Statistics, order

Both ZK4 and faujasite exhibit short range statistical order which can be rationalized in terms of a competition between the statistics of the step in which the sodalite cage is formed and the tendency of the system to avoid Al, Al next nearest neighbor interactions. The basic difference between the two systems can be understood in terms of the difference in the severity of the constraint imposed by Loewenstein s rule for the formation of a sodalite cage from four ordered 6R s in the case of faujasite or from six 4R s or D4R s in the case of ZK4. The local order in faujasite can be considered as "frozen in" by the topological constraint imposed by the ordered 6R sub-units. As such this order is more a property of the sub-unit than a manifestation of next nearest neighbor interactions in the faujasite crystal. By contrast the next nearest neighbor interaction in ZK4 cannot be... 

No crystallographic structure is currently available for GPV-type single-shelled dsRNA virus capsids. Two cryo-EM reconstructions have been performed, which reveal much the same architecture as the orthoreovirus core a single layer with turrets at the 5-fold vertices, statistically ordered layers of dsRNA, and a clamping protein at 2-fold axes (Hill et al, 1999 ... 

Figure 4.49. Plot of the statistical entropy 5 (per unit volume, in units of Boltzmann s constant) as a function of the statistical order parameter 5 for two representative concentrations in the range Cj < 0.1. The upper curve displays the results for nine configurations characterized by Cj = 6/256, while for the different set of nine configurations noted on the lower curve, Ct = 16/256. Several configurations noted on the upper curve are diagrammed in Figure 4.50a and several configurations noted on the lower curve are shown in Figure 4.50b the remaining configurations are specified in ref. 113.

In smectic liquid crystals, not only is there a statistical ordering parallel to the long axis of the molecules but, in addition, the molecules are ordered into layers, within which there is some freedom of motion. A schematic representation of the structure is given in Fig. 8.5(b). The type of motion possible, coupled with an orthogonal or tilted arrangement of the molecules in the layers allows a subdivision into several smectic subtypes. As the... 

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

We start with the Helmholtz integral and we use the Kirchhoff treatment, as Beckmarm and Spizzichino did [10]. Likewise, we shall make almost the same assumptions about the statistics of the radii h(cp,z) in order to find a way to deal with the integrals involved in the calculation. 

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

The leading order quantum correction to the classical free energy is always positive, is proportional to the sum of mean square forces acting on the particles and decreases with either increasing particle mass or mcreasing temperature. The next tenn in this expansion is of order This feature enables one to independently calculate the leading correction due to quanmm statistics, which is 0(h ). The result calculated in section A2.2.5.5 is... 

The first temi is the classical ideal gas temi and the next temi is the first-order quantum correction due to Femii or Bose statistics, so that one can write... 

Onsager L 1944 Orystal statistics I. A two-dimensional model with an order-disorder transition Phys. Rev. 65 117... 

Onsager L and Kaufman B 1949 Orystal statistics III. Short range order in a binary Ising lattice Phys. Rev. 65 1244... 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

Case 1. The particles are statistically distributed around the ring. Then, the number of escaping particles will be proportional both to the time interval (opening time) dt and to the total number of particles in the container. The result is a first-order rate law. 

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

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