Heterogeneous random walk in one dimension

In studies of 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. G ¯ i j ( s ) = Γ ¯ i j ( s ) Φ ¯ ( s , L ~ ) Φ ¯ ( s , L ) Ψ ¯ i ( s ) . {displaystyle {ar {G}}_{ij}(s)={ar {Gamma }}_{ij}(s){frac {{ar {Phi }}(s,{ ilde {L}})}{{ar {Phi }}(s,L)}}{ar {Psi }}_{i}(s).}     (1) Γ ¯ i j ( s ) = ∏ c = 0 i ∓ 1 ψ ¯ c ± 1 c ( s ) , {displaystyle {ar {Gamma }}_{ij}(s)=prod _{c=0}^{imp 1}{ar {psi }}_{cpm 1c}(s),}     (2) Φ ¯ ( s , L ) = 1 + ∑ c = 1 [ L / 2 ] ( − 1 ) c h ¯ ( s , c ; L ) {displaystyle {ar {Phi }}(s,L)=1+sum _{c=1}^{}(-1)^{c}{ar {h}}(s,c;L)}     (3) h ¯ ( s , i ; L ) = ∏ c = 1 i ∑ k c = 2 + k c − 1 L − 1 − 2 ( i − c ) f ¯ k c ( s ) {displaystyle {ar {h}}(s,i;L)=prod _{c=1}^{i}sum _{k_{c}=2+k_{c-1}}^{L-1-2(i-c)}{ar {f}}_{k_{c}}(s)}     (4) f ¯ k j ( s ) = ψ ¯ k j k j + 1 ( s ) ψ ¯ k j + 1 k j ( s ) . {displaystyle {ar {f}}_{k_{j}}(s)={ar {psi }}_{k_{j}k_{j}+1}(s){ar {psi }}_{k_{j}+1k_{j}}(s).}     (5) G i j ( t ; L ) = ∫ 0 ∞ Ψ i ( t − τ ) W i j ( τ ; L ) d τ , {displaystyle G_{ij}(t;L)=int _{0}^{infty }Psi _{i}(t- au )W_{ij}( au ;L),d au ,}     (6) W i j ( τ ; L ) = ∑ n = 0 ∞ w i j ( τ , 2 n + γ i j ; L ) . {displaystyle W_{ij}( au ;L)=sum _{n=0}^{infty }w_{ij}( au ,2n+gamma _{ij};L).}     (7) w ¯ 1 L ( s , 2 n + γ 1 L ; L ) = δ n 0 Γ ¯ 1 L ( s ) + ∑ c = 1 [ L / 2 ] ( − 1 ) c + 1 w ¯ 1 L ( s , 2 ( n − c ) + γ 1 L ; L ) h ¯ ( s , c ; L ) . {displaystyle {ar {w}}_{1L}(s,2n+gamma _{1L};L)=delta _{n0}{ar {Gamma }}_{1L}(s)+sum _{c=1}^{}(-1)^{c+1}{ar {w}}_{1L}(s,2(n-c)+gamma _{1L};L){ar {h}}(s,c;L).}     (8) w ¯ ^ 1 L ( s , 2 z + γ 1 L ; L ) = ∑ n = 0 ∞ w ¯ 1 L ( s , 2 n + γ 1 L ; L ) z n = Γ ¯ 1 L ( s ) [ 1 − ∑ c = 1 [ L / 2 ] ( − 1 ) c + 1 h ¯ ( s , c ; L ) z i ] − 1 . {displaystyle {hat {ar {w}}}_{1L}(s,2z+gamma _{1L};L)=sum _{n=0}^{infty }{ar {w}}_{1L}(s,2n+gamma _{1L};L)z^{n}={ar {Gamma }}_{1L}(s){ig }(-1)^{c+1}{ar {h}}(s,c;L)z^{i}{ig ]}^{-1}.}     (9) w ¯ 1 L ( s , 2 n + γ 1 L ; L ) = Γ ¯ 1 L ( s ) ∑ k 0 = a 0 , n n h ¯ ( 1 , s ; L ) k 0 c k 0 ( s ; L ) . {displaystyle {ar {w}}_{1L}(s,2n+gamma _{1L};L)={ar {Gamma }}_{1L}(s)sum _{k_{0}=a_{0,n}}^{n}{ar {h}}(1,s;L)^{k_{0}}c_{k_{0}}(s;L).}     (10) c ¯ k 0 ( s ; L ) = ∏ c = 0 [ L / 2 ] − 1 ∑ k c = a c , n n − ∑ j = 0 c − 1 k j g ¯ k c ( s ; L ) , {displaystyle {ar {c}}_{k_{0}}(s;L)=prod _{c=0}^{-1}sum _{k_{c}=a_{c,n}}^{n-sum _{j=0}^{c-1}k_{j}}{ar {g}}_{k_{c}}(s;L),}     (11) g ¯ k i ( s ; L ) = ( k i − 1 k i ) ( − h ¯ ( s , i + 1 ; L ) h ¯ ( s , i ; L ) ) k i . {displaystyle {ar {g}}_{k_{i}}(s;L)={inom {k_{i-1}}{k_{i}}}left(-{frac {{ar {h}}(s,i+1;L)}{{ar {h}}(s,i;L)}} ight)^{k_{i}}.}     (12) a i , n = [ n − ∑ j = 0 i − 1 k j + [ L / 2 ] − 1 − i [ L / 2 ] − i ] ; i > 0 , {displaystyle a_{i,n}=left-1-i}{-i}} ight];i>0,}     (13) a i , 0 = [ n + [ L / 2 ] − 1 [ L / 2 ] ] . {displaystyle a_{i,0}=left-1}{}} ight].}     (14) In studies of 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 ightarrow 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. Random walks appear in the description of a wide variety of processes in biology, chemistry and physics. Random walks are used in describing chemical kinetics and polymer dynamics. In the evolving field of individual molecules, random walks supply the natural platform for describing the data. Namely, we see random walks when looking on individual molecules, individual channels, individual biomolecules, individual enzymes, 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. The actual random walk obeys a stochastic equation of motion. Yet, the probability density function (PDF) obeys a deterministic equation of motion. Formulation of PDFs 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.