Reflecting boundary

For the combination (BC2) of a reflecting boundary at x = 0 and an absorbing boundary at x = L, the same procedure as above can be carried out to find the following results. 

The asymptotic limits of this result for the weak and strong strengths of the drift with respect to diffusion follow as 

For the combination (BC3) of an absorbing boundary at x = L and a mixed boundary condition at x = 0, the same procedure as above can be carried out to And the various quantities associated with the FPT. For the sake of completeness with respect to different boundary conditions, we provide the formulas for the probability distribution function and the unconditional mean FPT. Other quantities can also be calculated in an analogous manner. 

The above results for mixed boundary condition reduce to those of the absorbing and reflecting boundary conditions discussed above, for It oo and h = 0, respectively. Other moments of the FPT can be similarly written down for the general boundary condition. 

The motion of polymer molecules in a solution under chemical potential gradients or externally imposed electric fields is an example of the drift-diffusion stochastic processes. We have introduced several equivalent formalisms for studying these processes biased random walk, master equation, and Langevin equation of motion. In each of these lines of arguments, we have arrived at the Fokker-Planck-Smoluchowski equation. 

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A problem obviously exists in trying to characterise anomalies in concrete due to the limitations of the individual techniques. Even a simple problem such as measurement of concrete thickness can result in misleading data if complementary measurements are not made In Fig. 7 and 8 the results of Impact Echo and SASW on concrete slabs are shown. The lE-result indicates a reflecting boundary at a depth corresponding to a frequency of transient stress wave reflection of 5.2 KHz. This is equivalent to a depth of 530 mm for a compression wave speed (Cp) of 3000 m/s, or 706 mm if Cp = 4000 m/s. Does the reflection come from a crack, void or back-side of a wall, and what is the true Cp ... 

It is worth noting that within a range of 20 %, five different methods of analyzing the crystallite size, viz., (a) microscopic inspection, (b) application of Eq. (3.1.7) for restricted diffusion in the limit of large observation times, (c) application of Eq. (3.1.15) to the results of the PFG NMR tracer desorption technique, and, finally, consideration of the limit of short observation times for (d) reflecting boundaries [Eq. (3.1.16)] and (e) absorbing boundaries [Eq. (3.1.17)], have led to results for the size of the crystallites under study that coincide. 

Reflecting Boundary. The reflecting boundary may be represented as an infinitely high potential wall. Use of the reflecting boundary assumes that there is no probability current behind the boundary. Mathematically, the reflecting boundary condition is written as... 

Natural Boundary Conditions. If the Markov process is considered in infinite interval, then boundary conditions at oo are called natural. There are two possible situations. If the considered potential at +oo or —oo tends to —oo (infinitely deep potential well), then the absorbing boundary should be supposed at Too or —oo, respectively. If, however, the considered potential at Too or — oo tends to Too, then it is natural to suppose the reflecting boundary at Too or —oo, respectively. 

Integrating Eq. (3.16) and taking into account the reflecting boundary conditions (probability current is equal to zero at the points d), we get... 

Figure 7.4. Sketch of vectors used in reflective boundary conditions. 

When the reaction is represented by a sink term, a reflective boundary condition must be imposed at a contact distance d. This boundary condition leads to the following boundary condition on the escape probability... 

Calculation of the electric field dependence of the escape probability for boundary conditions other than Eq. (11b) with 7 = 0 poses a serious theoretical problem. For the partially reflecting boundary condition imposed at a nonzero R, some analytical treatments were presented by Hong and Noolandi [11]. However, their theory was not developed to the level, where concrete results of (p(ro,F) for the partially diffusion-controlled geminate recombination could be obtained. Also, in the most general case, where the reaction is represented by a sink term, the analytical treatment is very complicated, and the only practical way to calculate the field dependence of the escape probability is to use numerical methods. 

FIM experiments show that a plane boundary acts mostly as a reflective barrier.86 In a few cases plane boundaries are slightly absorptive.95 Three effects will be considered here, namely on the mean square displacement, on the frequency of encountering a reflecting boundary, and on the mean adsorption time if the boundary is absorptive. 

Consider here a one-dimensional lattice, or surface channel, with M equilibrium rest sites, numbered from 1 to M. The boundaries are assumed to be perfectly reflective. The probability that an atom initially at site i is found at site / after N jumps, Pb(i, N), is given by summing the probabilities of displacements from site i to site / and all the images of site y.113 The mirror images of site / are sites with indices 2kM + 1 — / and 2kM + j where k = 0, 1, 2,. .., etc. Thus for the symmetric random walk in a one-dimensional lattice of M sites and with reflective boundaries, Pb(i,j, N) is given by... 

The Smoluchowski reactive (or inner) boundary condition [eqn. (5)] is implied in the partially reflecting boundary condition [eqn. (22)]. When reaction between A and B at the encounter separation is very fast compared with the rate of diffusive approach of A and B, kaci -> °°. However, the concentration gradient remains finite, so that the density of B about reactant A tends to zero. 

The width of the encounter pair reactivity zone, 672, is to be considered small. There is no reason for this choice, save convenience. Probably rather larger widths would be more appropriate following work on gas-phase collision kinetics or long-range transfer processes (Chap. 4). In such circumstances, the partially reflecting boundary condition is no longer suitable and other techniques have to be used (see Chap. 8 Sect. 2.4 and Chap. 9 Sect. 4). 

To solve the diffusion equation (9) or (10) for the density p(r, f) with the random initial condition (3), the outer boundary condition (4) and the partially reflecting boundary condition (22) is straightforward. Again, the solution follows from eqn. (12), but the Laplace transform of eqn. (22) is... 

Finally, for completeness, the Green s function corresponding to a pair of reactants initially formed with separation r0 and subjected to the partially reflecting boundary condition, is quoted (Pagistas and Kapral [37], Naqvi et al. [38]. 

THE RATE COEFFICIENT FOR A PARTIALLY REFLECTING BOUNDARY CONDITION... 

To find the second-order rate coefficient for the reaction of A and B subject to the encounter pair reacting with a rate coefficient feact, the method developed in Sect. 3.7 can be used. Using eqn. (19), the rate coefficient, k(t), can be defined in terms of the diffusive current of B towards the central A reactant. But the partially reflecting boundary condition (22) equates this to the rate of reaction of encounter pairs. The observed rate coefficient is equal to the rate at which the species A and B could react were diffusion infinitely rapid, feact, times the probability that A and B are close enough together to react, p(R). 

Thus the partially reflecting boundary condition reduces the effective encounter distance by a factor of fcact (4nRD + fcact) 1 for both the steady-state and transient terms in the rate coefficient. 

Because the diffusive flux is enhanced by this drift of a charge under the influence of the coulomb potential [as represented in eqn. (142)], the partially reflecting boundary condition (127) has to be modified to balance the rate of reaction of encounter pairs with the rate of formation of encounter pairs [eqn. (46)]. However, the rate of reaction of ion-pairs at encounter is usually extremely fast and the Smoluchowski condition, eqn. (5), is adequate. The initial and outer boundary conditions are the same as before [eqns. (131) and (128), respectively], representing on ion-pair absent until it is formed at time t0 and a negligibly small probability of finding the ion-pair with a separation r - ... 

Furthermore, the initial and outer boundary conditions are effectively identical [eqns. (3), (4) and (165)] as are also the partially reflecting boundary conditions [eqns. (46) and (165)]. This can be shown by substituting p by exp — p p in the boundary conditions (165). Consequently, the relationship between the survival probability of an ion-pair at a time t0 after they were formed at time t and separation r and the density distribution of an initial (time t0) homogeneous distribution of the majority ion species around the minority ionic species, p(r, f f0), is an identity. 

Comparing eqns. (170) and (171) shows that the density distribution for the steady-state formation, recombination and scavenging, pss(r cs r0), is closely related to the Laplace transformed (time-dependent) density distribution for recombination and escape. The partially reflecting boundary condition [eqn. (46)] with p replaced by p or pss... 

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