Global variational principle

We start our analysis from the co-ordinate independent (global) variation principle... 

Global Variational Principle The Concept of Local-Scaling Self-Consistent Field... 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

It has been amply demonstrated elsewhere [10-15] that many classical problems of theoretical chemistry can be approached afresh using this novel IT perspective. For example, displacements in information distribution in molecules relative to the promolecular reference consisting of nonbonded constituent atoms have been investigated [10-18], and the least biased partition of molecular electron distributions into subsystem contributions (e.g., densities of AIM) has been investigated as well [10,19-26], The IT approach has been shown to lead to the stockholder molecular fragments of Hirshfeld [27], These optimum density pieces have been derived from alternative global and local variational principles of IT. 

Total energy calculations of the quantum mechanical ground state have advanced significantly in recent years, and have been applied to an ever-increasing number of different systems and physical properties. The core of most of this work is the density-functional theory of Hohenberg, Kohn and Sham, and in particular the local-density approximation (see, e.g., Lundqvist and March, 1983). This theory is based on the variational principle and applies to the ground-state of a quantum system. It is, however, not restricted to the state of globally lowest... 

Show that from the minimum principle of the global entropy production (which as we seen, is a restricted variation principle) can be constructed by Lagrange s method of multipliers in the following free variational-task ... 

The variational principles used are either local minimum or absolute(global) minimum principles. Absolute minimum principles may provide more robust minimization algorithms. 

Another disadvantage of direct discrete methods is their requirement for the assignment of many weights. When the discrete variational form arises from a continuum variational principle most of the weights (except those controlling the global balance of the major terms of the functional such as total area or curvature) are deduced via the discretisation. 

All scientists agree that chemistry is the basis of life and with that preamble it seems almost certain that the first cells one sees in the Hadean stones are the first ones on earth their ancestors were the heat- and light-driven bio-reactors. The Darwinians see the same cells but because the (in principle unprovable) single origin is a creed of the model, they postulate that a single ancestor must have lived much earlier and mutated into all of the cells that are visible at this horizon. At this point the old hypothesis is defeated by the researchers in planetary sciences who do not see any biology-time before the time of the first massive invasion of the earth crust by cells. The hell fire of global accretion was too close for descent with variation . 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

Global and local functions depend upon a very large and generally indefinite number of parameters which may be quantifiable or not and which may account in principle for anything. The study of the evolution of the dynamical system upon variations of its control space parameters intends to answer two questions ... 

---