PH5619: Theory and Applications of Stochastic Processes

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Course Information

L-T-P-C3-0-0-3
Prerequisites None

Syllabus

Module 1: Random Variables

Lecture hours: 8 · Tutorial hours: 0 · Lab hours: 0

Basic notions of probability, Distributions, moments and cumulants, Generating functions, The joint Gaussian distribution, The central limit theorem .

Module 2: Random Events

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

Independent and non-independent events, Poisson process, Waiting time distributions .

Module 3: General Concepts of Stochastic Processes

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

Green's functions, Chapman-Kolmogorov equation, Markov property, Auto-correlation functions and power-spectral density, Wiener-Khinchin theorem, Wiener, Ornstein-Uhlenbeck and random telegraph processes .

Module 4: Discrete Markov Processes I

Lecture hours: 6 · Tutorial hours: 0 · Lab hours: 0

Master equation and methods of solution, Stationary solutions, Detailed balance, Non-equilibrium steady states .

Module 5: Discrete Markov Processes II

Lecture hours: 8 · Tutorial hours: 0 · Lab hours: 0

One-step processes, stationary states, Random walks, reflecting and absorbing boundaries, Survival probability, first passage time density, Backward Kolmogorov equation .

Module 6: Continuous Markov Processes

Lecture hours: 8 · Tutorial hours: 0 · Lab hours: 0

Kramers-Moyal and system size expansions, Fokker-Planck equation, Langevin equations for Brownian motion, Einstein-Smoluchowski relation and fluctuation-dissipation theorem .

Study Materials

Textbooks

  1. Stochastic Processes in Physics and Chemistry, N. G. Van Kampen, Elsevier, 3rd ed., ISBN: 9780444529657 (paperback); 9780080475363 (eBook)

References

  1. Stochastic Methods: A handbook for the natural and social sciences, C. W. Gardiner, Springer, 4th ed. (2009), ISBN: 978-3-540-70712-7
  2. The Fokker-Planck Equation: Methods of Solution and Applications, H. Risken, Springer, 2nd ed., ISBN: 978-3-642-61544-3