Heterogeneous random walk in one dimension

In dynamics, probability, physics, chemistry and related fields, a heterogeneous random walk in one dimension is a random walk in a one dimensional interval with jumping rules that depend on the location of the random walker in the interval. For example: say that the time is discrete and also the interval. Namely, the random walker jumps every time step either left or right. A possible heterogeneous random walk draws in each time step a random number that determines the local jumping probabilities and then a random number that determines the actual jump direction. Specifically, say that the interval has 9 sites (labeled 1 through 9), and the sites (also termed states) are connected with each other linearly (where the edges sites are connected their adjacent sites and together). In each time step, the jump probabilities (from the actual site) are determined when flipping a coin; for head we set: probability jumping left =1/3, where for tail we set: probability jumping left = 0.55. Then, a random number is drawn from a uniform distribution: when the random number is smaller than probability jumping left, the jump is for the left, otherwise, the jump is for the right. Usually, in such a system, we are interested in the probability of staying in each of the various sites after t jumps, and in the limit of this probability when t is very large, t → ∞ {\displaystyle t\rightarrow \infty } . Generally, the time in such processes can also vary in a continuous way, and the interval is also either discrete or continuous. Moreover, the interval is either finite or without bounds. In a discrete system, the connections are among adjacent states. The basic dynamics are either Markovian, semi-Markovian, or even not Markovian depending on the model. In discrete systems, 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. General solutions for heterogeneous random walks in 1d obey equations (1)-(5), presented in what follows.

Introduction

Random walks in applications Random walks can be used to describe processes in biology, chemistry, and physics, including chemical kinetics and polymer dynamics. In individual molecules, random walks appear when studying individual molecules, individual channels, individual biomolecules, individual enzymes, and quantum dots. 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.

Formulations of random walks The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF) obeys a deterministic equation. PDFs of random walks can be formulated in terms of the (discrete in space) master equation and the generalized master equation or the (continuous 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.

Results for random walks in one dimension

Simple systems 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. These 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 to reach a distance L from the starting position in a finite interval of length L, the time τ {\displaystyle \tau } for reaching this distance is exponential with the length L: τ = e L {\displaystyle \tau =e^{L}} . Here, the diffusion is against a linear potential.

Heterogeneous systems The solution for the Green's function G i j ( t ; L ) {\displaystyle G_{ij}(t;L)} for a semi-Markovian random walk in an arbitrarily heterogeneous environment in 1D was recently given using the path representation. (The function G i j ( t ; L ) {\displaystyle 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.) A semi-Markovian random walk in 1D is defined as follows: a random walk whose dynamics are described by the (possibly) state- and direction-dependent JT-PDFs, ψ i j ( t ) {\displaystyle \psi _{ij}(t)} , for transitions between states i and i ± 1, that generates stochastic trajectories of uncorrelated waiting times that are not-exponential distributed. ψ i j ( t ) {\displaystyle \psi _{ij}(t)} obeys the normalization conditions (see fig. 1)

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