Kirkwood complete

The molecular basis of the Rouse model (and its more sophisticated relatives), however, has remained somewhat obscure. The more complete and realistic treatment of Kirkwood,20 though itself rarely tractable save in a formal sense, could in principle offer a starting point for the derivation... 

The Fokker-Planck method was set forth in a series of papers by Kirkwood and collaborators.3 After taking into account a certain error in the original formulation of this method,4 the theory may be regarded as complete, in the sense that it provides a well-defined method of calculation. (There are reasons, however, for questioning the correctness of the model, i.e., point sources of friction in a hydrodynamic continuum, for which the theory was constructed. These reasons will be discussed in another place.)... 

Equations (19)—(23) provide a complete translation of Kirkwood s Fokker-Planck theory into Langevin terms, when the entire 3JV-dimen-sional configuration space is used. 

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

Kuzovkov and Kotomin [89-91] (see also [92, 93]) were the first to use the complete Kirkwood superposition approximation (2.3.62) in the kinetic calculations for bimolecular reaction in condensed media. This approximation allows us to cut off the infinite hierarchy of equations for the correlation functions describing spatial distribution of particles of the two kinds and to restrict ourselves to the treatment of minimal set of the kinetic equations which realistically could be handled (Fig. 2.21). In earlier studies [82, 84, 91, 94-97] a shortened superposition approximation was widely used... 

The applications of the many-particle densities will be demonstrated on a full scale in further Chapters. It should be only said here that the many-particle density formalism being combined with the shortened Kirkwood superposition approximation, equation (2.3.64), results in the well-known equations of the standard kinetics for both neutral [83] and charged particles [100] giving just another way of their derivation. On the other hand, the use of the full-scale (complete) Kirkwood s approximation, equation (2.3.62), permits us to take into account the many-particle (cooperative) effects [81, 91, 99-102] we are studying in this book. 

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K(t) we need to know either Y(r, t) or p. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

The shortened Kirkwood superposition approximation (2.3.64) differs from the complete one, equation (2.3.63), by the additional condition imposed on the correlation function of similar particles Xv(r,t) = 1 at any time t. Its substitution into equation (5.1.4) and taking into account equation (5.1.5) leads, as one could expect, to the linearized equation (4.1.23) for the correlation dynamics. Therefore, the applicability range of the linearized kinetic... 

Fig. 5.1. The interrelation between the two kinetics of concentrations and correlations treated in terms of complete Kirkwood superposition approximation.

Lastly, in conclusion of the Chapter 5 a study of the effect of the shortened and complete Kirkwood superposition approximations on asymptotic kinetics of a number of single-species reaction should be noted [112, 113]. The general conclusion was drawn that for irreversible reactions the complete superposition approximation, (2.3.62), gives better results than its shortened... 

The single-species A + A —> A reaction allows us also to test the applicability of the shortened and the complete Kirkwood superposition approximations, (2.3.62) and (2.3.64) [98], The calculated quantities of the saturation concentrations... 

In spite of claims to the contrary, to date no completely satisfactory method exists to calculate the polarity / polarizability parameter, n, as it applies to the equilibrium of solute between water and octanol. The excess molar refractivity of the solute (compared to an alkane of equal size) can be estimated separately from polarizability/dipolarity (Abraham, 1994) and seems an attractive approach to this problem, but it needs further verification. The dipole moment of the entire molecule has been used as a polarity parameter (Bodor, 1992), but there are good reasons to believe it has marginal value at best. The square of the dipole moment also has been used (Leahy, 1992), and it, at least, has some theoretical basis (Kirkwood, 1934). 

The radial distribution function was obtained by Pople,103 c.f., Harris and Alder,110 Haggis, Hasted, and Buchanan.111 Pople showed that Kirkwood s assumption of complete hydrogen bonding in the first shell with none in the second was oversimplified. The first shell dipoles are bonded to the second via bent hydrogen bonds of bending... 

However, the most recent discussions favour these high values of g although values of the order of 20% lower had ori nally been favoured. This is because the Frohlich equation [equation (1)] differs from the earlier version of Kirkwood, and treats the inner field in a more nearly correct manner. It is no longer necessary to make a calculation of the HjO dipole moment in its surroundings in the liquid, as had been necessary in the application of the Kirkwood equation. The dipole moment of the free molecule, /i = 1.84D, is used in equation (1), together with = 1.80 at 293 K. This leads to a value of = 2.82, which is sufficiently close to that calculated from the computer dynamics model to warrant optimism for future calculations. The exact choice of will continue to present difficulties until the far-i.r. data are complete over a wide range of temperature. 

Booth" calculated the deviation of As from quadratidty by OnsagCT s model. Calculations of non-linear As-variations of higher order have also been poformed by Kielich in the Kirkwood-Frohlich semi-macroscopic approach taking into consideration statistical molecular correlations. Results such as these can be derived with the non-linear polarization (282). This treatment, however, is not directly applicable to the description of complete electric saturation, and we shall not develop it furtho- here. It appears preferable, for simplicity, to proceed within the framework of classical Langevin-Debye theory, which yidds results wdl adapted to numerical computations. ... 

In the limiting case of complete solvent immobilization inside the molecule (according to Debye and Kirkwood) or at 6-temperature (according to Flory) the above-mentioned treatments give essentially identical results (see also Peterlin, 1959). 

Two theoretical estimates of end effects in fact suggest that they are negligible for segments smaller than this. Zimm et al. (1959), using the polarizability theory of Fitts and Kirkwood (1956a), found that most of the rotation characteristic of the helix should be present with one complete... 

The main-belt region is not evenly populated with asteroids, and several zones have been found in which virtually no asteroids reside (Fig. 1). The American astronomer Daniel Kirkwood (1814-1895) first noticed these empty regions, or gaps, in 1866. Now called Kirkwood gaps, these asteroid-devoid zones are located near orbits for which the time to complete one circuit around the Sun is a simple fraction (e.g., 1/2, 2/3, 3/4) of Jupiter s orbital period. For example, given that Jupiter orbits the Sun once every 11.86 years, an asteroid belt gap is expected at a distance of 3.3 astronomical units (AU) from the Sun, where any orbiting body would have a period of 5.93 years, one-half that of Jupiter. Such a gap does indeed exist. These Kirkwood gaps are produced by... 

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. 

In order to answer this question one has to find out what modifications are necessary in (a) the diffusion equation for the distribution function, and (b) the expression for the stress tensor. Kirkwood and coworkers (39,40,67) and Kotaka (42)w studied this problem for multibead dumbbells including complete hydrodynamic interaction. If one neglects the hydrodynamic interaction entirely, then from the articles cited above one concludes that all the results for rigid dumbbells can be taken over for the multibead dumbbells by replacing X — (,I / 2kT by XN — XN(N + l)/6(iV — 1) everywhere. For the case of complete hydro-dynamic interaction no such simple replacement is possible. 

---