Scalar variance definition

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

Note that from its definition, the scalar spatial correlation function is related to the scalar variance by... 

By definition, the scalar variance can be found directly from the scalar energy spectrum by integrating over wavenumber space ... 

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

Typical chemical concepts are not as sharp as typical physical concepts. A nice example is that of atomic charge densities which were analyzed [21] in terms of factor analysis and were found to be scalar quantities (different possible definitions do not lead to the same numerical values, but these correlate satisfactorily), at variance with aromaticity [22, 23] which turned out to be a multidimensional half-ordered concept [23]. Primas [17] suggested that molecular structure is a genuine chemical concept, which has no counterpart in rigorous quantum mechanics. On the other hand, situations where molecular structures become undefined do show up in chemistry, or at least in molecular spectroscopy. Quantum phenomena apparently have a stronger tendency to survive in chemistry than in mac-... 

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