Building construction statistics

Building costs per square meter for the construction of new buildings can readily be obtained from reference books such as Span s Architects and Builders Price Book and Paxton s Building Price Book. Both are updated and issued on an annual basis, with the costs quoted based on the previous year s statistics. 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

There is much to be admired in the analyses presented here. In my opinion, this serves as a shining example of many different elements in the quest to build effective theories of material response. First, it is to be noted again that the key structural features of these complex materials have been mapped onto a corresponding two-dimensional model, itself already an intriguing and useful idea. In addition, the construction of the cluster expansion illustrates the way in which first-principles calculations for a relatively complex material can be used to inform a more tractable (and enlightening) model. Finally, the tools of statistical mechanics can be used to explore the phase diagrams of these systems which helps explain observed features of these problems but also suggests further complexity as can be seen in fig. 3 of de Fontaine et al. (1990). 

Certain aspects of permanent gels are comparable if they are analyzed on the basis of the cluster structure of the system. A cluster is a particularly constructed, inhomogeneously linked, multichain macromolecule of a size generally similar to that of most macromolecules studied isolated in dilute solution. As a rule, the cluster is internally very heterogeneous in structure and composition, but, overall, the cluster is the statistically similar repeating unit that builds up the gel. In swelling equilibrium, all parts of a cluster, but its periphery in particular, are in equilibrium with an excess of pure solvent medium. 

To summarize, we have constructed state vectors that obey Fermi-Dirac statistics through introducing creation and annihilation operators that fulfill the anticommutation relations of Eqs. (1.2), (1.4), and (1.5). The anticommutation relations allow us to build the Slater-Condon rules directly into the operators in the second-quantized language. The operators thereby lose their dependence on the electron number N. The only dependence on N in the second-quantized language appears in the state vectors 0>. In contrast, in the first-quantized language the dependence on N appears in both the operators and the wavefunctions. 

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