User:Flomenbom/Heterogeneous random walk in 1d

Figure 1 A part of a semi-Markovian chain in 1D with directional JT-PDFs. A way for simulating such a random walk is when first drawing a random number out of a uniform distribution that determines the propagation direction according to the transition probabilities, and then drawing a random time out of the relevant JT-PDF.

Heterogeneous random walks in one dimension (also termed: a random walk in 1d, one-dimensional random walk, etc.) is the diffusion of a random walker in a one dimensional interval. The time is either discrete or continuous. The interval is also either discrete or continuous, and it is either finite or without bounds. In a discrete system, the connections are among adjacent states. The dynamics are either Markovian, Semi-Markovian, or even not-Markovian depending on the model. Heterogeneous random walks in 1d have jump probabilities that depend on the location in the system, and/or different jumping time (JT) probability density functions (PDFs) that depend on the location in the system.
Known important results in simple systems include:

• In a symmetric Markovian random walk, the Green's function (also termed the PDF of the walker) for occupying state i is a Gaussian in the position and has a variance that scales like the time. This is correct for a system with discrete time and space, yet also in a system with continuous time and space. This results is for systems without bounds.
• When there is a simple bias in the system (i.e. a constant force is applied on the system in a particular direction), the average distance of the random walker from its starting position is linear with time.
  
• When trying reaching a distance L from the starting position in a finite interval of length L, the time $\tau$ for reaching this distance is exponential with the length L: $\tau =e^{L}$
  

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 id defined with, ${\bar {f}}(s)=\int _{0}^{\infty }e^{-st}f(t)dt$). Here, the system is defined through the jumping time (JT) PDFs: $\psi _{ij}(t)$ 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:

${\bar {G}}_{ij}(s)={\bar {\Gamma }}_{ij}(s){\frac {{\bar {\Phi }}(s,{\tilde {L}})}{{\bar {\Phi }}(s,L)}}{\bar {\Psi }}_{i}(s).$ (1)

Here, ${\bar {\Psi }}_{i}(s)=\Sigma _{j}{\bar {\Psi }}_{ij}(s)$ and ${\bar {\Psi }}_{ij}(s)={\frac {1-{\bar {\psi }}_{ij}(s)}{s}}$. Also, in Eq. (1),

${\bar {\Gamma }}_{ij}(s)=\Pi _{c=0}^{i\mp 1}{\bar {\psi }}_{c\pm 1c}(s),$ (2)

and,

${\bar {\Phi }}(s,L)=1+\Sigma _{c=1}^{[L/2]}(-1)^{c}{\bar {h}}(s,c;L)$ (3)

with,

${\bar {h}}(s,i;L)=\Pi _{c=1}^{i}\Sigma _{k_{c}=2+k_{c-1}}^{L-1-2(i-c)}{\bar {f}}_{k_{c}}(s)$ (4)

and,

${\bar {f}}_{k_{j}}(s)={\bar {\psi }}_{k_{j}k_{j}+1}(s){\bar {\psi }}_{k_{j}+1k_{j}}(s).$ (5)

For L=1, ${\bar {\Phi }}(s;L)=1$. 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 $\Phi (s,{\tilde {L}})$ in eq. (1) has the same form as in ${\bar {\Phi }}(s;L)$ in eqs. (3)-(5) , yet it is calculated on a lattice ${\tilde {L}}$ . Lattice ${\tilde {L}}$ 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 ${\tilde {L}}$ is the longer fragment. When each fragment is a single state, ${\bar {\Phi }}(s;{\tilde {L}})=1$.

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.

Introduction

Random walks appear in the description of a wide variety of processes in biology, chemistry and physics. Enzymatic activity , chemical kinetics and polymer dynamics are familiar relevant examples. In the evolving field of individual molecules (see all the references from and on), random walks supply the natural platform for describing the data, which is made of stochastic events. Importantly, PDFs and special correlation functions can be easily calculated from single molecule measurements but not from ensemble measurements. This unique information can be used for discriminating between distinct random walk models that share some properties , and this demands a detailed theoretical analysis of random walk models. In this context, utilizing the information content in single molecule data is a matter of ongoing research.

Formulation of random walks can be done in terms of the discrete (in space) master equation and the generalized master equation or the continuum (in space and time) Fokker Planck equation and its generalizations . Continuous time random walks , renewal theory , and the path representation are also useful formulations of random walks. The network of relationships between the various descriptions provides a powerful tool in the analysis of random walks. Arbitrarily heterogeneous environments make the analysis difficult, especially in high dimensions. In 1D, however, the solution for the Green's function $G_{ij}(t;L)$ for a semi-Markovian random walk in an arbitrarily heterogeneous environment in 1D was recently given using the path representation . ($G_{ij}(t;L)$ is the PDF for occupying state i at time t given that the process started at state j exactly at time 0.) We define a semi-Markovian random walk in 1D as follow: a random walk whose dynamics are described by the (possibly) state- and direction-dependent JT-PDFs, $\psi _{ij}(t)$, for transitions between states i and i±1, that generates stochastic trajectories of uncorrelated waiting times that are not-exponential distributed. $\psi _{ij}(t)$ obeys the normalization conditions, $\Sigma _{j}\int _{0}^{\infty }\psi _{ij}(t)=1$ (see fig. 1). The dynamics can also include state- and direction-dependent irreversible trapping JT-PDFs, $\psi _{iI}(t)$, with I=i+L. The environment is heterogeneous when $\psi _{ij}(t)$ depends on i. The above process is also a continuous time random walk and has an equivalent generalized master equation representation for the Green's function $G_{ij}(t)$ .

Path representation

Clearly, ${\bar {G}}_{ij}(s;L)$ 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 ${\bar {G}}_{ij}(s;L)$ also manifests its utility when numerical inversion of the generalized master equation is unstable. Moreover, using ${\bar {G}}_{ij}(s;L)$ in simple analytical manipulations gives , (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 $G_{ij}(t)$, and this supplies further information on the process. The path representation follows:

$G_{ij}(t;L)=\int _{0}^{\infty }\Psi _{i}(t-\tau )W_{ij}(\tau ;L)d\tau ,$ (6)

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

$W_{ij}(\tau ;L)=\Sigma _{n=0}^{\infty }w_{ij}(\tau ,2n+\gamma _{ij};L).$ (7)

$W_{ij}(t;L)$ 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 $2n+\gamma _{ij}$ transitions that connect states j with i. Two different path types contribute to $w_{ij}(\tau ,2n+\gamma _{ij};L)$ : paths made of the same states appearing in different orders and different paths of the same length of $2n+\gamma _{ij}$ 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, ${\bar {W}}_{ij}(s;L)={\bar {W}}_{1L}(s;L)/{\bar {W}}_{1{\tilde {L}}}(s;{\tilde {L}})$. Using this relation, we focus in what follows on solving ${\bar {w}}_{1L}(s;L)$.

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 . Also, analytical properties of the Green’s function are clarified only in path PDF analysis. Here, presented is the recursion relation for $w_{ij}(\tau ,2n+\gamma _{ij};L)$ in the length n of path PDFs for any fixed value of L. The recursion relation is linear in path PDFs with the ${\bar {h}}(s,i;L)$s in Eq. (5) serving as the n independent coefficients, and is of order [L / 2]:

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

${\hat {\bar {w}}}_{1L}(s,2z+\gamma _{1L};L)=\Sigma _{n=0}^{\infty }{\bar {w}}_{1L}(s,2n+\gamma _{1L};L)z^{n}={\bar {\Gamma }}_{1L}(s){\big [}1-\Sigma _{c=1}^{[L/2]}(-1)^{c+1}{\bar {h}}(s,c;L)z^{i}{\big ]}^{-1}.$ (9)

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

${\bar {w}}_{1L}(s,2n+\gamma _{1L};L)={\bar {\Gamma }}_{1L}(s)\Sigma _{k_{0}=a_{0,n}}^{n}{\bar {h}}(1,s;L)^{k_{0}}c_{k_{0}}(s;L).$ (10)

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

${\bar {c}}_{k_{0}}(s;L)=\Pi _{c=0}^{[L/2]-1}\Sigma _{k_{c}=a_{c,n}}^{n-\Sigma _{j=0}^{c-1}k_{j}}{\bar {g}}_{k_{c}}(s;L),$ (11)

where,

${\bar {g}}_{k_{i}}(s;L)={\binom {k_{i-1}}{k_{i}}}{\big (}-{\frac {{\bar {h}}(s,i+1;L)}{{\bar {h}}(s,i;L)}}{\big )}^{k_{i}}.$ (12)

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

$a_{i,n}={\big [}{\frac {n-\Sigma _{j=0}^{i-1}k_{j}+[L/2]-1-i}{[L/2]-i}}{\big ]};i>0,$ (13)

and,

$a_{i,0}={\big [}{\frac {n+[L/2]-1}{[L/2]}}{\big ]}.$ (14)

References

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