Action principle

Laplace s equation, 146 Least action, principle of, 69, 304 Line of heterogeneous states, 172 Liquefaction of gases, 167, 173 of mixtures, 428... 

The form of the action principle given above was first applied to quantum mechanics to describe the time evolution of pure states (i.e. 

The equations of motion for the nuclei are obtained from Hamilton s least action principle. The nuclei total kinetic energy, K, is given by the sum of individual nucleus kinetic energy, (l/2)Mk(dXk/dt)2. The time integral of the Lagrangian L(X,dX /dt,t) = K-V is the action S of the system. For different paths (X=X(t)) the action has different numerical values. 

Further analysis from the minimum action principle shows that the exchange (xc) potential is then the functional derivative of that quantity in terms of the density ... 

To make quantitative predictions of DDI for the new compound as perpetrator, a reliable estimate of a relevant in vivo concentration is needed. What is tmly needed is knowledge of the concentration of the inhibitor available to bind to the enzyme. For liver, if the well accepted free-dmg hypothesis (which underwrites fundamental drug action principles in pharmacology) is applied for DDI, then the use of a free intracellular liver concentration is needed. For inhibitors that are permeable through membranes, the free concentration in the portal vein should serve as the closest proxy for free intracellular concentration in the liver. Diminished permeability as well as active uptake and efflux from liver cells can confound this relationship. Nevertheless, use of estimates of unbound portal vein concentrations (which can be estimated from... 

Van Boxtel CJ, Holford NHG, Danhof M, editors. The in vivo study of drug action. Principles and applications of kinetic-dynamic modelling. Amsterdam Elsevier 1992. 

In formulating QED a least-action principle involving a Lagrangian is often used [9,18,20]. This involves the potentials in various forms. Not only is relativistic invariance (Lorenz potentials) desired, but also gauge invariance. At least in the current state of QED, gauge invariance is included as a fundamental part [21,22]. 

Christoph, T. and Buschmann, H. Zwei komplexe Wirkprinzipien in einer Struktur - Gemischte opioide Agonisten/Antagonisten und partielle Agonisten (Two complex action principles in one structure -mixed opioid agonists/antagonists and partial agonists), Pharm. i. u. Zeit 2002, 31, 40-43. 

The bolt-action principle was rapidly adopted by many nations at the end of the 19th century. Outstanding military weapons of this type are the Mauser and Springfield rifles, brought to a high state of perfection in the early years of the 20th century and since used thruout the world... 

Generic Name Trade Name(s) Onset of Action Duration of Action Principle Use(s)... 

Most gases dissolve monatomically in liquid metals. For example, the solution reactions may be written as H2 -> 2H, N2 —> 2N, O2 —> 20, where the underlining signifies the element is in solution. Sievert s law for diatomic gases is an application of the mass action principle. It states that the solubility is proportional to the square root of the partial pressure of the gas. For example,... 

Euler s proof of the least action principle for a single particle (mass point in motion) was extended by Lagrange (c. 1760) to the general case of mutually interacting particles, appropriate to celestial mechanics. In Lagrange s derivation [436], action along a system path from initial coordinates P to final coordinates Q is defined by... 

Thus, for a toxicant (ligand) interacting with a single (or homogeneous population) of receptor(s), mass-action principles require ... 

The relevant kinetic model for competition experiments with a radiolabeled drug [D] and an unlabeled competitor [I] is shown in the two equations in (19.15). When both sets of reactions have proceeded to equilibrium, the net rate of formation of both (DR) and (IR) are zero, and the following Eq. (19.16) can be derived from mass-action principles. 

Flow measurements using nonintrusive or low mechanical action principles are desired, such as magnetic, vortex-shedding, or Coriolis-type flowmeters. Orifice plates are easy to use and reliable but have a limited range and may not be suitable for streams which are not totally clean. Rotameters with glass tubes should not be used. 

Once again, this 1-1 correspondence is valid for a fixed initial many-body state F(ri,. ..fjv, R/i to)- Besides this HK-type statement, one can derive a stationary-action principle and a set of coupled TDKS equations for electrons and nuclei. The latter read as follows ... 

Equation (225) was first obtained by Vignale [107] from invariance properties of the xc action functional aj/xc defined in Eq. (43). The derivation given here [108, 109] has the advantage of being independent of the stationary action principle. 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

At the risk of being redundant, we may state here the salient features of the TD-functional formalism. The first requirement is a variational principle, and for a time-dependent quantum description only a stationary action principle is available. With this a mapping theorem is established which turns the action functional into a functional of relevant physical quantities (which are the expectation values), and the condition of stationarity is now in terms of these variables instead of the entire density matrix. Thus the stationary property with respect to the density matrix now becomes one with respect to all the variables... 

The time derivative operator in the H-space now becomes a superoperator, d in view of Eq. (31b), defined in the sense shown on the left-hand side of Eq. (32). This was missed in earlier work [21] even though its existence was surmised. It is here found to be crucial in the development of the stationary action principle in the L-space. When the system is considered to be in a pure state, this gives a transparent reduction to the pure-state TD-functional theory. We may then state the usual normalization condition on the density matrix in the form of a matrix element in the superspace ... 

Liouvillean space the expectation values become simple matrix elements, no longer a trace, and may even be viewed as thermal vacuum expectation values with respect to the given initial thermodynamic state. We can now restate Eq. (32) as a stationary action principle in this superspace ... 

In Eq. (38), the first term is the well-known free energy expression in the usual //-space. The variation of Q with respect to D,. 0) yields the equilibrium state and the value of Q at the minimum is the free energy of the system given initially. Therefore, a functional theory holds for this equilibrium state also and thus the two principles together lead to a new procedure which maintains self-consistency at all levels and treats in tandem the stationary action principle and its initial state specification on equal footing. 

Dirac-Frenkel stationary action principle involving the functional W determines the exact system density matrix. 

We have also pointed out here the formal connection between our formalism and the existing numerical algorithms in special cases (CP-algorithm and time-dependent optimized potential) as well as avenues to go beyond these to include non-adiabatic processes. It should be stressed that unlike the existing theories, our framework is based on a stationary action principle, which facilitates incorporation of the initial constraint of thermodynamic equilibrium. This development is made feasible by working in a superspace formalism. This work thus provides a practical theoretical framework for studying the non-equilibrium statistical mechanics of systems initially in thermodynamic equilibrium. 

In this appendix, we establish the important result that the TD stationary action principle used in the derivation of the TD-functional theory [Eq. (54)] is the same as the stationary principle for the effective action functional of Baym [Bq. (89)]. Thus the action functionals appearing in the two formulations are different representations of the same quantum action leading to two different optimization strategies. We give a version of the work of Jackiw and Kerman [54] (hereafter referred to JK) on this subject, adapted here to LQD. Consequences of this important result in the development of the functional theory presented here are given. 

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