Local order theory

C2/C1 decreased with interchain distance (expressed as cross-sectional area per unit chain), appears consistent with both an entanglement theory and a local order theory. In the paper, arguments for both theories are critically examined. The suggestion given above that the relevant modulus should be C2/A would favour the entanglement theory rather than the local order theory. The reason is that an increase in A would decrease C2/A but would be expected to increase local order. 

An important hierarchy of parameterization schemes based on successive refinements to a mean-field theoretic description, called local structure theory, has been developed by Gutowitz, et. al. ([guto87a], [guto87b], and [guto88]), and is discussed in detail in chapter five. Below we summarize results of what is essentially a zeroth order local structure theory developed by Langton [lang90]. 

While there is, at present, no known CA analogue of a Froebenius-Perron construction, a systematic n -order approximation to the invariant probability distributions for CA systems is readily obtainable from the local structure theory (LST), developed by Gutowitz, et.al. [guto87a] LST is discussed in some detail in section 5.3. 

Ip = hG/c y is the Planck length. Since c6t < 6x, A and B are outside of each other s light cone and local field theory assures us that these two experiments can be performed completely independently of one another. Heisenberg s uncertainty principle, however, asserts that these two measurements will also yield an energy fluctuation on the order of AE > Ip. We know that the gravitational... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

It would be important to find analogous mechanism also for description of the main (librational) absorption band in water. After that it would be interesting to calculate for such molecular structures the spectral junction complex dielectric permittivity in terms of the ACF method. If this attempt will be successful, a new level of a nonheuristic molecular modeling of water and, generally, of aqueous media could be accomplished. We hope to convincingly demonstrate in the future that even a drastically simplified local-order structure of water could constitute a basis for a satisfactory description of the wideband spectra of water in terms of an analytical theory. 

The liquid state is intermediate between the gaseous and solid states. It presents neither the structural regularity of solid crystals nor the typical disorder of gases. For that reason, theoretical studies of the liquid phase are based on the methods and results of the research of real gases (van der Waals) but also on the theory of disordered solids, like the results of X-ray experiments which show local ordering in liquids. 

Kirkwood took a more rigorous statistical-mechanical approach in an attempt to incorporate the effect of local ordering. His theory is only valid for rigid dipoles, and it was left to Frohlich to extend the treatment properly to a system of polarisable dipolar molecules. The work is well described in Frohlich s classic text (1949). The final outcome was the following formula,... 

The data plots of Fig. 15b (silica) are differentiated for the use of methyl- -butyl ether (MTBE) or acetonitrile (ACN) as localizing solvent C in the mobile phase. It is seen that for some solute pairs (Fig. 15a and c) the open squares (MTBE) fall on a different curve than the closed squares (ACN). This implies that the constant in Eq. (31a) is solvent-specific, rather than being constant for all solvents (as first-order theory would predict). A similar behavior is observed for alumina as well. Figure 15a plots data for 18 different polar solvents B or C, and some scatter of these plots of log a versus m is observed here, as in Fig. 15b for silica. The variation of Q with the localizing solvent C used for the mobile phase has been shown (18) to correlate with the relative basicity of the solvent, or its placement in the solvent classification scheme of Refs. 40 and 41. Thus, for relatively less basic solvents (groups VI or VII in Refs. 40 and 4/),... 

K. S. Schweizer and J. G. Curro, Macromolecules, 21, 3070 (1988). Integral Equation Theory of Polymer Melts Intramolecular Structure, Local Order, and the Correlation Hole. 

In the next few pages we shall discuss the question of local interfacial structures bounding idealised aggregates, tiled by blocks of fixed dimensions. The model represents one extreme idealisation of the molecular constituents that form the aggregate, most applicable to small surfactant molecules. At the other extreme, the block dimensions are not set a priori, they must be determined as a function of the temperature, concentration, etc. This case will be dealt with later. The welding of two concepts, a fluid-like mixture of hydrocarbons, with that of an idealised block is at first sight contradictory. However it can be shown to be consistent in a first order theory [2]. 

One way of conceptualizing a liquid is as a very disordered solid. If one disrupts the structure of the nearest neighbors around a reference molecule in a molecular crystal, the effect of the disruption extends quite far. As a result, there is some local order around the reference point but the extent of order falls off rapidly with distance so that at distances equivalent to four or five molecular diameters the system does not possess order with respect to the reference point. Theories of the liquid state based on an approach involving disordered solids were pursued from the 1930s to the 1960s but did not meet with much success. On the other hand, the liquid may be regarded as an extremely imperfect gas. In this approach, which has been quite successful, the statistical mechanical techniques used to describe the properties of non-ideal gases are extended to liquids. 

Etzler (1983) has proposed a first-order theory of vicinal water based on the percolation theory treatment of bulk water developed by Stanley and Teixeira (1980a,b). Etzler assumes two distinct types of H-bond connectivity, namely, a perfect four-connected set of water molecules and the remainder, with three, two or one (or no) H-bonds. Based on the idea that two distinct populations of water molecule environments exist, Etzler calculated the increase in the amount of ice-like (four-coordinated) local environments and proposed that the tendency to create such more nearly tetrahedral arrangements is somehow induced by proximity to the surface. The vicinal water thus represents an enhancement of the ice-like structure. Etzler then calculated the density of the water in silica pores. This was... 

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically [36-38]. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as... 

The sign of C(t) can now be adjusted such that the vibrational energy in the ground state increases (or decreases) monotonically, while the sum of the populations on the ground and first excited states is kept constant. As in the general formulation of local control theory, in order to avoid unphysical divergences, one choice for C(t) is... 

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