Modeling Stochastic phenomena for Engineering applications: Part-1

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 methodsINDUSTRY SUPPORT: Research Organizations such as BARC, IGCAR

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 usefulnessLecture -7: Statement of Central Limit theorem and its relevanceLecture -8: Conditional probability; Derivation of Central Limit theorem (CLT)Lecture -9: Cauchy distribution and Central limit theoremLecture -10: Implications of CLT to random walk models
Week 3:
Lecture -11: Definition and examples of Markov processesLecture -12: Constructing transition MatrixLecture -13: Chapman-Kolmogorov Equation- implicationsLecture -14: N-Step transition Matrix, StationarityLecture -15: Absorbing, transient and Recurrent states
Week 4:
Lecture -16: Ergodicity, Equilibrium,non-Markovian examplesLecture -17: Unbiased Random walk on a lattice: Formulation with and without pauseLecture -18: Exact solutionLecture -19: Biased Random walk: Formulations and solutionsLecture -20: Random-walk in higher dimensions
Week 5:
Lecture -21: Probability of return to origin – Generating function formulationLecture -22: Proof of Polya’s theoremLecture -23: Random walk in the presence of absorbers and reflectorsLecture -24: Continuous time Random walkLecture -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 variablesLecture -27: Solution to Random walk problems in finite domainLecture -28: Survival probability estimatesLecture -29: Gambler’s ruin problem and recurrence equationLecture -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 examplesLecture -33:, Solution for velocity and displacement, limiting behavior,Lecture -34: fluctuation dissipation theorem and practical implicationsLecture -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 particleLecture -37: Derivation of Fick's law, diffusion approximation,Lecture -38: Conditions of validity, some examples in high friction limitLecture -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 & timeLecture -42: Key assumption on transition probabilities, distinguishing features, Poisson representation, Ehrenfest’s flea modelLecture -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 PerspectiveLecture -47: Birth & Death processes, Malthus-Verhulst process, Stability analysis of the deterministic counter-partLecture -48: General solution for the distribution function, Extinction ProbabilityLecture -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 processesLecture -52: 1-member transition probabilities and their generating functionsLecture -53: Proof of k-member transition probabilityLecture -54: Markov model of occupancy probabilityLecture -55: Population extinction-Proof of criticality theorem
Week 12:
Lecture -56: Examples and implications of criticality theoremLecture -57: Numerical simulation of Central Limit theoremLecture -58: Numerical approaches to master equationLecture -59: Numerical simulations: Markov processesLecture -60: Numerical Simulation of Random Walk