1. Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process

2. anomalous diffusion Recall derivation of Fokker-Planck equation:

3. anomalous diffusion Recall derivation of Fokker-Planck equation:

4. anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ?

5. anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ? And what if the distribution of time steps has infinite mean?

6. anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ? And what if the distribution of time steps has infinite mean? Go back and reformulate the problem:

7. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t)

8. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t)

9. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t) Let η(x,t) = probability density of x at a t right after a jump

10. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t) Let η(x,t) = probability density of x at a t right after a jump

11. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t) Let η(x,t) = probability density of x at a t right after a jump

12. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t) Let η(x,t) = probability density of x at a t right after a jump Then

13. continuous-time random walk distribution of jump sizes r(x), distribution of waiting times w(t) can have joint distribution ψ(x,t) ; here, ψ(x,t) = r(x)w(t) Let η(x,t) = probability density of x at a t right after a jump Then ______________  prob to survive from t ’ to t without a jump

14. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time:

15. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time:

16. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

17. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

18. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

19. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

20. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

21. Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

22. Fourier-Laplace inversion 2 ways:( D = 1 )

23. Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

24. Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

25. Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

26. Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

27. Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

28. other way: 2. Invert the Fourier transform first:

29. other way: 2. Invert the Fourier transform first:

30. other way: 2. Invert the Fourier transform first:

31. other way: 2. Invert the Fourier transform first:

32. other way: 2. Invert the Fourier transform first:

33. anomalous diffusion: long waiting times:

34. anomalous diffusion: long waiting times: long jumps:

35. anomalous diffusion: long waiting times: long jumps: =>

36. anomalous diffusion: long waiting times: long jumps: =>

37. anomalous diffusion: long waiting times: long jumps: =>

38. anomalous diffusion: long waiting times: long jumps: =>

39. Subdiffusion: long wait time distribution

40. Invert Fourier transform first:

41. Subdiffusion: long wait time distribution Invert Fourier transform first:

42. Subdiffusion: long wait time distribution Invert Fourier transform first:

43. Subdiffusion: long wait time distribution Invert Fourier transform first:

44. Subdiffusion: long wait time distribution Invert Fourier transform first:

45. Subdiffusion: long wait time distribution Invert Fourier transform first:

46. Subdiffusion: long wait time distribution Invert Fourier transform first:

47. Subdiffusion: long wait time distribution Invert Fourier transform first: α < 1 : subdiffusion

48. long-tailed jump distribution: ( α = 1, σ < 2 )

49. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

50. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

51. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

52. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ

53. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ

54. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ

55. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ),

56. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞.

57. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞. fractional moments:

58. long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞. fractional moments:

59. Fractional Wiener process For an ordinary Wiener process,

60. Fractional Wiener process For an ordinary Wiener process, How can we get ?

61. Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider

62. Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then

63. Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then Laplace-transformed:

64. Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then Laplace-transformed: so choose

65. fractional derivatives

68. or

69. fractional derivatives or

70. fractional derivatives or i.e., or

71. fractional derivatives or i.e., or nonlocal!