Equation local

Kinetic Equations, Local Potential Methods Used in the Study of... 

In an attempt to place calculations of liquid-phase activity coefficients on a simpler, yet more theoretical basis, Abrams and Prausnitz used statistical mechanics to derive a new expression for excess free energy. Their model, called UNIQUAC (universal qua si-chemical), generalizes a previous analysis by Guggenheim and extends it to mixtures of molecules that differ appreciably in size and shape. As in the Wilson and NRTL equations, local concentrations are used. However, rather than local volume fractions or local mole fractions, UNIQUAC uses the local area fraction 0,j as the primary concentration variable. 

In more general constitutive models the Gibbs equations (local equilibrium) are not valid and therefore explicit calculations of entropy are impossible. This seems to correspond to the nonuniqueness of entropy or to irreversibility of processes between nonequilibrium states [see below (1.37) and Rem. 20]. Such are some constitutive models in Sects. 2.1-2.3, but in particular models with long range memory [17, 23, 48]. Even the usefulness of entropy in situations far from equilibrium [11, 101, 114-120] seems questionable, the entropy inequality deduced and used in... 

In the E and the lower F regions of the ionosphere, photochemical processes dominate, and the electron density canbe calculated simply by equating local production aiKl loss rates,... 

At low altitudes, below about 250 km, the electron temperature can be obtained simply by equating local heating and cooling rates. However, above this altitude, electron thermal conduction is important, and the equation governing the electron temperature becomes more complicated (see Schunk and Nagy, 1978, 2000). 

From the values shown in Table 1, the following conclusions of general validity can be drawn. The FT equation (localized, with interactions) can be substituted with a Toth equation (localized, no interactions) applied to a surface with less heterogeneity (ij- < and with a lower value of jydiff(0 those corresponding to the FT relationship. It means that the energies of... 

Adsorption without lateral interactions is particularly interesting because the topographic structure of the surface does not affect the integral equation. Localized adsorption without lateral interactions is even more interesting because the Langmuir isotherm (19) has the following additional features ... 

By taking the time derivative of this integral, one obtains a set of coupled equations for the density matrix of the system Ps t) and the n auxiliary matrices Pi(t). The main point is that this technique allows us to carry out non-Markovian dynamics by a system of coupled equations local in time. 

Any point defect modifies the crystal potential of the structure [i.e. U(r) in the Schrodinger Equation] locally and therefore perturbs the bands near it. For defects with atomic orbitals very similar to those of the matrix atoms this generally leads to a... 

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