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Lecture 6: Langevin equations

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

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Lecture 6: Langevin equations

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  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 noiseξ(t):

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

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

  6. Stochastic differential equations Differential equations which contain (“are driven by”) random functions Langevin equations: the random function is Gaussian white noiseξ(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 noiseξ(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 noiseξ(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 solution:

  30. Solution using Fourier transform solution:

  31. Solution using Fourier transform solution:

  32. Solution using Fourier transform solution:

  33. Solution using Fourier transform solution: inverse FT:

  34. Solution using Fourier transform solution: inverse FT:

  35. Solution using Fourier transform solution: inverse FT:

  36. Solution using Fourier transform solution: inverse FT:

  37. Solution using Fourier transform solution: 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:

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