1. Lecture 6: Langevin equations
Outline:
linear/nonlinear, additive and multiplicative noise
soluble linear example w/ additive noise: Ornstein-Uhlenbeck process
general 1-d nonlinear equation with multiplicative noise
relation to Fokker-Planck equation
Ito formulation, relation between Ito & Stratonovich approaches

2. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions

3. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):

4. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):

5. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
etc.

6. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):

7. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):
Solution v(t) is random (because it depends on ξ(t))

8. Stochastic differential equations
Differential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):
Solution v(t) is random (because it depends on ξ(t))
Want to know P[v], averages over distribution of ξ(t)

9. More generally,
multivariate:

10. More generally,
multivariate:
higher-order:

11. More generally,
multivariate:
higher-order:
nonlinear:

12. More generally, multivariate: higher-order: nonlinear: multiplicative
noise:

13. Brownian motion

14. Brownian motion
solution (with m = 1):

15. Brownian motion
solution (with m = 1):
averages:

16. Brownian motion
solution (with m = 1):
averages:

17. Brownian motion
solution (with m = 1):
averages:

18. Brownian motion
solution (with m = 1):
averages:

19. Brownian motion
solution (with m = 1):
averages:

20. Brownian motion
solution (with m = 1):
averages:

21. Brown (2)
equal-time correlation:

22. Brown (2)
equal-time correlation:
but from equilibrium stat mech:

23. Brown (2)
equal-time correlation:
but from equilibrium stat mech:

24. Brown (2) equal-time correlation: but from equilibrium stat mech:
(another Einstein relation)

25. Brown (2) equal-time correlation: but from equilibrium stat mech:
(another Einstein relation)
Note: OU model also applies with v -> x (position) to overdamped
motion in a parabolic potential

26. Solution using Fourier transform

27. Solution using Fourier transform

28. Solution using Fourier transform

29. Solution using Fourier transform

30. Solution using Fourier transform

31. Solution using Fourier transform

32. Solution using Fourier transform

33. Solution using Fourier transform
inverse FT:

34. Solution using Fourier transform
inverse FT:

35. Solution using Fourier transform
inverse FT:

36. Solution using Fourier transform
inverse FT:

37. Solution using Fourier transform
inverse FT:
(as in direct calculation)

38. Damped oscillator

39. Damped oscillator
FT:

40. Damped oscillator
FT:

41. Damped oscillator
FT:
inverse FT:

42. General OU process

43. General OU process
damped oscillator:

44. General OU process
damped oscillator:

45. General OU process
damped oscillator:
Is a 2-d OU process with

46. General OU process damped oscillator: Is a 2-d OU process with
x(t) is not a Markov process (2nd order equation),
but (x(t),p(t)) is (1st order equation).

47. Formal solution by FT

48. Formal solution by FT

49. Formal solution by FT

50. Formal solution by FT
damped oscillator case:

51. Formal solution by FT
damped oscillator case:

52. Formal solution by FT
damped oscillator case:

53. General 1-d Langevin equation
nonlinear:

54. General 1-d Langevin equation
nonlinear:
ex: overdamped pendulum

55. General 1-d Langevin equation
nonlinear:
ex: overdamped pendulum
with multiplicative noise

56. General 1-d Langevin equation
nonlinear:
ex: overdamped pendulum
with multiplicative noise
ex: geometric Brownian
motion

57. Fokker-Planck for nonlinear case

58. Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,

59. Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,

60. Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
in terms of Kramers-Moyal expansion coefficients,

61. Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
in terms of Kramers-Moyal expansion coefficients,
=> FP equation

62. Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
in terms of Kramers-Moyal expansion coefficients,
=> FP equation
(FP equation is still linear, though Langevin equation is nonlinear)

63. with multiplicative noise

64. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x

65. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?

66. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting

67. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention

68. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention
An alternative approach is to take
and take the limit τ -> 0 at the end.

69. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention
An alternative approach is to take
and take the limit τ -> 0 at the end. This is
the Stratonovich convention

70. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention
An alternative approach is to take
and take the limit τ -> 0 at the end. This is
the Stratonovich convention
It is equivalent to evaluating G at the midpoint of the jump.

71. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention
An alternative approach is to take
and take the limit τ -> 0 at the end. This is
the Stratonovich convention
It is equivalent to evaluating G at the midpoint of the jump.
With multiplicative noise, these 2 conventions lead to different
FP equations

72. with multiplicative noise
But ξ(t) is composed of δ-functions
each δ-function causes a jump in x
is the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
the Ito convention
An alternative approach is to take
and take the limit τ -> 0 at the end. This is
the Stratonovich convention
It is equivalent to evaluating G at the midpoint of the jump.
With multiplicative noise, these 2 conventions lead to different
FP equations. (For additive noise, they are 2 different ways to do
the problem but must give the same answer.)

73. FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):

74. FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):

75. FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
Stratonovich (it can be shown that):

76. FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
Stratonovich (it can be shown that):
or,
equivalently,

77. FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
Stratonovich (it can be shown that):
or,
equivalently,
 “anomalous drift”

78. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

79. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

80. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

81. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

82. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

83. Where does the anomalous drift come from?
From Stratonovich midpoint prescription:

84. formalism using differentials
SDE is written

85. formalism using differentials
SDE is written

86. formalism using differentials
SDE is written

87. formalism using differentials
SDE is written

88. formalism using differentials
SDE is written
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)?
Expand:

89. formalism using differentials
SDE is written
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)?
Expand:

90. formalism using differentials
SDE is written
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)?
Expand:

91. formalism using differentials
SDE is written
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)?
Expand:
______________
“because dW = O(Δt)”