Partition function canonical ensemble

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 canonical ensemble partition function is the phase space integral... 

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 density of states in turn, is connected with the canonical ensemble partition function Z T) through the formal weighted sum over states,... 

The canonical ensemble partition function for a binary solution is — v/VT... 

Making use of the random mixing theory, the canonical ensemble partition function becomes... 

This canonical ensemble partition function predicts a first-order two dimensional phase transition as shown by Hill (8). 

Three of the eight thermodynamic potentials for a system with one species are frequently used in statistical mechanics (McQuarrie, 2000), and there are generally accepted symbols for the corresponding partition functions V[T = A = — RTlnQ, where Q is the canonical ensemble partition function ... 

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. 

Q canonical ensemble partition function zi charge number of ion j... 

These relations will be established specifically in Chapter 3. j3 = kT, where k is the Boltzmann constant, and A is the thermal de Broglie wavelength. Q Ua = 1) = ISa is the canonical ensemble partition function of a system... 

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. 

The previous expression should be compared with Eq. (C.29a) where we emphasize that, unlike Eq. (3.55), Eq. (C.29a) was derived without employing a specific form of the canonical ensemble partition function Q. 

B.5 Basis sets and the canonical ensemble partition function... 

Now, let us apply the same general scheme, bnt this time to the derivation of the van der Waals isotherm, which corresponds to nonlocalized adsorption of interacting molecules. (Expressions corresponding to the Volmer isotherm can be obtained by setting p = 0 in the respective expressions for the van der Waals isotherm.) Now the adsorbed molecules are considered a two-dimensional gas. The corresponding expression for the canonical ensemble partition function is... 

Here iVi is the number of particles in the cell, from 0 to Qi Ni) is the canonical ensemble partition function of the cell if it contains... 

Since Ti=K. + V, the canonical ensemble partition function factorizes into ideal gas and excess parts, and as a consequence most averages of interest may be split into corresponding ideal and excess components, which sum to give the total. In MC simulations, we frequently calculate just the excess or configurational parts in this case, y consists just of the atomic coordinates, not the momenta, and the appropriate expressions are obtained from equation b3.3.2 by replacing Efby the potential energy V. The ideal gas contributions are usually easily calculated from exact... 

Statistical mechanics may be used to derive practical microscopic formulae for thermodynamic quantities. A well-known example is the virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition function... 

Given these assumptions the corresponding-states principle may be easily derived from scaling arguments applied to the residual canonical-ensemble partition function, = QIQ" - Using standard notation ... 

With the help of statistical mechanics an expression is obtained, say, for the canonical ensemble partition function, Q, from which the surface free energy is determined [11] ... 

This quantity q is called the partition function. It plays a central role in statistical thermodynamics. Because we defined our system as a canonical ensemble, q is commonly called the canonical ensemble partition function. 

ZEab ZEf — ZEbb 2Zw The canonical ensemble partition function then becomes... 

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