About: Heterogeneous random walk in one dimension

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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.

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rdfs:label	Heterogeneous random walk in one dimension (en)
rdfs:comment	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. (en)
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dcterms:subject	Statistical mechanics Variants of random walks
Wikipage page ID	31907287 (xsd:integer)
Wikipage revision ID	1100285570 (xsd:integer)
Link from a Wikipage to another Wikipage	Quantum dots Uniform distribution (continuous) Dynamics (mechanics) Statistical mechanics Chemical kinetics Green's function Variants of random walks Physics Master equation Fokker–Planck equation Probability density function Probability Renewal theory Chemistry Markov process Random walk Stochastic differential equation Semi-Markov process
sameAs	Heterogeneous random walk in one dimension Heterogeneous random walk in one dimension Heterogeneous random walk in one dimension
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has abstract	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, . 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-, presented in what follows. (en)
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