Order parameter partial

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

Although the underlying physics and mathematics used to convert relaxation rates into molecular motions are rather complex (Lipari and Szabo, 1982), the most important parameter obtained from such analyses, the order parameter. S 2, has a simple interpretation. In approximate terms, it corresponds to the fraction of motion experienced by a bond vector that arises from slow rotation as a rigid body of roughly the size of the macromolecule. Thus, in the interior of folded proteins, S2 for Hn bonds is always close to 1.0. In very flexible loops, on the other hand, it may drop as low as 0.6 because subnanosecond motions partially randomize the bond vector before it rotates as a rigid body. 

To calculate d4/d , we need to evaluate partial derivatives, such as U->4/7) , which measures the rate of change in energy with the order parameter. To do so we need to define generalized coordinates of the form ( , qi, , qN-1). Classical examples are spherical coordinates (r, 6, o), cylindrical coordinates (r, 0, z) or polar coordinates in 2D. Those coordinates are necessary to form a full set that determines... 

The composition or the number fraction of component B, Xb, is an example of an order parameter that is conserved in a closed system. Figure 17.6 shows a molar free energy versus composition curve for a binary solution. The molar free energy for a solution at any composition Xb can be written in terms of its partial molar quantities, Fa(Xb) and Fb(Xb) 4... 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

The use of an electric field is not the only effective way to influence the LC polymer structure, magnetic fields displays a closely similar effect167 168). It is interesting as a method allowing to orient LC polymers, as well as from the viewpoint of determining some parameters, such as the order parameter, values of magnetic susceptibility, rotational viscosity and others. Some relationships established for LC polymer 1 (Table 15), its blends with low-molecular liquid crystals and partially deuterated polyacrylate (polymer 4, Table 15) specially synthesized for NMR studies can be summarized as follows ... 

The reorientation of the local chain axis gives rise to the partial narrowing of the amorphous REV-8 spectra and is of a random nature. In a short period of time, a particular segment of the macromolecular chain assumes a distribution of directions which deviate from its orientation at rest. In analogy to the description of molecular ordering in liquid crystals—, we use the concept of a local order parameter for the quantitative characterization of the extent of these fluctuations. 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

In some liquid crystal systems, broad, partially resolved spectra with some structure attributed to proton dipole-dipole interaction have been observed (16, 17, 18). On the other hand, such spectra were not observed in this study, so that the ordering parameters of the samples employed would be expected to be very small. This result is compatible with that of the ESK study reported by Y. Yamada et al. (19). 

Malcherek T, Salje EKH, Kroll H (1997) A phenomenological approach to ordering kinetics for partially conserved order parameters. J Phys Cond Matter 9 8075-8084 Matsui M, Todo S, Chikazumi S (1977) Magnetization of the low temperature phase of Fe304. J Phys Soc Japan 43 47-52... 

Linear first order hyperbolic partial differential equations are solved using Laplace transform techniques in this section. Hyperbolic partial differential equations are first order in the time variable and first order in the spatial variable. The method involves applying Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes an initial value problem (IVP) in the spatial direction with s, the Laplace variable, as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 2.1 for solving linear initial value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). This is best illustrated with the following example. 

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