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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