Statistical mechanics grand partition function

S. Adair, H. S. Sinuns, K. Linderstrom-Lang, and, especially, J. Wyman. These treatments, however, were empirical or thermodynamic in content, that is, expressed from the outset in terms of thermodynamic equilibrium constants. The advantage of the explicit use of the actual grand partition function is that it is more general it includes everything in the empirical or thermodynamic approach, plus providing, when needed, the background molecular theory (as statistical mechanics always does). 

The statistical mechanical verification of the adsorption Equation 11 proceeds most conveniently with use of the expression for y given by Equation 5. An identical starting formula is obtained via the virial theorem or by differentiation of the grand partition function (3). We simplify the presentation, without loss of generality, by restricting ourselves to multicomponent classical systems possessing a potential of intermolecular forces of the form... 

Gerdanian (245) has developed a statistical-mechanical model for UO2 + X with small departure from stoichiometry to interpret the experimental values of AHq (244). His model is based on three types of defects—(a) f + 2Uy [Kroger and Vink notation (262)] in U02 [denoted by (a)], (b) 2 1 2 Willis cluster and two [denoted by (/)] in U02+ ,., and (c) 2 2 2 Willis clusters [denoted by (ii)] in U02+ (. The semi-grand partition function (SGPF) related to the defects... 

The connections of defined macro-parameters with molecular structure system provide statistical mechanics. The route proceeds from the partition (or better to say grand partition function) Zof the system going to the free energy G and finally to y by means of the general relationship ... 

We round out this introduction to the virial equation of state by reference to its theoretical foundation. Thus statistical mechanics permits deduction of an expression for pVin terms of either the grand partition function or the radial distribution function. The leading term in the expansion of the latter function corresponds to pairwise interaction between molecules, and indicates the following relation between the second virial coefficient and the potential energy (r) of the interacting pair, when this depends only on the distance r between molecular centres ... 

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... 

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... 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

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