Grand partition function

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... 

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... 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

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 reader who is less familiar with the theory of grand partition functions may directly proceed to Eqs. 12a and 13. The physical basis of these formulas and the significance of the quantities CK% will then become apparent in the subsequent paragraph is the vapor pressure (or fugacity) of solute K and y i is the probability of finding a K molecule in a cavity of type i. 

The function S is a grand partition function with respect to the solutes, but an ordinary partition function with respect to the solvent it is linked to thermodynamics by the relation... 

The relation is the same as that for grand partition functions (cf., e.g., Rushbrooke37), evidently with the exception of the term... 

The eigenvalues of this operator are n E0u, and those of the operator exp HjlcT) are therefore exp nuE0JlcT). Using Eq. (8-220) for the grand partition function, we then find... 

To obtain a similarly explicit expression for the grand partition function Z, we proceed in a somewhat different way. Thus we have... 

This, combined with Eq. (8-238), yields the general explicit form for the grand partition function operator. 

Particles spin Vz, 517 Dirac equation, 517 spin 1, mass 0,547 spin zero, 498 Partition function, 471 grand, 476 Parzen, E., 119,168 Pauli spin matrices, 730 PavM, W., 520,539,562,664 Payoff, 308 function, 309 discontinuous, 310 matrix, 309... 

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... 

The same approach applies to alternate partition functions the WL output is used directly in the macroscopic probability scheme. For example, in the previous grand canonical scenario, the WL simulation would yield which would be substituted directly into the macrostate probabilities as Q exp(J ) for subsequent results generation. 

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

In the basis of energy eigenfunctions the grand partition function is... 

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. 

Having defined the system to be studied, we proceed to characterize the states of the system and the corresponding energies. These are the fundamental building blocks for constructing the grand partition function (GPF) of the system (Section 1.4). 

We first treat the most general case where all sites are different. The grand partition function of a single adsorbent molecule is... 

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