Fractional and High Order Asymptotic Results of the MFPT

The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point have a variety of dependencies in respect to its size N. For example, for the case of birth and death processes the MFPT~N for the case of symmetric random walks the MFPT~N2 [13,14]; and for the case of biased random walks, the MFPT~αN where α is a constant that depends on the system’s rates [13-15].

The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point have a variety of dependencies in respect to its size N. For example, for the case of birth and death processes the MFPT~N for the case of symmetric random walks the MFPT~N 2 [13,14]; and for the case of biased random walks, the MFPT~α N where α is a constant that depends on the system's rates [13][14][15].
This work presents an analytic derivation of the transition rates matrix for continuous-time nearest-neighbor random walks in a finite one-dimensional system, with a trap at the origin and a reflecting barrier at the other end.
The transition rates matrix is designed, in such a way that the calculated MRTs of this system would be (m+1) d , where m is the site number and d is any arbitrary number satisfying: d>0.
Since the MFPT is the sum of MRTs, then, in this case, the asymptotic result of the MFPT would be N 1+d where the number of sites is N.
Thus, one can decide the asymptotic results of the MFPT such as fractional or any high order dependency with respect to N and based on it obtain the corresponding transition matrix. Figure 1 presents a schematic illustration of a well-known continuous-time random walk in a finite one-dimensional system, in the presence of a trap and an absorbing barrier [13][14][15][16].
The matrix presentation of the above set is: Where the transition matrix A is: is the survival probability vector expressed as The general solution of eqn. (1), which describes the survival population at the mth site at time t, starting from the nth site is [14,16,17]: Now we can iterate eqn. (12) to yield: and in general: Note that if Since 0<R 1 <1 from eqn. (14), then R m remains positive and bounded between 0<R m <1.
The rates toward the trap, T m are obtained by using eqns. (10) and (16) as follows: Using eqns. (16) and (17) one can obtain transition rates matrix A for any arbitrary d>0, which yields MRTs of site m equal to (m+1) d , and asymptotic results of the MFPT proportional to N 1+d .
The following figures present the transition matrices and the inverse transition matrices for the cases of d=4, d=2.5, with 7 sites (N=7). Note that in this case, for a large finite system, the asymptotic result of the MFPT is proportional to N 5 and N 3.5 respectively. Figure  Note that for a large system R m converge since: Which means that R m has an asymptotic result and the sum of R m +R (m-1) for a large m is:

Deriving the Exact Transition Rate Matrix, A, Starting at Site n=N
In this section an analytic transition rate matrix is derived, where where ( ) The expression of the MRTs as a function of the transition rates can be given as follows [15]:

Deriving the Exact Transition Rate Matrix, A When Starting At Site n=1
In this section, an analytic transition rate matrix is derived based on eqn. (6)-(8) [15], in which its MRTs is equal to (m+1) d , where m describes the site and d is any arbitrary number satisfying: d>0.
Consider a transition rates matrix A where its MRTs, τ(m,n), are: τ(m,n=1)=(m+1) d (9) and consider the following relation mainly to reduce the number of degrees of freedom: where R m and T m are the rates toward the reflecting barrier and toward the trap respectively; Substituting eqn. (10) into eqn. (7) yields the following recurrent: And after substituting the MRTs described in eqn. (9) into eqn. (11) and rearranging yields: To calculate an arbitrary R m we need R 1 .
Substituting in eqn. (6) τ(1,1)=2 d that is given by eqn. (9) and m=1 yields: and by using the relation of eqn. (10) R 1 is obtained as follows: and consider the following relations between the transition rates in a similar way to eqn. (10): where r m and t m are the rates towards the reflecting barrier and the trap correspondingly, for the case of starting at the reflecting barrier.
Finding r 1 is similar to finding R 1 , as describes above, and yields: Now we can iterate eqn. (23) to obtain: and for an odd m: Note that for a large m the asymptotic results of r m and t m are the same as R m and T m , since the last term of both r m and t m tends to zero, as seen from eqns. (26) and (29).
Using eqns. (27) and (29) one can obtain a transition rate matrix A which yields the MRTs of site m equal to (m+1) d for any arbitrary d>0, for the case of starting at the reflecting barrier.
In the following Figure 4 the transition matrices and the inverse transition matrices are presented for d=4, d=5, and in these cases, for a large finite system, the asymptotic results of the MFPT are proportional to N 5 and N 6 respectively.

Conclusion
This paper presents a derivation of transition matrix A in nearestneighbor random walks in finite-one dimensional system. The transition matrix is derived in such a way that the MRT of the system is equal to (m+1) d where m is the site number, and d is any arbitrary number satisfying: d>0.
As a consequence one can determine the asymptotic result of the MFPT to be N 1+d for any fractional or integer, and based on it, obtain the corresponding transition matrix.