Absorbing barrier

To find the distribution of a busy period, one must compute [dPQ(t)]jdt from the system of equations (5-3) with c = 2 with an absorbing barrier at the origin, i.e., the first two equations are altered for this purpose to read ... 

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... 

This corresponds to a one-dimensional random walk with an absorbing barrier, with the transition probabilities calculated from quantum mechanics. The time dependent distribution of the reactant molecules among... 

Consider a one-dimensional random walk on a lattice of N sites j = 0,1,2,... N — 1 with absorbing barriers at j = — 1 and j = N. Let fj t denote the probability that the walker is at site j after / steps, fjfi being the initial probability distribution. Suppose that P(j i) = Pt-j is the probability of making a step from site j to site i (ignoring the barriers). Then, taking account of the absorption, the distribution after / steps is easily seen to be given by... 

Fig. 2.24. Schematic diagram showing the potential energy of the methyl + methyl reaction as a function of separation. Ec is the position of the absorbing barrier at a temperature of...

The observed association rate is the overall flux from reactants through to deactivated product. This rate can be modelled using a ME method given a suitable definition of deactivation. In the calculations presented here the molecule is considered to be deactivated if the energy reaches at least 10/cbT below threshold. The deactivation rate is therefore modelled as the flux into an absorbing barrier placed at this energy. 

The solution for this irreversible system is characterized by a fast transient period followed by a steady-state regime in which the shape of the population distribution remains constant. The steady-state solution can be obtained from the ME, equation (2.49), by setting the time derivative to zero and by setting all reactivation rates from the states in the absorbing barrier to zero. This effectively reduces the size of the matrix by removing all states more than 10 A bT below the threshold. 

Viewed from the product, the overall rate of association is equivalent to the rate into the absorbing barrier, hence the rate coefficient is given... 

Example 2.17 is a simple random walk between two absorbing barriers. It is characterized by the behavior of the moving bird, the system, that when it occupies states Si and S9, it remains there from that time on (pn = P99 = 1). In this case the transition matrix is given by ... 

Fig.2-11. Probability distribution of occupying various states for the two absorbing barriers Si and S9...

Example 2.41 demonstrates a game [6, p.21] related to a simple random walk between two absorbing barriers, considered also in example 2.17. Assume that Jacob and Moses have five shekels (Israeli currency) divided between them. On cme side there is a symbol of a lectf and on the other side appears the number one. The shekels are assumed fair ones, so that the probability of a leaf on a toss equals the probability of a one on a toss, equals 1/2. Jacob tosses a coin first and records the outcome, feqfor one. Then Moses tosses a coin. If Moses matches... 

Let A be the transition probability matrix of the set of differential equations (analogous to Eq. IV.5) which describe the variation of level concentration with time in the presence of an absorbing barrier. Then the set of equations in matrix form is... 

This corresponds to a totally absorbing barrier (a = 1) at level... 

This differs by a factor Z from the estimate derived in Eq. VI. 7 by the assumption that the characteristic value of the transition probability matrix A can be determined by perturbation theory while the corresponding characteristic vector remains unchanged by the existence of an absorbing barrier. The significance of the error will be discussed presently. 

Tufeote. [E-A-R] Polyedier urethane foam for absorbers, barriers, and composite prods. 

Ghild, M.S. (1991) Analysis of a complex absorbing barrier, Molec. Phys. 72, 89-93. 

Humans can protect themselves from the effects of high-energy radiation in two ways (1) by the use of an absorbing barrier of some kind, and (2) by remaining as far away from the radiation source... 

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