Classical partition function, matrix

Using the matrix method of calculation (Li), it is possible to show that the classical partition function of the system is ... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

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. 

Equation (21) is a description of the sorption isotherm. It provides the total dissolved concentration c at any pressure p. Its usefuln lies in the fact that for any temperature T the local classical configuration partition function Zj can be evaluated for each site j, once the distribution function p(r) is known for a solute motecule as a function of its position r in the polymer matrix. 

Here a and b are occupied MO s of systems A and B. Equation (6,32) is easily expressible in terms of integrals over atomic basis functions and elements of the density matrix. In eqn. (5.31) two terms may be distinguished. The first one is due to single electron excitations of the type a r") and (b —->s"), where a and r", respectively, are occupied and virtual MO s in the system A, and b and s" are occupied and virtual MO s in the system B, Contribution of these terms corresponds to the classical polarization interaction energy, Ep, Two-electron excitations (a r", b — s"), i.e. simultaneous single excitations of either subsystem, may be taken as contributions to the second term - the classical London dispersion energy, Ep, If the Mjiller-Ples-set partitioning of the Hamiltonian is used, Ep may be expressed in... 

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