Modeling Stochastic phenomena for Engineering applications: Part-1

Lectures

week-01

1. Week1: Lecture1: Stirling’s Approximation Video
2. Week 1: Lecture 2: Fourier Transforms and characteristic function. Video
3. Week 1: Lecture 3: Dirac Delta function Video
4. Week 1: Lecture 4: Applications of delta function and Generating functions. Video
5. Week 1: Lecture 5: Laplace Transforms & Convolution theorem Video

week-02

6. Week 2: Lecture 6: Generating function for discrete variables and Binomial distribution Video
7. Week 2: Lecture 7: Bernoulli and Poisson distributions Video
8. Week 2: Lecture 8: Waiting time distributions; Gaussian approximation to Poisson distribution Video
9. Week 2: Lecture 9: Introduction to Central Limit Theorem Video
10. Week 2: Lecture 10: Proof of Central Limit Theorem (CLT) Video

week-03

11. Week 3: Lecture 11: Universality of Normal distribution and Exceptions Video
12. Week 3: Lecture 12: Introduction to Random Walk: Extension of Central Limit Theorem Video
13. Week 3: Lecture 13: Random walk and Diffusion coefficient: Conditional and Transition Video

week-04

14. Week 4: Lecture 14: Characteristics of Stochastic Phenomena: Markov Processes Video
15. Week 4: Lecture 15: Propagating Markov processes via Transition Probability Matrix with Video
16. Week 4: Lecture 16: Chapman-Kolmogorov Equation for Multistep Transition probability and solution... Video
17. Week 4: Lecture 17: Transient solutions and Continuous time Markov process Video

week-05

18. Week 5: Lecture 18: Exact solution to Symmetric (or unbiased) one-dimensional Random walk (1-D RW).. Video
19. Week 5: Lecture 19: Properties of the solution for 1-D unbiased RW Video
20. Week 5: Lecture 20: 1-D unbiased RW: Asymptotic form of occupancy probability and transition ... Video

week-06

21. Week 6: Lecture 21: Solution to the problem of 1-D Random Walk with Bias Video
22. Week 6: Lecture 22: Generalized Random Walk with Bias and Pausing Video
23. Week 6: Lecture 23: Effect of Pausing on Mean and Variance of Random walk Video
24. Week 6: Lecture 24: Random-walk in the presence of reflecting barrier Video
25. Week 6: Lecture 25: Boundary conditions for reflected Random-Walk and formulating absorbing ... Video
26. Week 6: Lecture 26: The survival probability and first-passage time distribution for Random walker.. Video
27. Week 6: Lecture 27: Random Walk with Bias and Absorber Video
28. Week 6: Lecture 28: Drift and Survival probability for Random walk with bias and absorber Video

week-07

29. Week 7: Lecture 29: Introduction to gambler’s ruin problem Video
30. Week 7: Lecture 30: Solution for ultimate winning probability in Gambler’s ruin problem Video
31. Week 7: Lecture 31: Solution to gambler’s ruin problem with site dependent jump probabilities Video
32. Week 7: Lecture 32: Fourier transform method of solving lattice Random walks Video
33. Week 7: Lecture 33: Two and higher dimensional Random walks Video
34. Week 7: Lecture 34: Formulating the problem of Probability of Return to the origin Video
35. Week 7: Lecture 35: Relationship between occupancy probability and first-time-return probability Video

week-08

36. Week 8: Lecture 36: Proof of Polya’s theorem on the probability of return Video
37. Week 8: Lecture 37: Return probability estimates in various dimensions and effect of bias in 1-D Video
38. Week 8: Lecture 38: Dependence of first time return probability (Fk) on steps Video
39. Week 8: Lecture 39: Equilibrium solutions in lattice random walk models Video
40. Week 8: Lecture 40: Equilibrium solution to Ehrenfest’s flea model Video

week-09

41. Week 9: Lecture 41: Differential equation formulation of stochastic phenomena Video
42. Week 9: Lecture 42: Derivation of Fokker-Planck equation Video
43. Week 9: Lecture 43: Generalized transition probability functions for Fokker-Planck equation Video
44. Week 9: Lecture 44: Solution to 1-D Fokker-Planck equation for free particle: Method of Fourier... Video
45. Week 9: Lecture 45: General non-gaussian solution to translationally invariant Chapman-Kolmogorov... Video

week-10

46. Week 10: Lecture 46: Cauchy distribution, power-law and other non-gaussian solutions Video
47. Week 10: Lecture 47: Wiener process and solution to absorbing barrier problems from Fokker-Planck... Video
48. Week 10: Lecture 48: Application of Fourier Sine transform for single absorber problem Video
49. Week 10: Lecture 49: Setting up Langevin equation for velocity fluctuations of Brownian particles Video
50. Week 10: Lecture 48: Application of Fourier Sine transform for single absorber problem Video
51. Week 10: Lecture 49: Setting up Langevin equation for velocity fluctuations of Brownian particles Video
52. Week 10: Lecture 50: Understanding the origin of systematic and random parts of force from kinetic.. Video

week-11

53. Week 11: Lecture 51: Kinetic derivation of a formula for delta-correlated random force Video
54. Week 11: Lecture 52: Mean square velocity, thermal equilibrium and relationship between relaxation.. Video
55. Week 11: Lecture 53: Velocity autocorrelation in Brownian motion Video
56. Week 11: Lecture 54: Derivation of Stokes-Einstein relationship between diffusion coefficient and... Video
57. Week 11: Lecture 55: Alternative derivation of Stokes-Einstein relationship & Brownian motion with.. Video

week-12

58. Week 12: Lecture 56: Numerical simulation of the Langevin equation Video
59. Week 12: Lecture 57: Derivation of Klein-Kramers equation from Langevin equation for joint... Video
60. Week 12: Lecture 58: Illustrative solutions to the Klein-Kramers equation Video
61. Week 12: Lecture 59: Numerical simulation: Sampling from general distributions and Central... Video
62. Week 12: Lecture 60: Numerical simulation of Random walk trajectories and method of solving Fokker.. Video

Live Session

63. Live Session 05-03-2024 Video