Modeling Stochastic phenomena for Engineering applications: Part-1

By Prof. Yelia Shankaranarayana Mayya | IIT Bombay

Learners enrolled: 82 | Exam registration: 2

ABOUT THE COURSE:
Mechanics deals with deterministic laws to describe phenomena. However, the real world is replete with examples involving randomness. Brownian Motion, chemical reactions, pandemic propagation, eco-dynamics, material aggregation, nucleation, weather and climate, market fluctuations are some of the examples where randomness plays a key role. While the classical probability theory deals with characterising the random events: i.e., how to describe outcomes, what are the distributions etc, it does not deal with dynamical evolution of these probabilities. In contrast stochastic processes deal with the temporal evolution of the outcome of random events. In these lectures, we learn how to choose the variables, how to formulate the problems, what underlying assumptions are to be made and how best one can extract useful information in evolving probabilistic systems. We explore these aspects from an engineering, rather than a from formal theoretical perspective, by limiting ourselves to physical systems.

INTENDED AUDIENCE: B.Tech/BE/Masters/Ph.D in Chemical, Mechanical, Electrical and Environmental engineering, Physics.

PREREQUISITES: Probability theory, Integral transforms, differential equations, Mathematical methods

INDUSTRY SUPPORT: Research Organizations such as BARC, IGCAR

Summary

Course Status :	Upcoming
Course Type :	Elective
Language for course content :	English
Duration :	12 weeks
Category :	Chemical Engineering
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	20 Jan 2025
End Date :	11 Apr 2025
Enrollment Ends :	27 Jan 2025
Exam Registration Ends :	14 Feb 2025
Exam Date :	26 Apr 2025 IST

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

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

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

Week 3:
Lecture - 11: Universality of Normal distribution and Exceptions
Lecture - 12: Introduction to Random Walk: Extension of Central Limit Theorem
Lecture - 13: Random walk and Diffusion coefficient: Conditional and Transition probabilities
Lecture - 14: Characteristics of Stochastic Phenomena: Markov Processes
Lecture - 15: Examples of Propagating the Markov process via Transition probability matrix

Week 4:
Lecture - 16: Chapman-Kolmogorov Equation for Multistep Transition probability and solution methods
Lecture - 17: Transient solutions and Continuous time Markov process
Lecture - 18: Exact solution to Symmetric (or unbiased) one-dimensional Random walk (1-D RW) using Generating function method.
Lecture - 19: Properties of the solution for 1-D unbiased RW
Lecture - 20: 1-D unbiased RW: Asymptotic form of occupancy probability and transition to continuous variables

Week 5:
Lecture - 21: Solution to the problem of 1-D Random Walk with bias
Lecture - 22: Generalized Random Walk with Bias and Pausing
Lecture - 23: Effect of Pausing on Mean and Variance of Random walk
Lecture - 24: Random-walk in the presence of reflecting barrier
Lecture - 25: Boundary conditions for reflected Random-Walk and formulating absorbing barrier problem

Week 6:
Lecture - 26: The survival probability and residence time distribution for Random walker in the presence of an absorber
Lecture - 27: Random Walk with Bias and Absorber
Lecture - 28: Drift and Survival probability for Random walk with bias and absorber.
Lecture - 29: Introduction to gambler’s ruin problem.
Lecture - 30: Solution for ultimate winning probability in Gambler’s ruin problem

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

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

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

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

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

Week 12:
Lecture - 56: Numerical simulation of the Langevin equation
Lecture - 57: Derivation of Klein-Kramers equation from Langevin equation for joint position-velocity fluctuations of Brownian particle.
Lecture - 58: Illustrative solutions to the Klein-Kramers equation
Lecture - 59: Numerical simulation: Sampling from general distributions and Central Limit theorem
Lecture - 60: Numerical simulation of Random walk trajectories and method of solving Fokker-Planck Equation in bounded domain

Books and references

1. P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Stochastic Processes, UBS Publishers

2. S. Ross, Introduction to Stochastic Models, Harcourt Asia, Academic Press.

3. D.S. Lemons, An Introduction to Stochastic Processes in John Hopkins (2002)

4. H. Risken, The Fokker-Planck equation, Methods of solution and applications, 2nd edition, Springer (1989)

5. Nelson Wax, Selected papers in Noise and stochastic processes (Dover ) 1954

Instructor bio

Prof. Yelia Shankaranarayana Mayya

IIT Bombay

Prof. Yelia Shankaranarayana Mayya joined the department of Chemical Engineering, IIT Bombay as an adjunct faculty after superannuating from the post of Head Radiological Physics Division at the Bhabha Atomic Research Centre, in 2012. He is an aerosol Physicist who has contributed, over the past 49 years, to diverse aspects of fine particle science and their technological applications including theoretical studies on Brownian Motion and particle charging characteristics using stochastic methods. He teaches Aerosol Technology and Stochastic processes courses at IIT Bombay and is associated with various research activities in the department of chemical engineering and the department of environmental science. He has co-authored 195 papers in refereed Journals.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: April 26, 2025 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Please note that assignments encompass all types (including quizzes, programming tasks, and essay submissions) available in the specific week.

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Bombay .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team