External potential origin density with

FIG. 10 Nomalized unbalanced surface charge in a cylindrical pore with R = 5din the presence of an external potential The results, from left to right, are for original surface charge densities of —0.001, —0.005, —0.01, —0.02, —0.04, —0.05, —0.07123 C/m respectively. The x-intercepts are values of the corresponding equilibrium Donnan potentials. 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

Figure J Density at origin for Is-state of the hydrogen atom in R3 with an external potential. The external potential has the form U w r — Rq) with the Heaviside function w. Note the monotony of density with respect to U. According to Section 5.4 for Ro =0 and Rq = oo wave functions are the same p(0) = 4.0 for any U. The region of small Rq values is not presented (the computational procedure is unstable in this region).

Figure 7 plots the surface pressure as a function of the separation H between two paralleled slit wall filled with one-component HS duid. The external potential for the confined HS fluid is zero if 0 < zsurface pressure, i.e., can be calculated from the integration of one-body density distribution multiplying the external force over the distance ar. In the circumstance of hard-waU, the external force recovers to a Dirac delta function and thus the surface pressure is directly related to the fluid contact density p z = 0). The predicted results from MFMT and original FMT are compared with simulation results. This comparison shows that FMT, especially the modified version, can yield very accurate results. 

Consider a flat gas-liquid surface lying in the x, y-plane and take the origin of the coordinate system to lie on a convenient dividing surface, e.g. the equimolar surface, Fp = 0. Let u(r) be an external potential which deforms the flat surface into a spherical one with radius of curvature R (Fig. 4.2). The potential can be arbitrarily weak since we shall be interested only in the limiting behaviour R ->0. If Ap(t) is the change of density on applying the potential... 

These equations vividly demonstrate that apart Irom the potential energy density originating from the interaction of each quantum particle with the external field, the surface virial also contributes to the basin energy. Assuming C1 = R the surface virial vanishes and the equations are indistinguishable from those derived for the total system independently. 

FIGURE 6.2 Variation of the dimensionless surface potential <6(x) against the coordinate x whose origin is taken at the midpoint between plates with separation dt = 20 nm. The analysis is for the two-charged-plate system (N= 1). The concentration of the external soaking solution is c = 0.001 mol T1, and the surface potential is 4>s = -4.0 in the Dirichlet model, which corresponds to the surface charge density Z0 = -0.084 nmr2 in the Neumann model. The two plates are located at = -10 nm and x2 = 10 nm. 

Facilitated diffusion has certain general characteristics. As already mentioned, the net flux is toward a lower chemical potential. (According to the usual definition, active transport is in the energetically uphill direction active transport may use the same carriers as those used for facilitated diffusion.) Facilitated diffusion causes fluxes to be larger than those expected for ordinary diffusion. Furthermore, the transporters can exhibit selectivity (Fig. 3-17) that is, they can be specific for certain molecules solute and not bind closely related ones, similar to the properties of enzymes. In addition, carriers in facilitated diffusion become saturated when the external concentration of the solute transported is raised sufficiently, a behavior consistent with Equation 3.28. Finally, because carriers can exhibit competition, the flux density of a solute entering a cell by facilitated diffusion can be reduced when structurally similar molecules are added to the external solution. Such molecules compete for the same sites on the carriers and thereby reduce the binding and the subsequent transfer of the original solute into the cell. 

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