Partition function grand canonical ensemble

G[T P] = G = —RTIn A, where A is the isothermal-isobaric partition function and U[T,n ] = —RTInE, where S is the grand canonical ensemble partition function. When a system involves several species, but only one can pass through a membrane to a reservoir, L/(7 jux] = — PTlnT, where T is the semigrand ensemble partition function. The last chapter of the book is on semigrand partition functions. 

These relations are obtained from the grand canonical ensemble partition function. 

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and... 

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

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... 

The starting point is the definition of the partition function, S, in the grand canonical ensemble ... 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

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] ... 

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 partition functions contain all the information defining the system, they can also be used to derive the surface pressure. For the canonical and grand canonical ensemble we have... 

The one-field model gives a description of the approximate grand canonical ensembles introduced in Chapter 11, Section 5. These ensembles, called equilibrium ensembles , depend on only two fugacities, which determine respectively the average number of polymers and the average number of monomers (constituting the polymers). In the continuous case, a connected partition function J(/, a) can be associated with this ensemble it is defined by... 

This system, consisting of z identical open subsystems, is best represented by a grand canonical ensemble. From the grand ensemble partition function 3... 

The average is taken with respect to the grand canonical ensemble whose partition function is Z(j(j8, A) = Trexp(-j8 ) [see eq. (107)]. These Green s functions are computed jn a self-consistent perturbation expansion to all orders in VIVN. For a constant V this amounts to a 1/iV-expansion of G and D to infinite order where vertex corrections are neglected (Coleman 1984). Their Fourier transforms are then given by... 

There are numerous other applications and extensions of the Fukui function. For instance, we have not included the extension of the Fukui function formalism to conductors (where there are bands of occupied states rather than occupied orbitals) [58, 59]. One can also fruitfully consider condensed Fukui functions [12, 45, 60-64], where one affixes a Fukui index to each atomic center (by partitioning the molecule into regions in either real space or function space and integrating the Fukui function over that region). The relationships between the Fukui function and the grand canonical ensemble of DFT and the relationships between the Fukui function and the softness kernel and local softness are also important [24, 51, 65]. 

Keywords Configuration interaction Thermodynamics Partition function Temperature Canonical ensemble Grand canonical ensemble Fermi-Dirac statistics... 

A theory of polymer systems near the 0 point (by Uuplantier) has been worked out with the grand canonical ensemble of polymer chains, the grand canonical partition function being written as... 

To calculate the concentration or the number of molecules present in an open system at defined ju, V, T we must consider the grand canonical ensemble which is simply an ensemble in which each subsystem is itself a canonical ensemble. The grand partition function Z is simply the sum of the Boltzmann factors for each of the j canonical ensembles, weighted according to e / ... 

Formally, an expanded grand canonical ensemble can be defined through the partition function ... 

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