Random Walk - Variants of Random Walks - Heterogeneous Random Walks in One Dimension

Heterogeneous Random Walks in One Dimension

Heterogeneous random walks in one dimension can have either discrete time or continuous time. 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 result holds in a system with discrete time and space, yet also in a system with continuous time and space. This result 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 with a constant force, the time for reaching this distance is exponential with the length L:, when moving against the force, and is linear with the length L:, when moving with the force. Without force: .

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, ). Here, the system is defined through the jumping time (JT) PDFs: 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, and . Also, in Eq. (1), $bar{Gamma}_{ij}(s)=Pi_{c=0}^{imp1}bar{psi}_{cpm 1c}(s),$

(2)

and, $bar{Phi}(s,L)=1+Sigma_{c=1}^{}(-1)^cbar{h}(s,c;L)$

(3)

with, $bar{h}(s,i;L)=Pi_{c=1}^iSigma_{k_c=2+k_{c-1}}^{L-1-2(i-c)}bar{f}_{k_c}(s)$

(4)

and, $bar{f}_{k_j}(s)=bar{psi}_{k_jk_j+1}(s)bar{psi}_{k_j+1k_j}(s).$

(5)

For L=1, . The symbol that appears in the upper bound in the in eq. (3) is the floor operation (round towards zero). Finally, the factor in eq. (1) has the same form as in in eqs. (3)-(5), yet it is calculated on a lattice . Lattice 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 is the longer fragment. When each fragment is a single state, .

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.