Absorbing boundaries

Neuhauser D and Baer M 1989 The time dependent Schrodinger equation application of absorbing boundary conditions J. Chem. Phys. 90 4351... 

Kosloff R and Kosloff D 1986 Absorbing boundaries for wave propagation problems J. Comput. Phys. 63 363... 

Mandelshtam V A and Taylor H S 1995 A simple recursion polynomial expansion of the Green s function with absorbing boundary conditions. Application to the reactive scattering J. Chem. Phys. 102... 

Seideman T and Miller W H 1992 Quantum mechanical reaction probabilities via a discrete variable representation-absorbing boundary condition Green function J. Chem. Phys. 97 2499... 

B. Engquist and A. Majda, Absorbing Boundary Conditions for the Numerical Simulation of Waves, Math. Comput. 31, No. 139 (1977). 

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. 

However, most concrete tasks (see examples, listed above) are described by smooth potentials that do not have absorbing boundaries, and thus the moments of FPT may not give correct values of timescales in those cases. 

Absorbing Boundary. The absorbing boundary may be represented as an infinitely deep potential well just behind the boundary. Mathematically, the absorbing 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. 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

One can obtain an exact analytic solution to the first Pontryagin equation only in a few simple cases. That is why in practice one is restricted by the calculation of moments of the first passage time of absorbing boundaries, and, in particular, by the mean and the variance of the first passage time. 

As discussed in the previous section, the first passage time approach requires an artificial introduction of absorbing boundaries therefore, the steady-state... 

The Transition Probability. Suppose we have a Brownian particle located at an initial instant of time at the point xo, which corresponds to initial delta-shaped probability distribution. It is necessary to find the probability Qc,d(t,xo) = Q(t,xo) of transition of the Brownian particle from the point c 0 Q(t,xo) = W(x, t) dx + Jrf+ X W(x, t) dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary. Here we suppose that c and d are arbitrary chosen points of an arbitrary potential profile (x), and boundary conditions at these points may be arbitrary W(c, t) > 0, W(d, t) > 0. 

The main distinction between the transition probability and the probability to pass the absorbing boundary is the possibility for a Brownian particle to come back in the considered interval (c, d) after crossing boundary points (see, e.g., Ref. 55). This possibility may lead to a situation where despite the fact that a Brownian particle has already crossed points c or d, at the time t > oo this particle may be located within the interval (c, d). Thus, the set of transition events may be not complete that is, at the time t > oo the probability Q(t,xo) may tend to the constant, smaller than unity lim Q(t, x0) < 1, as in the case... 

It is easy to check that the normalization condition is satisfied at such a definition, wT(t.xoj dt = 1. The condition of nonnegativity of the probability density wx(f,x0) > 0 is, actually, the monotonic condition of the probability Q(t, xq). In the case where c and d are absorbing boundaries the probability density of transition time coincides with the probability density of the first passage time wT(t,x0y. 

Note that previously known time characteristics, such as moments of FPT, decay time of metastable state, or relaxation time to steady state, follow from moments of transition time if the concrete potential is assumed a potential with an absorbing boundary, a potential describing a metastable state or a potential within which a nonzero steady-state distribution may exist, respectively. Besides, such a general representation of moments dn(c,xo,d) (5.1) gives us an opportunity to apply the approach proposed by Malakhov [34,35] for obtaining the mean transition time and easily extend it to obtain any moments of transition time in arbitrary potentials, so i9n(c, xo, d) may be expressed by quadratures as it is known for moments of FPT. 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

Here we are interested in escape out of the domain L specified by a single cycle of the potential that is out of a domain of length n that is the domain of the well. Because the bistable potential of Eq. (5.42) has a maximum at x = n/2 and minima at x = 0, x = 7t, it will be convenient to take our domain as the interval —7t/2 < x < n/2. Thus we will impose absorbing boundaries at x = —n/2, x = n/2. Next we shall impose a second condition that all particles are initially located at the bottom of the potential well so that x0 = 0. The first boundary condition (absorbing barriers at —n/2, n/2) implies that only odd terms in p in the Fourier series will contribute to Y (x). While the second ensures that only the cosine terms in the series will contribute because there is a null set of initial values for the sine terms. Hence... 

For the decision interval extended to the absorbing boundary (d = X2) from (5.110), one obtains... 

The moments of transition time of a dynamical system driven by noise, described by arbitrary potential cp(x) such that cp( oo) = oo, symmetric relatively to some point x = d, with initial delta-shaped distribution, located at the point xo < d [Fig. A 1(a)], coincides with the corresponding moments of the first passage time for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile [Fig. A 1(b)]. 

Let us prove that formula (4.19) (in this particular case for c = — oo) not only gives values of moments of FPT of the absorbing boundary, but also expresses moments of transition time of the system with noise, described by an arbitrary symmetric with respect to the point d potential profile. 

Thus, we have proved the principle of conformity for both probability densities and moments of the transition time of symmetrical potential profile and FPT of the absorbing boundary located at the point of symmetry. 

Cumulative Reaction Probability via a Discrete Variable Representation with Absorbing Boundary Conditions. 

Recursion Polynomial Expansion of the Green s Function with Absorbing Boundary Conditions. Application to the Reactive Scattering. 

Reaction Probabilities via a Discrete Variable Representation- Absorbing Boundary Condition Green Function. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

W.C. Chew, J.M. Jin and E. Michelsen, Complex coordinate stretching as a generalized absorbing boundary condition, Microwave and Opt. Technol. Lett. 15, 363-369 (1997). 

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