to incorporate the above assumptions. The canonical partition function is written as the product of three factors the water lattice, the guest distribution within the cages, and the states of the guest molecules themselves assuming that they behave as ideal gas molecules, as follows ... 

We must use the grand canonical partition function E because N (a natural variable to Q) cannot be held constant with the insertion of guests in the hydrate. To obtain E from the canonical function Q, we use the standard statistical mechanics transformation... 

Our aim is to derive the chemical potential of water to enable phase equilibria calculations. Note that, while Equation 5.7 is the grand canonical partition function (2guest) an(j with respect to the solute (guest) molecule, it is the canonical partition function (<2host) with respect to the host (water) because k = 1, so that we have... 

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12] ... 

Making use of the grand canonical partition function, Carnie, Chan, Mitchell and Ninham [2] have derived the expressions corresponding to Eq. (3) for a multicomponent inhomogeneous fluid. 

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) ... 

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

Here e is the grand canonical partition function and serves as a normalizing factor in Eq. (2.20). Many useful statistical results don t depend on the ensemble used in evaluating an average, typically including averages of quantities not constrained by specification of the ensemble. In those cases, we will typically not use the subscripts that explicitly indicate the ensemble used. 

Taking the derivatives of this grand canonical partition function with respect to the Za with T and V fixed produces... 

The remaining factors of Eq. (7.16) will make up the integrand involved in a canonical partition function. But we will prefer to evaluate a grand canonical partition function. In doing this, we will select the appropriate m factors of the activity out of the activities available, and denote that combination by z . The ) feature will be zero for cases that N isn t sufficient to supply the ligand set m. Therefore, all nonzero contributions will be proportional to the factor of z . 

This reduction is not yet possible with [1.5.36] because of restriction [1.5.35b], but we can get rid of this by considering the system to be open, that is by changing to the grand canonical partition function ... 

Another option is to derive equations for the pressures at which condensation takes place in narrow capillaries. Let us illustrate this for slits. As before, the characteristic function -kTlnS of the grand canonical partition function equals -pV + y A, with V = Ah. Let liquid and vapour coexist Inside the capillary and assume that we have only these two phases (i.e. the contribution of the (thin) inhomogeneity at the phase boundary to is Ignored). Equilibrium... 

This is a central equation in the theoretical evaluation of interfacial tensions when the grand canonical partition function can be obtained from some model, / can immediately be found by differentiation with respect to A. Because V is kept constant, only interfacial work is considered. 

However, we now have to amend this expression by the condition that Nd must not exceed for Eq. (3.59) to be meaningful because the pore is completely filled if = -s. To implement this additional constraint into our statistical thermodynamic treatment, we replace the grand canonical partition function derived in Eq. (3.12) for the (infinitely large) bulk fluid by... 

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