Canonical partition function definition

With the benefit of the a and a definitions, it becomes possible to express the canonical partition function for the A-particle system as a simple (/) integral (Stillinger and Weber, 1982),... 

Estimation of the probability for a given quantum state leads to the definition of the canonical partition function of the macrosystem ... 

With these definitions we may now write for the canonical partition function for hydrogen bonding the following equation ... 

The approximation consists in giving a simple form to the fugacity fN which appears in the definition of the grand canonical partition function. Thus,/, is represented a priori by the expression... 

Calculate the canonical partition function for Qxp(—Pq Kq/2) for the case where q e Q = where is a symmetric positive definite matrix. 

The canonical partition function is linked, firstly to the molecular canonical functions, and secondly to the thermodynamic functions that define the phase on the macroscopic level (U, F, G, S, etc.). These two types of relation mean that the canonical partition function forms the link between the microscopic definition of the phase and its macroscopic thermodynamic properties. 

This consideration should then be added to the grand canonical partition function. Leaving the definition of C in the grand canonical partition function, Eq. (93), as the same for the adsorbent except that the localized adsorbate molecules are excluded, i.e. 

We have used the statistical thermodynamics definition of the chemical potential as a derivative with respect to the number of molecules instead of the amount in moles. We have replaced this derivative by a quotient of finite differences as in Eq. (26.1-24), and have attached a subscript to the canonical partition function to indicate the number of particles in the system. 

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

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

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