Inhomogeneous behavior

Within the DFT the electrons of a pair of electrons or a bond can be considered as belonging to an inhomogeneous continuum gas. In analytical terms this was translated as the ELF (5.233) index as combining the homogeneous and inhomogeneous behaviors of a many-electronic-nuclei system. 

An intuitive explanation of the oscillatory volume variations is given by Yoshida in Ref [8] it is based on hydrophobic effects that induce a difference of swelling rate between the oxidized and the reduced states of the catalyst. For a small piece of gel, the oscillations occur homogeneously [39]. But when the chemical wavelength is smaller than the system size, some inhomogeneous behavior in the form of waves may arise [8]. 

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

A final comment has to do with the concept of effective viscosity In strongly Inhomogeneous fluids. For these systems the definition of the effective viscosity depends on the type flow, hence different effective viscosities will be measured for different flow situations In the same system with the same density profile. Therefore, the effective viscosity Is a concept of limited value and measurements of this quantity do not provide much information about the effects of density structure on the flow behavior. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

The capacitance determined from the initial slopes of the charging curve is about 10/a F/cm2. Taking the dielectric permittivity as 9.0, one could calculate that initially (at the OCP) an oxide layer of the barrier type existed, which was about 0.6 nm thick. A Tafelian dependence of the extrapolated initial potential on current density, with slopes of the order of 700-1000 mV/decade, indicates transport control in the oxide film. The subsequent rise of potential resembles that of barrier-layer formation. Indeed, the inverse field, calculated as the ratio between the change of oxide film thickness (calculated from Faraday s law) and the change of potential, was found to be about 1.3 nm/V, which is in the usual range. The maximum and the subsequent decay to a steady state resemble the behavior associated with pore nucleation and growth. Hence, one could conclude that the same inhomogeneity which leads to pore formation results in the localized attack in halide solutions. 

There are other sources of noise, whose behavior cannot be described analytically. They are often principally due to the sample. A premier example is the variability of the measured reflectance of powdered solids. Since we do not have a rigorous ab initio theory of diffuse reflectance, we cannot create analytic expressions that describe the variation of the reflectance. Situations where the sample is unavoidably inhomogeneous will also fall into this category. In all such cases the nature of the noise will be unique to each situation and would have to be dealt with on a case-by-case basis. 

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