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Book Project

Together with Prof. Benjamin Friedrich, I am turning the notes of his lecture on ‘Stochastic Processes’ into a book.

I will publish parts of the book as blog posts on The Planet. Feel free to subscribe and follow our progress.

Our preliminary outline is here:

1 Prerequisites: Probability Theory

1.1  Mathematical Foundations

Random Variables

1.2  Probability in Physics

1.3  Important Probability Distributions

1.4  The Central-Limit Theorem

1.5 Stochastic Processes

2  Diffusion & Random Walk

2.1 Random Walker

2.2 Continuum Limit: Diffusion Equation

2.3 Random Force Model

3  Langevin Equation and Fokker-Planck Equation

3.1 Langevin Equation

3.2 Fokker-Planck-Equation

3.3 Master Equation

4  Dynkin Equation

4.1 Mean First-Passage Times and Dynkin Equation

4.2 KramersEscapeRateTheory

4.3 DiffusiontoCapture

4.4 Polya’stheorem

5  Active Oscillators and Synchronization

5.1 Van-der-Pol Oscillator

5.2 Hopf-Oscillator with Noise

5.3 TwocoupledPhaseOscillators

5.4 Synchronization in the Presence of Noise

6  Stochastic Resonance

7  Ito- versus Stratonovich Calculus

7.1 Numerical Motivation

7.2 Different Interpretations

7.3 Rotational Diffusion in 3D

7.4 How to derive a correct Langevin equation?

7.5 Numerical Integration of nonlinear SDE

8 Fluctuation-Dissipation-Theorem

8.1 Historical Examples

8.2 FDT for Classical Systems

9 A Link to Statistical Physics

9.1 Detailed Balance

9.2 Increase of Relative Entropy

9.3 Equilibrium vs. Non-Equilibrium

9.4 Thermal Fluctuations: Space-dependent Diffusion

9.5 Entropy Production

10 Decision Theory

10.1 StatisticalTesting

10.2 Student’s t-test

10.3 Fischer’s Exact: Testing categorial data for correlations

10.4 EstimationTheory

10.5 KalmanFilters