Statistical mechanics perturbation

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

Application of Statistical Mechanical Perturbation Theory to Compute Relative Free Energies of Ligand-Receptor Interactions. ... 

In the work of Haymet and Oxtoby the direct correlation function is approximated through statistical mechanical perturbation theory about its value for a uniform liquid of density Pq. This approach relies on the earlier work of Ramakrishnan and Youssouff who showed that such a jjerturbation approach gave rather accurate results for the equilibrium phase diagram for atomic liquids. One advantage of this approach is that the only external input to the functional 2 is the direct correlation of the liquid, which can be related to the structure factor, a quantity measurable by x-ray or neutron scattering... 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

Several types of equations of state are modifications of the VDW equation with different expressions for the attractive and repulsive interactions. Most authors focused on either a or b. However, Ihm, Song, and Mason (ISM) used a statistical mechanical perturbation method with the intermolecular pair potential to obtain a VDW-type equation of state [Song and Mason, 1989,1990, 1992 Ihmet al., 1991] ... 

The first of these developments is perturbation theory. Its application to solution theory was perhaps first made by H. C. Longuet-Higgins in his conformal solution theory (Longuet-Higgins 1951). The formal theory of statistical mechanical perturbation theory is very simple in the canonical ensemble. If denotes the intermo-lecular potential energy of a classical A-body system (not necessarily the sum of pair potentials), the central problem is to evaluate the partition function. 

This is the entire formal structure of classical statistical mechanical perturbation theory. The reader will note how much simpler it is than quantum perturbation theory. But the devil lies in the details. How does one choose the unperturbed potential, y How does one evaluate the first-order perturbation It is quite difficult to compute the quantities in Equation P5 from first principles. Most progress has been made by some clever application of the law of corresponding states. It is not the aim of this chapter to follow this road to solution theory any further. 

Perhaps the simplest application of the statistical mechanical perturbation theory of fluids is a derivation of the van der Waals equation. To derive the van der Waals equation, we first write the two-body intermolecular potential as the summation of a hard sphere part Ujjg(r) and an attractive part u r),... 

Fig. 10. Densities of coexisting phases for a fluid obeying the 6-12 potential according to a statistical mechanical perturbation theory developed by Barker and Henderson. The points are a mixture of machine calculations and actual experimental data (from Barker and Henderson, 1967).

We have so far ignored quantum corrections to the virial coefficients by assuming classical statistical mechanics in our discussion of the confignrational PF. Quantum effects, when they are relatively small, can be treated as a perturbation (Friedman 1995) when the leading correction to the PF can be written as... 

Smith W R 1972 Perturbation theory in the classical statistical mechanics of fluids Specialist Periodical Report vol 1 (London Chemical Society)... 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

In both examples discussed in this section, the second-order approximation to AA turned out to be satisfactory. We, however, do not want to leave the reader with the impression that this is always true. If this were the case, it would imply that probability distributions of interest were always Gaussian. Statistical mechanics would then be a much simpler field. Since this is obviously not so, we have to develop techniques to deal with large and not necessarily Gaussian-distributed perturbations. This issue is addressed in the remainder of this chapter. 

Just as in classical statistical mechanics, the different pictures of electronic changes are related by Legendre transforms. The state function for closed systems in the electron-following picture is just the electronic ground-state energy, /i v AT The total differential for the energy provides reactivity indicators for describing how various perturbations stabilize or destabilize the system,... 

Resibois, P., A Perturbative Approach to Irreversible Statistical Mechanics in Many Particle Physics, E. Meeron Ed., B. Gordon and Breach, New York, to be published, 1965. 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

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