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The aim of this course is to cover the graduate-level students the subject which explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities. Topics will be covered such as central limit theorem, asymptotic approximations, drift and dispersion, Fokker-Planck equation, non-identical steps, persistence and self-avoidance, interacting walkers and electrolytes.

Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

Certification

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Course Credit: MIT

Course Curriculum

Module: 01
Lecture 1: Introduction to Random Walks and Diffusion 00:05:00
Lecture 2: Moments, Cumulants, and Scaling 00:05:00
Lecture 3: Central Limit Theorem 00:05:00
Lecture 4: Asymptotics in the Central Region – Part I 00:05:00
Lecture 4: Asymptotics Outside the Central Region – Part II 00:10:00
Module: 02
Lecture 5: Asymptotics with Fat Tails 00:05:00
Lecture 7: Asymptotics of the Bernoulli Random Walk 00:10:00
Lecture 8: The Continuum Limit  00:05:00
Lecture 9: Kramers Moyall Cumulant Expansion 00:05:00
Lecture 10: Persistent Random Walks and the Telegrapher Equation 00:05:00
Module: 03
Lecture 11: More on Persistence and Self Avoiding Walk 00:10:00
Lecture 12: Levy Flights (σ=∞) 00:10:00
Lecture 13: Discrete and Continuous Stochastic Processes 00:05:00
Lecture 14: Non-identicall Distributed Steps 00:05:00
Lecture 15: Non-identically Distributed Steps and Random Waiting Times 00:10:00
Module: 04
Lecture 16: Continus Time Random Walks 00:10:00
Lecture 17: Anomalous (Sub) Diffusion Scaling Laws 00:10:00
Lecture 18: Non-Markovian Diffusion Equations 00:10:00
Lecture 19: Polymer Models Persistence and Self Avoidance 00:05:00
Lecture 20: (Physical)Brownian Motion 00:10:00
Module: 05
Lecture 22: L´evy Distributions 00:05:00
Lecture 23: Continuous Time Random Walks 00:05:00
Lecture 24: Laplacian Growth II 00:05:00
Lecture 25: Large Steps and Long Waiting Times 00:05:00
Lecture 26: Leapers and Creepers 00:10:00
Assessment
Submit Your Assignment 00:00:00
Certification 00:00:00

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