Explicit expressions for heterogeneous random walks in 1D

In a completely heterogeneous semi-Markovian random walk in a discrete system of L (> 1) states, the Green's function was found in Laplace space (the Laplace transform of a function is defined with, ${\displaystyle {\bar {f}}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt}$). Here, the system is defined through the jumping time (JT) PDFs: ${\displaystyle \psi _{ij}(t)}$ connecting state i with state j (the jump is from state i). The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths:

${\displaystyle {\bar {G}}_{ij}(s)={\bar {\Gamma }}_{ij}(s){\frac {{\bar {\Phi }}(s,{\tilde {L}})}{{\bar {\Phi }}(s,L)}}{\bar {\Psi }}_{i}(s).}$

(1)

Here,

${\displaystyle {\bar {\Psi }}_{i}(s)=\sum _{j}{\bar {\Psi }}_{ij}(s)}$

and

${\displaystyle {\bar {\Psi }}_{ij}(s)={\frac {1-{\bar {\psi }}_{ij}(s)}{s}}.}$

Also, in Eq. (1),

${\displaystyle {\bar {\Gamma }}_{ij}(s)=\prod _{c=0}^{i\mp 1}{\bar {\psi }}_{c\pm 1c}(s),}$

(2)

and

${\displaystyle {\bar {\Phi }}(s,L)=1+\sum _{c=1}^{[L/2]}(-1)^{c}{\bar {h}}(s,c;L)}$

(3)

with

${\displaystyle {\bar {h}}(s,i;L)=\prod _{c=1}^{i}\sum _{k_{c}=2+k_{c-1}}^{L-1-2(i-c)}{\bar {f}}_{k_{c}}(s)}$

(4)

and

${\displaystyle {\bar {f}}_{k_{j}}(s)={\bar {\psi }}_{k_{j}k_{j}+1}(s){\bar {\psi }}_{k_{j}+1k_{j}}(s).}$

(5)

For L = 1, ${\displaystyle {\bar {\Phi }}(s;L)=1}$. In this paper, the symbol [L/2], as appearing in the upper bound of the sum in eq. (5) is the floor operation (round towards zero). Finally, the factor ${\displaystyle \Phi (s,{\tilde {L}})}$ in eq. (1) has the same form as in ${\displaystyle {\bar {\Phi }}(s;L)}$ in eqs. (3)-(5), yet it is calculated on a lattice ${\displaystyle {\tilde {L}}}$ . Lattice ${\displaystyle {\tilde {L}}}$ is constructed from the original lattice by taking out from it the states i and j and the states between them, and then connecting the obtained two fragments. For cases in which a fragment is a single state, this fragment is excluded; namely, lattice ${\displaystyle {\tilde {L}}}$ is the longer fragment. When each fragment is a single state, ${\displaystyle {\bar {\Phi }}(s;{\tilde {L}})=1}$.

Equations (1)-(5) hold for any 1D semi-Markovian random walk in a L-state chain, and form the most general solution in an explicit form for random walks in 1d.

Path representation of heterogeneous random walks

Clearly, ${\displaystyle {\bar {G}}_{ij}(s;L)}$ in Eqs. (1)-(5) solves the corresponding continuous time random walk problem and the equivalent generalized master equation. Equations (1)-(5) enable analyzing semi-Markovian random walks in 1D chains from a wide variety of aspects. Inversion to time domain gives the Green’s function, but also moments and correlation functions can be calculated from Eqs. (1)-(5), and then inverted into time domain (for relevant quantities). The closed-form ${\displaystyle {\bar {G}}_{ij}(s;L)}$ also manifests its utility when numerical inversion of the generalized master equation is unstable. Moreover, using ${\displaystyle {\bar {G}}_{ij}(s;L)}$ in simple analytical manipulations gives,^[6][8][9] (i) the first passage time PDF, (ii)–(iii) the Green’s functions for a random walk with a special WT-PDF for the first event and for a random walk in a circular L-state 1D chain, and (iv) joint PDFs in space and time with many arguments.

Still, the formalism used in this article is the path representation of the Green's function ${\displaystyle G_{ij}(t)}$, and this supplies further information on the process. The path representation follows:

${\displaystyle G_{ij}(t;L)=\int _{0}^{\infty }\Psi _{i}(t-\tau )W_{ij}(\tau ;L)\,d\tau ,}$

(6)

The expression for ${\displaystyle W_{ij}(t;L)}$ in Eq. (6) follows,

${\displaystyle W_{ij}(\tau ;L)=\sum _{n=0}^{\infty }w_{ij}(\tau ,2n+\gamma _{ij};L).}$

(7)

${\displaystyle W_{ij}(t;L)}$ is the PDF of reaching state i exactly at time t when starting at state j exactly at time 0. This is the path PDF in time that is built from all paths with ${\displaystyle 2n+\gamma _{ij}}$ transitions that connect states j with i. Two different path types contribute to ${\displaystyle w_{ij}(\tau ,2n+\gamma _{ij};L)}$:^[8][9] paths made of the same states appearing in different orders and different paths of the same length of ${\displaystyle 2n+\gamma _{ij}}$ transitions. Path PDFs for translation invariant chains are mono-peaked. Path PDF for translation invariant chains mostly contribute to the Green's function in the vicinity of its peak, but this behavior is believed to characterize heterogeneous chains as well.

We also note that the following relation holds, ${\displaystyle {\bar {W}}_{ij}(s;L)={\bar {W}}_{1L}(s;L)/{\bar {W}}_{1{\tilde {L}}}(s;{\tilde {L}})}$. Using this relation, we focus in what follows on solving ${\displaystyle {\bar {w}}_{1L}(s;L)}$.

Path PDFs

Complementary information on the random walk with that supplied with the Green’s function is contained in path PDFs. This is evident, when constructing approximations for Green’s functions, in which path PDFs are the building blocks in the analysis.^[8][9] Also, analytical properties of the Green’s function are clarified only in path PDF analysis. Here, presented is the recursion relation for ${\displaystyle w_{ij}(\tau ,2n+\gamma _{ij};L)}$ in the length n of path PDFs for any fixed value of L. The recursion relation is linear in path PDFs with the ${\displaystyle {\bar {h}}(s,i;L)}$s in Eq. (5) serving as the n independent coefficients, and is of order [L / 2]:

${\displaystyle {\bar {w}}_{1L}(s,2n+\gamma _{1L};L)=\delta _{n0}{\bar {\Gamma }}_{1L}(s)+\sum _{c=1}^{[L/2]}(-1)^{c+1}{\bar {w}}_{1L}(s,2(n-c)+\gamma _{1L};L){\bar {h}}(s,c;L).}$

(8)

The recursion relation is used for explaining the universal formula for the coefficients in Eq. (1). The solution of the recursion relation is obtained by applying a z transform:

${\displaystyle {\hat {\bar {w}}}_{1L}(s,2z+\gamma _{1L};L)=\sum _{n=0}^{\infty }{\bar {w}}_{1L}(s,2n+\gamma _{1L};L)z^{n}={\bar {\Gamma }}_{1L}(s){\big [}1-\sum _{c=1}^{[L/2]}(-1)^{c+1}{\bar {h}}(s,c;L)z^{i}{\big ]}^{-1}.}$

(9)

Setting ${\displaystyle z=1}$ in Eq. (9) gives ${\displaystyle {\bar {W}}_{1L}(s;L)}$. The Taylor expansion of Eq. (9) gives ${\displaystyle {\bar {w}}_{1L}(s,2n+\gamma _{1L};L)}$. The result follows:

${\displaystyle {\bar {w}}_{1L}(s,2n+\gamma _{1L};L)={\bar {\Gamma }}_{1L}(s)\sum _{k_{0}=a_{0,n}}^{n}{\bar {h}}(1,s;L)^{k_{0}}c_{k_{0}}(s;L).}$

(10)

In Eq. (10) ${\displaystyle {\bar {c}}_{k_{0}}(s;L)}$ is one for ${\displaystyle L=2,3}$, and otherwise,

${\displaystyle {\bar {c}}_{k_{0}}(s;L)=\prod _{c=0}^{[L/2]-1}\sum _{k_{c}=a_{c,n}}^{n-\sum _{j=0}^{c-1}k_{j}}{\bar {g}}_{k_{c}}(s;L),}$

(11)

where

${\displaystyle {\bar {g}}_{k_{i}}(s;L)={\binom {k_{i-1}}{k_{i}}}\left(-{\frac {{\bar {h}}(s,i+1;L)}{{\bar {h}}(s,i;L)}}\right)^{k_{i}}.}$

(12)

The initial number ${\displaystyle a_{i,n}s}$ follow:

${\displaystyle a_{i,n}=\left[{\frac {n-\sum _{j=0}^{i-1}k_{j}+[L/2]-1-i}{[L/2]-i}}\right];i>0,}$

(13)

and,

${\displaystyle a_{i,0}=\left[{\frac {n+[L/2]-1}{[L/2]}}\right].}$

(14)