Microscopic selforganization

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

Instead of disintegration of the initial state we find that the law in Eq.(23) leads to increasing (self)-organization during a finite number of life times x. It is in this context that we will speak of microscopic selforganization in what follows below. As we need a more detailed description we will proceed to discuss the density operator concept in general and the reduced density matrices in particular. 

This section is devoted to biological order, organization, and evolution. We have already seen in Appendix F that constructive integration of quantum and thermal correlations under appropriate conditions lead to a so-called CDS, i.e., an optimal spatiotemporal structure formed by the precise relations between time, size, and temperature scales. CDS suggests microscopic selforganization including Godel-like self-referential traits. 

E. Brandas, Complex Symmetry, Jordan Blocks and Microscopic Selforganization An Examination of the Limits of Quantum Theory Based on Nonself-adjoint Extensions with Illustrations from Chemistry and Physics, in N. Russo, V. Ya. Antonchenko, E. Kryachko (Eds.), Self-Organization of Molecular Systems From Molecules and Clusters to Nanotubes and Proteins, NATO Science for Peace and Security Series A Chemistry and Biology, Springer Science+Business Media B.V., Dordrecht, 2009, p. 49. 

E. Brandas, Dissipative systems and microscopic selforganization, Adv. Quant. Chem., 2002, 41, 121. 

Fig. 4. Representative investigations of molecular state properties, microscopic reaction pathways, and selforganization phenomena of silicon compounds by the Frankfurt Group since 1966 (cf. Summary) [ I ].

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