Distribution function, theory

TFL.5. R. L. Anderson, R. Herman, and 1. Prigogine, On the statistical distribution function theory... 

A similar situation exists in the molecular-distribution function theory of liquids and one usually resorts to a superposition approximation. This amounts to assuming that, e.g., = 2 or something similar. It will be seen shortly that, contrary to unimolecular reactions, for bi-molecular reactions the stochastic mean is not the same as the classical kinetic expression for the concentration. 

Gonzales-Tovar E, Lozada-Cassou M, Henderson D (1985) J Chem Phys 83 361 Das T, Bratko D, Bhuyan LB, Outhwaite CW (1997) J Chem Phys 107 9197 Kjellander R (2001) Distribution function theory of electrolytes and electrical double layers. In Holm C, Kekicheff P, Podgornik R (eds) Electrostatic Effects in Soft Matter and Biophysics. Kluwer, Dordrecht, p. 317... 

We adopt an alternative route to the distribution function theory. The approach is based on the density-functional theory. In this approach, the change of variables is conducted through Legendre transform from the solute-solvent interaction potential function to the solute-solvent distribution function or the solvent density around the solute. The (solvation) free energy is then expressed approximately by expanding the corresponding Legendre-transformed function with respect to the distribution function to some low order. 

Schlijper, A. G. and Kikuchi, R., A variational approach to distribution function theory. /. Slat. Phys. 61, 143-160 (1990). 

Kurata, M., M. Tamura, and T. Watari J. Chem. Phys. 23, 991 (1955). Stockmayer, W. H. MakromoL Chem. 35, 54 (1960) gives an interesting and short discussion of the theory of dilnte polymer solutions in general. He also compares the lattice and distribution function theories somewhat similarly to our discussion. 

The fluid microstructure is of central interest in the theoretical study of supercritical solutions for several reasons. Bulk fluid properties can be obtained from knowledge of the fluid structure and the intermolecular potential through distribution function theory. In addition, it has recently become clear, both from experimental evidence and theoretical analyses (1-5) that solvation structure changes rapidly near a CP. 

In previous work (5.8.9 we have shown that models based on distribution function theory yield a priori predictions of the qualitative behavior of supercritical solutions and yield empirical fits to data which show excellent accuracy for solubility and partial molar volume. 

McGuigan, D. B., and P. A. Monson. 1990. Analysis of infinite dilution partial molar volumes using a distribution function theory. 7. Fluid Phase Equil. 57 227-247. 

Sung, J., Lee, S. Relations among the modern theories of diffusion-influenced reactions. 1. Reduced distribution function theory versus memory function theory of Yang, Lee, and Shin. J. Chem. Phys. 1999, 111, 10159. 

Another statistical mechanical approach makes use of the radial distribution function g(r), which gives the probability of finding a molecule at a distance r from a given one. This function may be obtained experimentally from x-ray or neutron scattering on a liquid or from computer simulation or statistical mechanical theories for model potential energies [56]. Kirkwood and Buff [38] showed that for a given potential function, U(r)... 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

Born M and Green H S 1946 A general kinetic theory of liquids I. The molecular distribution functions Proc. R. Soc. A 188 10... 

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

Amadei, A., Apol, M. E. F., Di Nola, A., Berendsen, H. J. C. The quasi-Gaussian entropy theory Free energy calculations based on the potential energy distribution function. J. Chem. Phys. 104 (1996) 1560-1574... 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

Using the calculated phonon modes of a SWCNT, the Raman intensities of the modes are calculated within the non-resonant bond polarisation theory, in which empirical bond polarisation parameters are used [18]. The bond parameters that we used in this chapter are an - aj = 0.04 A, aji + 2a = 4.7 A and an - a = 4.0 A, where a and a are the polarisability parameters and their derivatives with respect to bond length, respectively [12]. The Raman intensities for the various Raman-active modes in CNTs are calculated at a phonon temperature of 300K which appears in the formula for the Bose distribution function for phonons. The eigenfunctions for the various vibrational modes are calculated numerically at the T point k=Q). 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

Let us proceed with the description of the results from theory and simulation. First, consider the case of a narrow barrier, w = 0.5, and discuss the pair distribution functions (pdfs) of fluid species with respect to a matrix particle, gfm r). This pdf has been a main focus of previous statistical mechanical investigations of simple fluids in contact with an individual permeable barrier via integral equations and density functional methodology [49-52]. 

Theoretical results of similar quality have been obtained for thermodynamics and the structure of adsorbed fluid in matrices with m = M = 8, see Figs. 8 and 9, respectively. However, at a high matrix density = 0.273) we observe that the fluid structure, in spite of qualitatively similar behavior to simulations, is described inaccurately (Fig. 10(a)). On the other hand, the fluid-matrix correlations from the theory agree better with simulations in the case m = M = S (Fig. 10(b)). Very similar conclusions have been obtained in the case of matrices made of 16 hard sphere beads. As an example, we present the distribution functions from the theory and simulation in Fig. 11. It is worth mentioning that the fluid density obtained via GCMC simulations has been used as an input for all theoretical calculations. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

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