Definition of an Order Parameter

In order to better understand the definition of an order parameter S characterizing the degree of anisotropy of an otherwise fast motion, we turn to the simpler case of heteronuclear dipole-dipole coupling, where the interaction tensor is s)un-metric and depends only on the polar angle 0. The spectral frequencies are... 

Definition of the order parameter. The first step in the TPS algorithm is to define an order parameter, i.e. a variable that describes whether the system is in the reactants, products or in an intermediate region, as shown schematically in Fig. 21. The pyruvate and lactate regions were defined by the values of the appropriate bond... 

An anlysis of this question meets complications since in a purely one-dimensional substance the fluctuations exclude the existence of an order parameter (i.e., a phase transition) at finite temperatures. The fluctuations can be suppressed by means of the intercahin hopping but the impurities may prevent this process. The study of the influence of a smooth random potential (the potential (l) in our notations) in the work ] shows that what concerns fluctuations both types of transitions are suppressed by impurities, i.e., there is no gain on this way. One must have in mind however, that the fluctuation are imposed on the purely one-dimensional pairing and the absence of the latter means definitely no pairing even with hopping. 

This is called the saddle point approximation, and the resulting theory is equivalent to the mean-field theory. In practice, the coarse grained states are defined in terms of an order parameter that by definition has a value 0 in the disordered phase and a nonzero value in the ordered state. In the case of AB diblock copolymers, this order parameter is defined as the deviation of the A monomer fraction from the average value. Since this order parameter is a function of position, it actually represents infinitely many variables. By definition, in the weak segregation regime, the order parameter is small and it is meaningful to expand Fi in a Taylor series. 

The molecular statistical definition of the order parameter can be clarified if one considers, for example, the magnetic susceptibility of the nematic phase. The anisotropic part of the susceptibility tensor Xap be considered as an order parameter [3] because it vanishes in the isotropic phase and is nonzero in the nematic phase. In addition, the macroscopic magnetic susceptibility can be written as a sum of contributions from individual molecules ... 

Since the total concentration a + r + s follows the time evolution d(a + r + s)/ dt = F - k(a + r + s), it approaches the steady state value F/k with a relaxation time 1 /k. This is a consequence of unbiased outflow (Eq. 47) of all reactants with the same rate k. Consequently, even though we are dealing with an open system under a flow, the analysis is similar to the closed system by replacing the total concentration c with the steady state value F/k. Instead of recycling, therefore, constant supply of the substrate allows the system to reach a certain fixed point with a definite value of the order parameter 0i, independent of the initial condition. 

This work is a contribution to the definition of an experimental protocol which aims in identifying the parameters involved in a description of crazing within a cohesive surface methodology. The results obtained for PMMA are valuable for the calibration to perform in connection to the numerical work of Estevez et al. [2]. The method of preparation needs to be improved for more ductile material in order to characterize the failure by crazing only. 

The abscissas of the correlation diagrams can be quantified. The quantification is thus far only an ex post facto device with power to classify but not yet to predict. It does not yet have a microscopic interpretation. Nevertheless it satisfies the needs of a theory of the equilibrium behavior of melting and freezing of clusters. Only the existence and not the definition of the nonrigidity parameter or the details of its origins is used in stage 2. However, we make a brief aside here to explain the definition in order to clarify just what information it carries. To understand the order parameter y (not to be confused with the surface tension), it is useful to examine how nonrigidity is traditionally characterized in diatomic and linear, particularly triatomic molecules. For... 

The normalization constant G is often conveniently defined by setting Qzz equal to unity in perfectly oriented system. By definition, Q is real, symmetric, and traceless. Q vanishes in the isotropic phase as per the requirement of its suitability for an order parameter to describe the I-N transition. The macroscopic tensor order parameter Q can always be diagonalized ... 

Suitable definitions of regions A and B may require considerable trial and error. Fortunately, it is straightforward to diagnose an unsuccessful order parameter. For example, most short trajectories initiated from the state Xq will quickly visit states with values of q characteristic of state B. In other words, the probability to relax into B is close to one. (This relaxation probability plays an important role in the analysis of transition pathways, as will be discussed in detail in Section V.) In contrast, the probability to relax into B from is negligible. When relaxation probabilities indicate that definitions of A and B do not exclude nonreactive trajectories, the nature and/or ranges of the order parameter must be refined. 

Conditions for Biiayer Formation. In order to obtain bilayers or vesicles for an amphiphile of length F, one needs to adjust the interfacial area imtil the surfactant parameter A approaches unity. With the optimum head group area ao from equation 2 and the definition of the packing parameter (eq. 4), one obtains a simple expression that can serve to rationalize the influence of different parameters on the packing parameter... 

In order to predict polymer-polymer miseibility, we might turn to the Flory-Fluggins theory, where eaeh lattiee site has an interaeting segment volume v. Dividing both sides of Eq. (9.3.20) by the total mixture volume Fand using the definition of the interaetion parameter given by Eq. (9.3.18) yields... 

Minimum exposure times must be observed in order to reach the requisite S/N ratio. As per EN 1435 and EN 584-1, for the different ranges of utilization (energy, wall thickness), definite film elasses are prescribed. They are characterized by the minimum gradient-to-noise ratios. Based on this, one can calculate the minimum values for the S/N ratio based on the IP systems. The exposure time and the device parameter sensitivity and dynamics (latitude) must be adjusted accordingly, with an availability of an at least 12 bit system for the digitalization. 

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