Natural repulsive boundary

If a boundary is natural the boundary condition is replaced by a convergence condition. For a natural repulsive boundary (i) one has Lx oo. If also L3 - oo there is no solution. If L3 < oo but L4 -+ oo, only a solution with J = 0 is possible, which is compatible with a reflecting boundary at the other end, but not with an absorbing boundary. If L3 < oo and L4 < oo no restriction results from the boundary, so that it is compatible with any boundary condition at the other end. 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

In this case one may use the symmetry of the Coulomb potential and apply the comparison theorem for the ground state wavefunction ir r) and the "reflected" function external potential t/i(r) = U(err). As comparison theorem one finds ir r) > Hellmann-Feynman force is oriented into E+, that is, its n-projection is positive. Practically the same discussion was used in [34] to prove the monotone nature of the adiabatic potential for the ground state of the one-electron diatomic molecule (in the absence of the internuclear repulsion term). This statement is also easily modified for the Dirichlet boundary value problem for some region 2. One may formally require the external potential U to be infinite out of 2 (for the analysis of this statement see [35]). 

Within the PB cell model, the counterion concentration at the outer boundary gives the osmotic pressure, which is a measure of the electrostatic repulsion between neighboring biomolecules. This pressure can also be experimentally determined. " " The Donnan coefficient, on the other hand, is strongly influenced by conditions at the macromolecular surface and can be used to provide key insight into the nature of polyelectrolyte-counterion interaction. " This interaction is important because of the salt-induced conformational changes DNA undergoes. The nature of this behavior is believed to arise from the partial collapse, or condensation. 

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