Random-Walk and Multi-Barrier Kinetics

A fairly general transport equation for the random walk of particles with absorbing and reflecting barriers and with sources can be represented as follows  

In the case of the degeneracy, v )scan be properly orthogonal-ized to satisfy (6). Further, A has the following properties  

If the matrix contains only the nearest-neighbor transition probabilities, the corresponding eigenvectors and eigenvalues are related by the equation  

Equation (8) follows from the following argument. Since. 4 is a non-singular matrix of rank A -1-1, the square matrix P defined in (10a) also is of the N - -l rank thus it has a unique inverse P which satisfies the relation P- P = PP- — I. From Equations (6), (10a), and P P = I, one obtains (10b). The substitutions of (10a) and (10b) into PP = / 5delds Equation (8). 

The transport equation (1) reduces to that proposed by E uing and co-workers at a steady state with only nearest-neighbor transition probabilities. By setting e = 0o no a nd dCJt)jdt= 0 at the steady state, a transport equation of the following form is obtained from (1)  

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