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← RandWalk 1.3 TypesofRandomWalks

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Showing Revision 4 created 01/23/2016 by rintu kutum.

  1. There are many types of random walks, and
  2. I hope... few of them, so you have and idea
  3. of the richness.
  4. First is the Pearson random walk, in which
  5. each step is a fix length, but in a random
  6. direction. What I am showing here, is a
  7. typical trajectory of such a Pearson random walk.
  8. Another example is a Lattice random walk,
  9. in which the random walks are constraint
  10. to move between nearest neighbour sites of
  11. some regular lattice. So, here the steps
  12. are fixed length and the direction are
  13. either in north, east, south or west.
  14. Another type of random walk is so called
  15. Levy flight. In the Levy flight, there is a
  16. broad distribution of single step lengths
  17. but each step is in random direction.
  18. Here, we will see that the displacement
  19. after many steps can be dominated by the
  20. longest single step of the walk.
  21. Another example that dear to my heart is
  22. the example of Shrinking steps that is a
  23. random walker getting lazier and lazier as
  24. time is going on and the length of nth-step
  25. is landed to end where lambda is less than 1.
  26. An amazing aspect of this type of random
  27. walk is diversity of type of probability
  28. distributions as a function of the Shrinking
  29. factor lambda (λ)
  30. Know the λ = 0.61, in fact most precisely
  31. is the golden ratio, (1 + sqrt(5))/2, the
  32. probability distribution is beautiful, self
  33. similar pattern that repeats on all scales
  34. so the middle blob is same as the entire
  35. distribution in inside the middle blob is
  36. something which reproduces the entire
  37. distribution again
  38. Another interesting special case is λ= 0.707 which is actual 1/sqrt(2)
  39. Here the probability distribution is
  40. made up of 3 linear segments, two tilted
  41. lines and one flat line. And there are many
  42. other beautiful special cases of this type
  43. of random walk Shrinking steps
  44. Another important example that appears in
  45. nature, turbulent diffusion or random walks
  46. that are moving in random convection field
  47. In this case, the typical step of length
  48. of random walk is a growing with time.
  49. And one can get plume like behaviour as you
  50. see here from smoke rising from oil fires
  51. in the ocean
  52. Here are the types of random walks we have
  53. just discussed. As we will see, the first 4-types
  54. lie in the domain of celebrated central limit theorem
  55. in which the probability distribution is
  56. asymptotically a Gaussian, independent of
  57. details of the microscopic motion. This
  58. universality is extremely useful principle
  59. in many collective phenomena