Diffusion eigenfunction

The diffusion eigenfunction /) > is given, to first order in gradients, as... 

If we ignore nonhydrodynamic contributions and project onto the diffusion eigenfunction [(10.19)], we may write... 

Construction of Diffusion Eigenfunction The eigenfunction Z)> is defined as the solution to the equation (cf. 

Recall that the bond vectors and not the loci of beads with respect to a common origin. Thus we must choose sines rather than cosines as eigenfunctions for chains with free ends. Also, since Eq. (10) is concerned only with the averages of the bond vectors, there are only N normal modes, the translational diffusion mode being automatically excluded. 

The analysis shows that the diffusion current quickly settles down to the value given by Eq. 13.22 during all but very early times and that the transients which occur at the early times due to the higher-order eigenfunctions can be neglected whenever the degree of precipitation is significant. Because the effects of any transients are... 

Reaction-diffusion systems provide a means to subdivide successively a domain at a sequence of critical parameter values due to size, shape, diffusion constants, or other parameters. The chemical patterns that arise are the eigenfunctions of the Laplacian operator on that geometry. The succession of eigenfunctions on geometries close to the wing, leg, haltere, and genital discs yield sequential nodal lines reasonably similar to the observed sequence and symmetries and geometries of the observed com-... 

We note that the diffusion operator with Neumann (or periodic) boundary conditions is symmetric and has a simple zero eigenvalue with a constant eigenfunction. Equivalently, the eigenvalue problem... 

Although the other two fimctions are only eigenfimctions in special situations (contact, singlet and triplet infinite separation, two doublets), each set is nevertheless a valid set of basis functions that can be used to describe the spin state. S) and I To) are most useful for that purpose because they are eigenfunctions at the start and at the end of a diffusive excursion. At in-between times, the spin state X)(f), which will usually be time dependent, is always describable by a linear combination... 

We now calculate by projecting onto the eigenstates of the independent diffusive motion of the AB pair. These eigenfunctions are constructed so that... 

This calculation is quite similar to that for the derivation of Stokes law from kinetic theory, where one has an equation for the distribution function similar to (10.23) for. To obtain Stokes law, one must project the kinetic equation onto the hydrodynamic eigenfunctions, and it is essential to retain terms to first order in the gradients if the proper numerical factor (f = 47TTj/ for specular reflection and = 6 rrr)R for diffuse reflection) is to be obtained. In our calculation, it is also essential to retain the 6(V) terms in the Z)> eigenfunction. If these 6(V) terms are dropped, the result for the rate coefficient ky(z = 0) still has the form of (3.7), kf =kj + k but the zf O result does not agree with (3.6) and (3.8). The gradient terms are essential if one is to obtain the simple z-dependence given by kp z) = k + ooab). How this comes about is clearly demonstrated by the calculation in Appendix E. 

This simplifies the solution of the rotational diffusion equation considerably. As is well known, the spherical harmonics Yim 0, [Pg.120]

This technique originated in the numerical solution of the diffusion equation and other areas of fluid mechanics where numerical derivatives are required. In many areas of physics the algebraic expansion method is seen historically as related to the expansion in eigenfunctions method and is known as the spectral method because of the common use of Fourier transform techniques. 

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