Distribution variables, mixing description

After a brief summary of the molecular and MO-communication systems and their entropy/information descriptors in OCT (Section 2) the mutually decoupled, localized chemical bonds in simple hydrides will be qualitatively examined in Section 3, in order to establish the input probability requirements, which properly account for the nonbonding status of the lone-pair electrons and the mutually decoupled (noncommunicating, closed) character of these localized a bonds. It will be argued that each such subsystem defines the separate (externally closed) communication channel, which requires the individual, unity-normalized probability distribution of the input signal. This calls for the variable-input revision of the original and fixed-input formulation of OCT, which will be presented in Section 4. This extension will be shown to be capable of the continuous description of the orbital(s) decoupling limit, when AO subspace does not mix with (exhibit no communications with) the remaining basis functions. 

Admittedly, these words are a mouthful. But fortunately, they are very intuitive and descriptive. CH refers to the differences in the constitution, or makeup, of the material how alike or different the individual particles or molecules are. DH refers to differences in how the material is distributed how well mixed or segregated the material is due to density, particle size, or other factors. Each of these two types of heterogeneity gives rise to a sampling error. Together they determine how variable our samples can be and how easy or hard it is to get consistently representative samples. Because an understanding and assessment of these two types of heterogeneity are important, we need to examine them in detail. 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

For the sake of clarity, the mathematical description (Equations 14.1 through 14.11) assumes that the systan behavior is defined in terms of DAEs, in a semiexplicit form, where y(t) can be implicitly eliminated through the algebraic variables ft. On the other hand, mixed lumped and distributed systems, described by general integral, partial differential, and algebraic equations in time and one or more space... 

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