Statistical mechanics approximate theories

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... 

The calculation of transport coefficients and inverse transport coefficients, such as conductivity and viscosity, is an aim of transport theory. Calculations from first principles in transport theory start from non-equilibrium statistical mechanics. Because of the difficulties involved in calculations in non-equilibrium statistical mechanics, transport theory uses approximate methods, including the kinetic theory of gases and kinetic equations, such as the Boltzmann equation. 

The reduction of thread PRISM with the R-MMSA closure for the idealized fully symmetric block copolymer problem to the well-known incompressible RPA approach " is reassuring. However, in contrast with the blend case, for copolymers that tend to microphase separate on a finite length scale, the existence of critical or spinodal instabilities is expected to be an artifact of the crude statistical mechanical approximations. That is, finite N fluctuation effects are expected to destroy all such spinodal divergences and result in only first-order phase transitions in block copolymers [i.e., Eq. (7.3) is never satisfied]. Indeed, when PRISM theory is numerically implemented for finite thickness chain models using the R-MMSA or R-MPY/HTA closures spinodal divergences do not occur. Thus, one learns that even within the simpler molecular closures, the finite hard-core excluded volume constraint results in a fluctuation effect that destroys the mean-field divergences. 

The closure approximation is the fundamental statistical mechanical approximation in PRISM theory. Determining the appropriate closure depends on the form of the potentials as well as the system parameters such as temperature and pressure [6]. The standard Percus-Yevick (PY) closure has been found to work well for repulsive force potentials in small molecule and macromolecular systems. The PY closure for atomic liquids can be derived using Percus method [79, 80] of a perturbative expansion of the density functional or by Stell s [8] graph summation method. The pair and direct correlation functions in PY theory are given by... 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

Our second goal is to introduce these simple phenomena in a statistical mechanical scheme such that the calculations keep a transparent significance at each step. Nowadays, the predictions of theoretical approaches depend on approximations of a high level of technicality in the domain of liquid state theory. These approximations seem to have a mathematical rather than... 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

The extent of the agreement of the theoretical calculations with the experiments is somewhat unexpected since MSA is an approximate theory and the underlying model is rough. In particular, water is not a system of dipolar hard spheres.281 However, the good agreement is an indication of the utility of recent advances in the application of statistical mechanics to the study of the electric dipole layer at metal electrodes. 

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. 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

Concerning statement 1,1 believe that one should first define what exactly is meant by approximation. In La Fin des Certitudes (p. 29), Prigogine rightly attacks the rather widely present view according to which statistical mechanics requires a (brute) coarse graining (i.e., a grouping of the microscopic states into cells, considered as the basic units of the theory). This process is, indeed, an arbitrary approximation that cannot be accepted as a basis of the fundamental explanation of the very real macroscopic irreversible processes. 

Thus, Prigogine and Petrosky (PP) introduced the model of a Large Poincare system (EPS). As stated above, the latter is, in fact, a large system, to which the operation of Thermodynamic limit is applied. Clearly, there exists no real system satisfying strictly the definition of a EPS This infinite system is an idealization, on which, by the way, all of statistical mechanics is based. One should thus be more specific about the statement The irreversible processes... cannot be interpreted as approximations of the fundamental laws (statement 1). Quite explicitly, the approximations that are avoided in the PP theory are (a) the arbitrary coarse-graining and (b) the restriction to small parameters. 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

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