Ensemble 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 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,... 

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy e can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,... 

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... 

The number of microscopic energy distribution states Q(N,V,U) in the system is also related with the ensemble partition function Z. According to statistical mechanics, the entropy S has been connected with the ensemble partition function Z in the form of Eq. 3.7 ... 

Given an ensemble of static electric dipole moments of magnitude fi and random orientation in an external static electric field E, we can use the microcanonical ensemble partition function to compute the average moment... 

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... 

Finally, the formalism presented above can be used to create a procedure for constructing the equilibrium ensemble partition function that is generated by a non-Hamiltonian d3mamical system. First, determine all of the conservation laws that are satisfied by the equations of motion. The distribution function, /(x), will then be written in the form of (68). Second, eliminate... 

The equilibrium properties of an adsorbed layer can be examined based on the chemical or electrochemical potentials of the constituents of this layer and the equilibrium equations derived in the section above. This is the simplest approach, although problems might appear in the description of the adsorbed layer properties during a surface phase transition [18]. Alternatively, the chemical potentials may be used for the determination of the grand ensemble partition function of the adsorbed layer, which in turn is used for the derivation of the equilibrium equations. This approach is mathematically more complex, but it leads to a better description of an adsorbed layer when it undergoes a phase transformation [18]. The present analysis for simplicity is restricted to the first approach. 

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... 

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