Model clock

Models lattice gas models, Potts models, clock models, etc. 94... 

In Eq. (17), each lattice site i can be one of g different states, described by the spin variable Sj. An energy J is won only if two neighboring sites are in the same state. The symbol means a summation over all nearest neighbor pairs. In contrast, the vector Potts model (or clock model) has the Hamiltonian ... 

Basic models Ising model, Potts model, clock model, ANNNI model, etc.. 184... 

The clock model with q = 4 is also called the Z(4) model. It can be represented in terms of two Ising spins j , r, associated with each lattice site. In this form it is known as the Ashkin-Teller model (1943)... 

We now return to the clock model [eq. (147)] and mention a variant called the chiral clock model . Particularly the 3-state chiral clock (CCj) model has been studied in detail (Ostlund, 1981 Huse, 1981 Schulz, 1983 Haldane et al., 1983). Its Hamiltonian is... 

In models with several ground states (3 state Potts model, clock models etc.) a further wetting phenomenon may occur at interfaces between coexisting domains e.g., in a model with an interface between domains in states 1 and 2 the third phase may intrude in the interface (Selke, 1984 Sega et al., 1985 Dietrich, 1988). 

Edmunds, L.N. Jr. 1988. Cellular and Molecular Bases of Biological Clocks. Models and Mechanisms for Circadian Timekeeping. Springer, New York. 

Discrete models. The most advanced are discrete models that explicitly take into account the interactions between nearest neighbour (NN) layers and even next to nearest neighbour (NNN) layers. Among those approaches the most successful are Ising models [24] and the XY models, particularly, the so-called clock model [6,26]. 

In the discrete clock model [26], one operates with the c-director lying in the XY plane and the tilt plane in layer 1 is allowed to be at a discrete angle (p with... 

The Zapas Strain-Clock Model and the Bernstein-Shokooh Stress-Clock Model. Although the K-BKZ theory has been highly successful, as discussed above, it also had some flaws, which lead to attempts to modify the model by incorporating material clock or reduced time concepts. In one form, Zapas introduced a strain-clock (118), ie, a change in the time scale that depended on the strain history. While the model is fully three-dimensional, for simplicity, consider... 

Fig. 62. Normal force response in step to zero type of deformation history for a PMMA polymer glass, showing the comparison between the experimental data (filled squares), the K-BKZ model predictions (crosses), and the predictions (filled circles) from the Zapas strain-clock model (118). Note that the clock terms for the normal force response were determined by fitting the shear stress response in the same experiment. After McKenna and Zapas (112).

While the strain-clock version of the K-BKZ model seems capable of describing fairly complex nonlinear behavior, it is, at least, an inconvenient model to use. The stress-clock model of Bernstein and Shokool (155) has two features that could make it very useful. It takes less data to determine the material properties (imder certain conditions) and, in the relatively small deformation regime, it can be inverted between creep and stress relaxation. 

An interesting aspect of the Bemstein-Shokooh model is that the structure of the nonlinear equations is somewhat different from either the Schapery model or the Zapas strain-clock model. In the Bernstein-Shokooh model, the... 

Fig. 65. Half-step torque (a) and normal force (b) responses for a polycarbonate glass comparing the experimental data (open circles) with the K-BKZ equation (closed circles) and the Bernstein-Shokooh stress-clock model (155) modified to an energy clock (inverted triangles) predictions. After Pesce and McKenna (162).

Effective carbon flow from malic acid to carbohydrate with CO2 as an intermediate could be explained by the postulate that PEP-C might be temporarily suppressed as a competitor for CO2, thus interference with RuDP-C would be avoided. Both, the biological clock model and the competition model (see above) account for this postulate. However, considering the obvious limitations of these... 

Figure 9.25. Molecular arrangements obtained by a spiral or clock model.

The XY model gives helical structures with a very short pitch consisting of several layers. Because of such an ultrashort pitch, the system should appear to be optically uniaxial with the optic axis along the layer normal, and to exhibit negligible circular dichroism (CD) and optical rotatory power (ORP). However, macroscopic helices have been observed experimentally not only in SmC [74] but also in the subphases. Actually, Miyachi et al. [75], [76] observed laser light diffraction in the SmCy and AF phases of (R)-MHPBC. In this respect, the XY model, particularly the clock model, is not a realistic model for various subphases possessing macroscopic helices. Future studies are necessary to fully understand the appearance of various subphases. 

As for the molecular orientation structures of the SmCy and AF phases discussed on p. 273, recent detailed ellipsometry measurements succeeded in modeling the distorted structures that are distorted from the Ising model and clock model (P.M. Johnson, D.A. Olson, S. Pankratz, T. Nguyen, J. Goodby, M. Hird, and C.C. Huang, Phys. Rev. Lett. 84, 4870 (2000)) and are essentially the same as a distorted Ising model [76]. 

The natures of the different subphases and their main characteristics are still the subject of active research. Some models state that in the subsequent layers the polarization is always either parallel or antiparallel, and the macroscopic behavior depends on the number of repeating units, which should be more than of two layers range. For example, in the so-called SmCp phase, the zero field polarization is 2/3 of the saturated vale, which in frame of the orthogonal model would mean interaction of six layers in five consecutive layers in one direction, and in the sixth the opposite ((5 - l)/(5 + 1) = 2/3). This long-range interaction seems to be very unlikely. For this reason, the so-called "clock model" was suggested. This assumes only interactions of two layers, but does not require that the polarization directions in the neighbor layers be either parallel or antiparallel. This seems to be supported by... 

Figure 24.7 Simulation of circadian clock model for varying values of [1.0 (solid), 1.1 (dashed), 1.5 (dash-dot), 2.0 (dotted), 4.0 (asterisk)].

Figure 24.8 Simulation of circadian clock model for entraining signal with period of 20 h.

The need to make modifications to these models comes out of the result of resonance X-ray diffraction experiments. In the first report, the SmCFi2 phase was concluded to be a four-layer structure and the SmCpn phase a three-layer structure [65]. This conclusion was similar to ours, but unlike our Ising model, the authors interpreted the data as a XY clock model [65]. However, characteristics such as the optical rotation cannot be explained by this, and we have proposed a deformed Ising model shown in Fig. 9.8 which fuses the XY and Ising models [66]. Further resonant X-ray diffraction experiments and optical analysis led to the current deformed clock model [67], which was equivalent to our deformed Ising model [68]. 

There are also several conventions to name the ferroelectric phases. When these phase stmctures are not exactly known, the names have Greek subscripts, a, p, or y. The SmC phase is left as it is, but the ferroelectric phases have commrMily FIl and FI2 as subscript. They were proposed around the time when the clock model appeared, but there is a sense of incongruity because SmCpE is basically... 

The Purpose, the Main Proposition and the Limitations of the Schumpeter Clock Model... 

The Schumpeter Clock model [5.13] stresses the existent, explicitly active pushing micro-economic forces and powerful supply side checks and balances in explaining the short term non-equilibrium motions of an economy, whereas other model designs primarily take into account the less definite macro-economic forces and weak demand side checks and balances. 

Thus, changes in the industrial economy can be formulated as equations of motion of these two components of investment behaviour. The equation of motion for the investors configuration and the equation of motion for the investors propensities, as presented in the next two sections, constitute essential parts in the design of the Schumpeter Clock model being presented. 

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