Reflecting wall

As may be shown, the same relaxation time (5.99) will take place if the initial distribution is located in an immediate vicinity of the origin (i.e., when xq = +0) and when the reflecting wall is sited at the point x = 0. 

In the case where the reflecting wall and the absorbing wall are at the ends of the decision interval [c, d, the decay time... 

If there is the symmetrical potential profile initial distribution is located at the origin (i.e., xo = 0), then all results concerning the relaxation times will be the same as for the potential profile in which the reflecting wall is at x = 0 and at xo = +0. This coincidence of the relaxation times may be proven in a common case taking into account that the probability current at x = 0 is equal to zero at any instant of time. 

Another possible protective scheme, although rarely used in the petrochemical industry, is a blast resistant barrier wall. A barrier wall can be used to provide protection from fragments and reduce reflected wall loads. However, it will not reduce overpressures on the roof and unprotected side walls. 

For the half-space above the wall (y > 0), the Green s function for a reflecting wall, in the absence of flow, becomes... 

Three analytical expressions for the spin-echo intensity as a function of the gradient in a pulsed field gradient NMR experiment for spins diffusing in a sphere with reflecting walls are reinvestigated. It is found that none of the published formulas are completely correct. By numerical comparisons the correct formula is found. 

Since the Murday-Cotts paper in 1968 much progress has been made toward the theoretical description of the spin-echo intensity E g, A) for spins diffusing in well-defined geometries. Tanner and Stejskal derived already in 1968 the exact expression of ii(g. A) for spins diffusing in a rectangular box. The derivation of an exact expression for E g, A) for diffusion in a sphere with reflecting walls is not a trivial mathematical problem and it took between 1992 and 1994 when three expressions were published. All three expressions are only valid in the short-gradient-pulse approximation (see below). 

When we wanted to numerically fit experimental PFGE data of water diffusion in a water-in-oil emulsion, we found that for a beginner in this field the literature is quite confusing. First, all three expressions for diffusion in a sphere with reflecting walls are somewhat different and lead to very different fitting results, especially when the formulas are combined with a radius distribution function. Since the derivation of the published expressions needs some tedious algebra (which has not been published), it is not trivial to check the derivation in order to establish which expression is the correct one. Here we use a numerical approach to decide which expression is correct. 

The normal mode expression for the propagator Eq. (16) is perfect for working out the propagator for diffusion in a rectangular box with reflecting walls between the x -planes at z = 0 and z = a. One of the boundary conditions here is (reflecting walls) ... 

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

We defer the discussion of the effects of (r) until Section VII.C and begin with the special case, referred as a force-free diffusion, with a uniform distribution of electron spins outside the distance of closest approach with respect to the nuclear spin. Under the assumption of the reflecting-wall boundary condition at rjs = d, Hwang and Freed found the closed analytical form of the correlation function for translation diffusion (138) ... 

Exercise. What is the condition for a reflecting boundary Formulate the first-passage problem when L is a reflecting wall. 

Fig. 4.10 Compressional standing wave created by proximity of the reflecting wall (adapated from Janshoff et al., 2000)...

Section VIII. We consider dielectric response of an ion moving inside a hydration sheath. Assuming perfect reflections of the ion from the sheath, we model the latter by means of infinitely deep potential well (a like approach is used in Section III for the protomodel). So, no charges are proposed to penetrate through the sheath. Two variants of the ionic model are described. In the first one a charge (ion) moves along a line between two reflection points, and in the second model an ion moves in a space inside a spherical cavity with elastically reflecting walls (see Fig. 2d). 

This idea has led long ago [20, 21, 45, 46] to a simple model termed the confined rotator model, where the dielectric response was found due to free planar librations performed without friction during the lifetime of a molecule in a site of near order. These librations occur between two reflective walls, the... 

The general solution of a random walk on a lattice between perfect reflecting walls was provided by Feller.80 In what follows, a simpler approximate solution will be constructed using the method of images.24 When the walk is confined between two reflecting walls, one at the origin and the other at the distance 2d, the probability for the constrained walk to reach the distance z, yd(z), can be written as a sum of probabilities y>u for the unconstrained walk to reach various image points, as follows. 

The electron is restricted to move in the half-space x > 0. There is a totally reflecting wall at x = 0. Since the Hamiltonian (8.1.1) of the kicked hydrogen atom and the Hamiltonian of microwave-driven surface state electrons are so similar, we can use many of the results that were derived in Chapter 6. The most important result is the transformation to action and angle variables I and 6, respectively, defined in (6.1.18). The... 

We will find it convenient to consider for the moment the special case of perfectly reflecting walls. In the long time-scale limit A a /D, the average propagator for fully restricted diffusion has a simple relationship to the pore geometry. This requirement on A, also known as the pore equilibration condition, implies that the time is sufficiently long that most molecules have collided with the walls. Under this condition the conditional probabilities are independent of starting position so that P(r, t I r, 0) reduces to p(r ), the pore molecular density function. 

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