Modeling Stochastic phenomena for Engineering applications: Part-1

By Prof. Yelia Shankaranarayana Mayya | IIT Bombay

Learners enrolled: 170 | Exam registration: 5

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 :	Completed
Course Type :	Elective
Language for course content :	English
Duration :	12 weeks
Category :	Chemical Engineering
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	22 Jan 2024
End Date :	12 Apr 2024
Enrollment Ends :	05 Feb 2024
Exam Registration Ends :	16 Feb 2024
Exam Date :	21 Apr 2024 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: Introduction to stochastic phenomena
Lecture -2: Examples of stochastic processes from various fields
Lecture -3: Probability distributions (Binomial, Poisson, Gaussian)
Lecture -4: Cauchy distribution, extreme value distributions
Lecture -5: Useful Mathematical Tools: Fourier Transforms, Dirac delta function, Sterling’s approximation

Week 2:
Lecture -6: Generating function and its inversion: examples and usefulness
Lecture -7: Statement of Central Limit theorem and its relevance
Lecture -8: Conditional probability; Derivation of Central Limit theorem (CLT)
Lecture -9: Cauchy distribution and Central limit theorem
Lecture -10: Implications of CLT to random walk models

Week 3:
Lecture -11: Definition and examples of Markov processes
Lecture -12: Constructing transition Matrix
Lecture -13: Chapman-Kolmogorov Equation- implications
Lecture -14: N-Step transition Matrix, Stationarity
Lecture -15: Absorbing, transient and Recurrent states

Week 4:
Lecture -16: Ergodicity, Equilibrium,non-Markovian examples
Lecture -17: Unbiased Random walk on a lattice: Formulation with and without pause
Lecture -18: Exact solution
Lecture -19: Biased Random walk: Formulations and solutions
Lecture -20: Random-walk in higher dimensions

Week 5:
Lecture -21: Probability of return to origin – Generating function formulation
Lecture -22: Proof of Polya’s theorem
Lecture -23: Random walk in the presence of absorbers and reflectors
Lecture -24: Continuous time Random walk
Lecture -25:Taylor expanded Random-walk equation : Concept of drift and diffusion

Week 6:
Lecture -26: Passage to differential equation (Fokker-Planck) for continuous space and time variables
Lecture -27: Solution to Random walk problems in finite domain
Lecture -28: Survival probability estimates
Lecture -29: Gambler’s ruin problem and recurrence equation
Lecture -30: Exact solution to Gamblers ruin problem

Week 7:
Lecture -31: Brownian Motion of colloidal particles: Historical context, Langevin equation formulation,
Lecture -32: Ornstein-Uhlenbeck process, meaning of Gaussian White-noise, autocorrelation function, non-white noise examples
Lecture -33:, Solution for velocity and displacement, limiting behavior,
Lecture -34: fluctuation dissipation theorem and practical implications
Lecture -35: Transition probability, Derivation of Klein-Kramer’s differential equation for probability density in position-velocity space

Week 8:
Lecture -36: Some exact solutions to velocity relaxation of a Brownian particle
Lecture -37: Derivation of Fick's law, diffusion approximation,
Lecture -38: Conditions of validity, some examples in high friction limit
Lecture -39: Crossing over potential barriers; escape rate modeling under high friction limit,
Lecture -40: Kramer's theory of escape from KKE, Practical applications

Week 9:
Lecture -41: Master-equation formulation of Stochastic processes: Derivation from Chapman-Kolmogorov equation for continuous space & time
Lecture -42: Key assumption on transition probabilities, distinguishing features, Poisson representation, Ehrenfest’s flea model
Lecture -43: Master equation for Discrete space-continuous time, Constructing Master equation from its deterministic counter-part,
Lecture -44: Illustration using pure birth Process (Poisson process)
Lecture -45: Study of pure death process

Week 10:
Lecture -46: Solution to random-walk problem from Master-equation Perspective
Lecture -47: Birth & Death processes, Malthus-Verhulst process, Stability analysis of the deterministic counter-part
Lecture -48: General solution for the distribution function, Extinction Probability
Lecture -49: Formulating master equations for Chemical kinetics, Equations for Mean and variance,
Lecture -50: Method of solving Master equation, Expansion of the master equation

Week 11:
Lecture -51: Introduction and examples to Branching process, Galton-Watson processes
Lecture -52: 1-member transition probabilities and their generating functions
Lecture -53: Proof of k-member transition probability
Lecture -54: Markov model of occupancy probability
Lecture -55: Population extinction-Proof of criticality theorem

Week 12:
Lecture -56: Examples and implications of criticality theorem
Lecture -57: Numerical simulation of Central Limit theorem
Lecture -58: Numerical approaches to master equation
Lecture -59: Numerical simulations: Markov processes
Lecture -60: Numerical Simulation of Random Walk

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: 21 April 2024 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

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