Klein density distribution

Klein, C. L., Stevens, E. D., Zacharias, D. E., and dusker, J. P. 7,12-dimethylbenz[ a ]anthracene refined structure, electron density distribution and endo-peroxide structure. Carcinogenesis 8, 5-18 (1987). 

Because of the unphysical feature of the Klein-Gordon density and the fact that spin does not emerge naturally (but would have to be included a posteriori as in the nonrelativistic framework) we are not able to deduce a fundamental relativistic quantum mechanical equation of motion for a freely moving electron. However, we may wonder which results of this section may be of importance for the derivation of such an equation of motion for the electron. Certainly, we would like to recover the plane wave solutions of Eq. (5.8) for the freely moving particle, but in order to introduce only a single integration constant (or the choice of a single initial value) for a positive definite density distribution we need to focus on first-order differential equations in time. These must also he first-order differential equations in space for the sake of Lorentz co-variance. 

Figure 9.2 Quantification of risk for environmental contaminants using probability density functions. The probability of impacts corresponds to the extent of overlap of the estimated environmental concentration (left curve) and the effective toxic concentration (right curve). The risk can be quantified from the distance between the mean values of the concentrations and the variance in the respective concentrations (due to measurement variability and/or extrapolation errors) represented by the width of the distributions. Reproduced from Nendza, Volmer and Klein (1990) with kind permission from Kluwer Academic Publishers, Dordrecht.

Another method is to try and estimate the composition of the reactors based only on bulk property information. This bulk property information typically refers to routinely measured properties such density, distillation curves, etc. Klein and co-workers [29] have used a much more sophisticated version of this approach to probabilistically sample candidate molecules and generate a very large list of molecules whose combined properties match the measured bulk properties. Hu et al. [24] use a probabibty distribution method to estimate to the PN A compositions for their approach towards refinery reactor modeling. The approach we describe is similar, but much simpler to use since it is targeted only for reformer feeds. 

---