Variational property

TNC.9. P. Glansdorf, I. Prigogine, and D. E. Hays, Variational properties of a viscous liquid at a nonuniform temperature, Phys. Fluids 5, 144-149 (1962). 

TNC. 14.1. Prigogine and P. Glansdorff, Variational properties and fluctuation theory, Physica 31, 1242-1256 (1965). 

Progogine I., Evaluation Criteria, Variational Properties and Fluctuations, Non-equlibrium Thermodynamics Variational Techniques and Stability, Ed. R. J. Donnely, R. Elerman, I. Prigogine, The University of Chicago Press, Chicago, 1966... 

The quantized results of Eqs. (2.8) and (2.10) negate, at least in principle, the variational property of classical RRKM theory, since both the one-dimensional tuimeling probabilities and the step function may exceed the true reactivities. In practice, however, variational minimizations still prove useful. 

The Lagrangian-based functional [i, 2] or 2] derives directly from the Lagrangian as employed in the quantum action principle. For a total system, both the Lagrangian- and Hamiltonian-based functionals yield identical variational results. This equivalence in variational behaviour is maintained for the corresponding subsystem functionals only if the subsystem is bounded by a zero-flux surface. Only an atomic region ensures an equivalence in both the values and the variational properties of the two types of functionals (eqns (E5.6a,b)) thereby preserving the properties obtained for a total system. 

Thus, the atomic Lagrangian and action integrals in the presence of an electromagnetic field, like their field-free counterparts, vanish as a consequence of the zero-flux surface condition (eqn (8.109)). These properties are common to the corresponding integrals for the total system and it is a consequence of this equivalence in properties that the action integrals for the total system and each of the atoms which comprise it have similar variational properties. 

Knapp, S. and Mandell, A. J., Mouse midbrain tryptophan hydroxylase Strain differences in variational properties, /. Physiol., 77, 281, 1981. 

We shall see that the introduction of two collective modes suffices to account for the so-called dynamic friction effects. In other words, the dynamic friction is in fact not dynamic at all but is accounted for in terms of a free-energy surface. To reach this goal we still lack a number of elements. The transformation coefficients have yet to be defined and one needs to solve for the optimal dividing surface. Both these goals are attained by making explicit use of the variational property of transition state theory. 

In view of the variational property of transition state theory it is evident that within the parabolic barrier estimate for the rate, the optimal transformation coefficients are those that minimize the parabolic barrier frequency WPP of the free-energy surface, subject to the constraint that the coefficients a, are normalized. 

After some algebra and using the variational property of the density fitting the following equation is derived ... 

The expression in Eq. 6, however, does not have the variational property given by Eq. 2 unless n is the exact fully self-consistent charge density. This is because the sum of electron eigenvalues, the first term on the right of Eq. 6, contains information on both the input charge and the output charge of a particular iterative cycle in the calculation. Thus, E otal a functional of... 

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