Partition function definition

This definition is based on and proportional to the g-expectation value. However, it is more useful since it is not necessary to evaluate the partition function to compute an average. 

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... 

The last equation follows from the definition of the partition function, eq. (16.2). Analogously to eq. (16.10) the free energy difference can be evaluated as an ensemble average. 

The formal definition of a partition function implies that it is a summation over an infinitely high number of terms. Explain why the partition function... 

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

We use this knowledge to derive preexponential factors from (2-20) for a few desorption pathways (see Fig. 2.15). The simplest case arises if the partition functions Q and Q in (2-20) are about equal. This corresponds to a transition state that resembles the ground state of the adsorbed molecule. In order to compare (2-20) with the Arrhenius expression (2-15) we need to apply the definition of the activation energy ... 

The summation is here over all possible configurations of the defects on the lattice, each defect being able to occupy any site on its particular sublattice. Configurations in which more than one defect is assigned to a particular lattice site are now included in the summation, but such configurations do not contribute to the partition function because of the definition of the functions ha,... 

In this book, we define cooperativity in probabilistic terms. This is not the most common or popular definition, yet it conveys the spirit and essence of what researchers mean when they use this term. Since the partition function embodies the probabilities of the occupancy events, the definition of cooperativity can... 

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of 5trans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a standard state (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, y° = 24.5 L. 

As noted previously in Chapters 3 and 10, statistical thermodynamics relates all thermodynamic observables to the partition function Q. For ease of reference, the definition of Q and the equations defining various thermodynamic variables as a function of Q, some of which have appeared previously, are as follows... 

Here we use the label i to denote a molecular energy level, which may denote at once the specific translational (t), rotational (r), vibrational (u), and electronic (e) energy level of the molecule. From Eq. 8.46 and the definition of the molecular partition function q,... 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

It is also useful to define the chemical potential in terms of the partition function. By the definition of p,k in Eq. 9.24 and the Helmholtz free-energy expression of Eq. 8.114,... 

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

The WL procedure can be applied to any chosen macrovariable, M. But while a good estimate G(E) is sufficient to allow multicanonical sampling in E [and a definitive one is enough to determine Z(p), Eq. (A10)], the M-density of states does not itself deliver the desired analogues we need, instead, the joint density of states G(E, M) which determines the restricted, single-phase partition functions through... 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

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