Extremum condition

While the subnucleus holes tend to decay (IV, decreases with decreasing / < / ), the supemucleus holes can grow spontaneously (IV, decreases with increasing i > / ). For this reason, the bilayer can rupture only after the fluctuation appearance of at least one nucleus hole per unit time and, accordingly, IV is the energy barrier for bilayer rupture. From Eq. (3.109) and the extremum condition dWJdi = 0 at / = /, provided % is / independent, it follows that for A/i > 0 (i.e. for C < Ce) [399,402,403]... 

All contributions to the forces from terms involving derivatives with respect to the dipoles are zero from the extremum condition of Eq. [19]. " ... 

By comparing Equation IV.5 with the equilibrium condition expressed by Equation IV.3, we see that dG for a system equals zero at equilibrium at constant temperature and pressure. Moreover, G depends only on U, P, V, T, and S of the system. The extremum condition, dG = 0, actually occurs when G reaches a minimum at equilibrium. This useful attribute of the Gibbs free energy is strictly valid only when the overall system is at constant temperature and pressure, conditions that closely approximate those encountered in many biological situations. Thus our criterion for equilibrium shifts from a maximum of the entropy of the universe to a minimum in the Gibbs free energy of the system. 

The optimal prototypes have to satisfy the extremum condition dJ P,L)... 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

The condition for an extremum is now given by the usual condition from differential calculus that (dl/da). . = 0. To obtain the extremum condition we multiply by da and evaluate all derivatives at a = 0,... 

From the extremum condition that AAva r should be zero for arbitrary variations hP and thus for arbitrary changes Ap/l1 in the density operator, we derive the expression for the optimized single-particle Hamiltonian... 

As explained in Section 11, the partition function can be written as a onedimensional integral over the basin depth (Stillinger and Weber, 1982). In the thermodynamic hmit, the integral in Eq. (15) is dominated overwhelmingly by basin depths in the neighborhood of a particular, temperature-dependent value, which satisfies the extremum condition... 

To reach the maximum of Z, one must find the optimal concentration from the extremum condition... 

Application of a Fourier transformation to (n,, rN) produces the wavefunction (pi, , pjv) in momentum space. We have used a tilde to indicate that this Fourier-transformed wavefunction does not necessarily correspond to the optimal wavefunction (pi, , pN) within the momentum orbit The latter satisfies the extremum condition of the variational minimization of the energy functional S[tt(p) W] subject to the normalization condition / d3pir(p) = N. 

For brevity we have used the subscript notation (y) in Eq. (1.48) to indicate the partial differentials. The last equality in Eq. (1.48) emerges from the extremum conditions = 0. The second derivative we are searching for turns out to be... 

The maximal value x,na is derived from the extremum condition applied to Equation 60... 

It is seen from Figure 3.97 that the maximum A (minimum B) of the critical line coincides with the maximum (minimum) of the upper (lower) cusp point line. Thi.s relation is valid generally. The extremum condition of the cusp point line requires dx, be equal to zero along the cusp line. Calculations involving Fxjiiations 14, 15, and 31 lead to an equation symmetrical with Equation 31... 

The concentration of chain carriers eorresponding to the extremum of Hamiltonian is determined from the classical extremum condition dHjdn = 0. Then, according to (5.33)... 

The extremum condition specifying the critical state of a reaction system is... 

The eonditions dH/dn=Q and dHldP=0 eorrespond to the classical extremum condition (5.45). Then from (5.47) we obtain the system of the algebraic equations ... 

The extremum condition at a fixed number of electrons can be expressed by the Euler-Lagrange equation... 

Attempting to account for the dynamical exchange effects, neglected in the RPA, we y>glied a variational technique, which consists in deriving a functional F[/(t(p, g, w)] with the property that the equation of motion (Eq. 11-13) follows from the extremum condition ... 

So far, we use the model that assumes the fact that critical nuclei of aU phases, allowed by the phase diagram, appear at once (the unlimited nucleation model). It is known that the growth of a new phase from the nucleus is energetically favorable only in the case of nucleus size exceeding some critical value la, determined from the extremum condition of Gibbs thermodynamic potential. In a one-component substance, the extremum condition is expressed simply by the derivative of G with respect to the nucleus size being equal to zero ... 

For an ordinary fluid two-phase system, minimization of the free energy at constant temperature flrst of all results in the well-known chemical potential condition for diffusive equilibrium with respect to the soluble components present. Furthermore, the mathematical extremum condition for the free energy contains the pressure difference AP across the interface as a parameter. Solving for AP, this condition takes the form of a generalized Laplace equation. Whether or not this equation signifies a stable equilibrium is, however, often a rather complex issue where the detailed system properties may enter in a crucial manner. 

What are the thermodynamic forces, conjugate fluxes and applicable extremum conditions for processes proceeding to or from non-equilibrium stationary states What is the dissipation for these processes ... 

At the stationary state we have the necessary and sufficient extremum condition... 

However, in terms of the known periodic structure, the Bohr interpretation cannot apply. In order to generate the periodic table, it is necessary to interpret the extremum condition as satisfied by the nodal surfaces of the spherical electron wave, as shown in Fig. 2. The detailed periodic structure, together with subshell... 

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