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Random walks with waiting times depending on the preceding jump length

Random walks with waiting times depending on the preceding jump length. V.Yu.Zaburdaev MPI for Dynamics and Self-Organization, Göttingen, Germany. Standard Random Walk Model. x. x. x+x’. The probability to jump into. is. The probability to jump away at. is.

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Random walks with waiting times depending on the preceding jump length

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  1. Random walks with waiting times depending on the preceding jump length V.Yu.Zaburdaev MPI for Dynamics and Self-Organization, Göttingen, Germany

  2. Standard Random Walk Model x x x+x’ The probability to jump into is The probability to jump away at is The probability to “survive” until t :

  3. Generalized Random Walk Model x x-y x x+x’ The probability to jump into is The probability to jump away at, provided it arrived from distance The probability to “survive” until t :

  4. Coupled transition kernel “Physiological” example: For a longer jump, more time is necessary to recover Mean resting time is a function of a jump distance In the simplest case analogy with M.F.Shlesinger, J.Klafter et al. (1982,1987), E.Barkai (2002), M.Meerschaert et al. (2002), S.A.Trigger et al. (2005)

  5. Microscopic details In the given point (x,t) there are particles which arrived there at different times and from different points, therefore they will fly away also at different times. (1) The outgoing flow of particles from a given point is not a simple function of the concentration alone. (2) (3)

  6. Coupled transition kernel (equations)

  7. Coupled transition kernel (equations II) “Finite velocity” Green’s function with

  8. Finite velocity x x x+x’ G.Zumofen, J.Klafter et al. (1993), V.Yu.Zaburdaev, K.V.Chukbar (2002),E.Barkai (2002), M.Meerschaert et al. (2002), I.M.Sokolov, R.Metzler (2003)

  9. Back to the coupled model

  10. Coupling & finite velocity together

  11. Conclusions • Introduction of microscopic details is necessary for the understanding and correct description of CTRW • They allow e.g. to solve a problem of coupled transition kernel and take into account the finite velocity of walking particles simultaneously • These features open a possibility for applications in biological problems where recovery processes and finite velocity of motion are presented: foraging movements of animals, motion of zooplankton, … V.Yu.Zaburdaev, J.Stat.Phys. (on-line first) (2006)

  12. Thank you for attention!

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