Perturbation theory, thermodynamic properties

In thermodynamic perturbation theory the properties of the real system, in which the pair potential is u° g, are expanded about the values for a reference system, in which the pair potential is ug3. Here we take the reference potential to be the (n,6) potential, so that the anisotropic parts of the potential in Eq. (1) are the perturbation. Expanding the Helmholtz free energy A in powers of the perturbing potential about the value Aq for the reference system gives... 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

Feg). Subsequently, thermodynamic properties of spins weakly coupled by the dipolar interaction are calculated. Dipolar interaction is, due to its long range and reduced symmetry, difficult to treat analytically most previous work on dipolar interaction is therefore numerical [10-13]. Here thermodynamic perturbation theory will be used to treat weak dipolar interaction analytically. Finally, the dynamical properties of magnetic nanoparticles are reviewed with focus on how relaxation time and superparamegnetic blocking are affected by weak dipolar interaction. For notational simplicity, it will be assumed throughout this section that the parameters characterizing different nanoparticles are identical (e.g., volume and anisotropy). 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

The truncation of the high temperature series in Eq. (4) at order is a vahd concern when applying the theory at low temperatures. This concern also extends to GD theory, which implicitly involves a truncation at order p. At some point, these perturbative treatments must simply fail, but we expect the lattice theories to identify faithfully the location of the entropy crisis at low temperatures, based on numerous previous comparisons between measurements and GD theory. Experience [96] with the LCT in describing equation of state [97] and miscibility [98] data indicates that this approach gives sensible and often accurate estimates of thermodynamic properties over wide ranges of temperatures and pressures. In light of these limitations, we focus on the temperature range above Tg, where the theory is more reliable. 

Donohue M.D., Prausnitz J.M., "Perturbed hard chain theory for fluid mixtures Thermodynamic properties for mixtures in natural gas and petroleum technology", AIChEJ. 1975, 24(5), 850. 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

The first chapter by Moszyliski presents in a systematic and comprehensive manner the current state-of-the-art theory of intermolecular interactions. Numerous examples illustrate how theoreticians and experimentalists working in tandem may gather valuable quantitative results related to intermolecular interactions, like accurate potential functions, interaction-induced properties, spectra and collisional characteristics or dielectric, refractive or thermodynamic properties of bulk phases. On the other hand the most advanced Symmetry Adapted Perturbation Theory (SAPT) enables validation of more approximate variation-pertubation models which could be applied to the analysis of specific interactions in much larger molecular systems, for example enzyme-drug interactions discussed in Chapter VIII by Berlicki et al. 

In the present paper we review recent advances in the symmetry-adapted perturbation theory calculations of interaction potentials and interaction-induced properties. We will give a brief description of the theoretical methods needed on the route from the intermolecular potential and properties to rovibrational spectra and collision-induced Raman spectra. We also discuss applications of the interaction potentials and interaction-induced polarizabilities to compute (thermodynamic and dielectric) second virial coefficients. Finally, we illustrate these theoretical approaches on several examples from our own work. 

As we described in Section III.G, perturbation theories can be extended in a systematic way using cluster expansion techniques. These techniques have recently been applied to the calculation of the thermodynamic properties and vapor-liquid equilibrium of 12-6 diatomics and seem to offer a clear improvement over the first-order perturbation theories. To illustrate this point. Table I shows values of the critical density and critical temperature predicted by the ISF-ORPA theory and the first-order perturbation theory together with results recently obtained from molecular dynamics... 

Perturbation theory has been applied to anharmonic calculations of spectroscopy from ab initio potentials in a large number of studies [19-25,115-121]. In nearly all cases so far, second-order perturbation theory was employed. The representation of the anharmonic potential generally used in these studies is a polynomial in the normal modes, most often a quartic force field. A code implementing this vibrational method was recently incorporated by V. Barone in gaussian [24]. Calculations were carried out for relatively large molecules, such as pyrrole and furan [25], uracil and thiouracil [118], and azabenzenes [119]. We note that in addition to spectroscopy, the ab initio perturbation theoretic algorithms were also applied to the calculation of thermodynamic properties... 

These equations provide a convenient and accurate representation of the thermodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

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