Stochastic Processes in Physics, Chemistry and Biology is an elective course for PhD coursework at The Institute of Mathematical Sciences and at the Homi Bhabha National Institute. The course was offered for the first time in Autumn 2007 and has been offered every alternate year since.
The course is aimed to provide a broad, application-agnostic, introduction to stochastic methods in the physical sciences and in biology. The first part, covering approximately three-fourths of the course duration, will deal with the algebra of stochastic variables, the calculus of stochastic processes, Markov processes, the master equation with techniques of exact and approximate solutions, the diffusion approximation of the master equation, the related Fokker-Planck and Langevin descriptions, and end with stochastic field theories. Depending on feedback, applications in statistical mechanics, chemical kinetics, and population dynamics may be discussed.
In the second part, an elementary introduction will be given to the Bayesian interpretation of probability, Bayesian inference, and the maximum entropy method. Depending on feedback, applications in data analysis, image restoration, and file compression may be given.
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Introduction to probability: The classical definition of probability. Examples: Brownian motion, chem- ical kinetics, population biology.
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Algebra and calculus of stochastic variables: Definition of stochastic variables, averages, addition and transformation of stochastic variables, the Gaussian distribution, the central limit theorem. Definition of stochastic processes, Fourier analysis of stationary stochastic processes, distribution functions describing a stochastic process, illustrative examples.
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Markov processes: The Markov property, Chapman-Kolmogorov equation, stationary Markov processes. Radioactive decay as a Markov process.
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The Master equation: Derivation, long time limit of the Master equation, increase of entropy of the distribution, proof of detailed balance, passage to the macroscopic equation. The Master equation for one-step processes, definition, Poisson processes, general properties, linear one-step processes, boundary conditions, the general solution of the linear one-step process. First passage problems. Monte Carlo simulations of the Master equation. The Gillespie algorithm for chemical kinetics.
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The Fokker-Planck and Langevin descriptions: Derivation, multivariate linear Fokker-Planck equa- tions, Langevin description. Applications to Brownian motion, barrier crossing. The problem with nonlinearity, the Ito/Stratonovich picture in non-linear Langevin equations.
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Hydrodynamic and continuous descriptions: Statement of the problem, general formulation using van Kampen’s Omega expansion, emergence of the macroscopic law with linear noise, application to the SIR model in epidemology. Generalised hydrodynamics, the fluctuation-dissipation relationa for continuous systems,the Landau-Lifshitz equations of fluctuating hydrodynamics, fluctuations in the Boltzmann equation.
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Bayesian probability theory: an historical introduction, origin, Bayes, Laplace; modern revival, Jeffreys, Jaynes.
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The axioms of probability theory: the approach of Richard Cox.
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Bayesian parameter estimation: the mean, the variance, the frequency of a noisy sinusoid.
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Model selection and Ockham’s razor: Illustrative examples from astrophysical data analysis.
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Markov Chain Monte Carlo : calculating the posterior in a high dimensional space. Applications in computational genomics and parameter estimation.
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The maximum entropy method for assigning probabilities: equilibrium statistical mechanics and image restoration.
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Other applications (optional, according to instructor’s choice): machine learning, file compression.
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Stochastic Processes in Physics and Chemistry by N. G. Van Kampen
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The Fokker-Planck Equation by H. Risken
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Handbook of Stochastic Methods by C. W. Gardiner
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Probability Theory: The Logic of Science by E. T. Jaynes
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Information Theory, Inference and Learning Algorithms by D. MacKay