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WS 2018/19: Lecture "Stochastic processes (with programming exercises)"

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**Abstract: **Stochastic dynamics is abound as thermal fluctuations in equilibrium systems, in chemical reactions, and even technical systems with random service requests and failures, or fluctuating stock prices. In this lecture, I will introduce the powerful framework of Markov chains and Langevin equations to predict the time-evolution of stochastic systems and make statistical predictions. We will discuss the link between stochastic dynamics and statistical physics. As a special feature, selected programming exercises will accompany the lecture, allowing you to get hands-on experience of key concepts by simulating exemplary stochastic processes in Python.

**Target audience:**- Physics students at the Master’s level

- Mathematics students interested in applications of stochastic processes

- Bioengineering students with strong background in quantitative methods

**Date:** Thursday 1pm (first lecture on October 11th)**Location:** BZW A120/P (Zellescher Weg 11)

https://navigator.tu-dresden.

**Every second Thursday (i.e. 18.10.2018, 01.11.2018, ...), we will have p****rogramming tutorials right after the lecture: **Every second thursday 2.50pm in the PC pool REC/B113 (3 min walk from BZW)

https://navigator.tu-dresden.de/etplan/rec/01/raum/217801.0320

**Python tutorials:**

Tutorial 0: A Python primer

Tutorial 1: Simulating diffusion

Tutorial 2: Probability theory live

Tutorial 3: Diffusion in a double-well potential

Tutorial 4: Integrating the Fokker-Planck equation

Tutorial 5: Kramer's escape rate

Tutorial 6: Power spectral density

Tutorial 7: Ito and Stratonovich calculus

Tutorial 8: Matrix representation of Fokker-Planck operator

Tutorial 9: Kalman filter

External link to infotaxis example [Vergassola et al. Nature 2007]

Download zip-file.

Python is open-source can be downloaded e.g. here. This distribution (Anaconda) includes already packages for scientific computing (numpy, scipy), for plotting (matplotlib), and a browser-based developing front-end (jupyter-notebook). Please install the new version Python 3.6, not the old one Python 2.7; Windows, MacOs, Linux are all supported.

**Lecture notes:**

Benjamin Wolba is texing his notes and kindly agreed to share these:** click here.**

Handwritten notes as backup:

Lecture 1: Diffusion

Lecture 2: Fundamentals of probability theory

Lecture 3: Langevin and Fokker-Planck equation and part ii

Lecture 4: Dynkin equation and Kramer's escape rate theory

Lecture 5: Synchronization of noisy oscillators

Lecture 6: Ito versus Stratonovich calculus

Lecture 7: Fuctuation-dissipation theorem

Lecture 8: Link to statistical physics

Lecture 9: Decision and estimation theory

Lecture 10: Stochastic resonance

**Relevant literature:**

[1] vanKampen: Stochastic Processes in Physics and Chemistry, North-Holland

[2] Philip Nelson: Physical Models of Living Systems, W. H. Freeman, 2015

[3] H. Risken: The Fokker-Planck Equation: Methods of Solution and Applications, Springer

[4] C. Gardiner: Stochastic Methods: A Handbook for the Natural and Social Sciences

[5] W. Paul und J. Baschnagel: Stochastic Processes: From Physics to Finance, Springer

**Contact:**

benjamin.m.friedrich@tu-dresden.de

**Course content:**

- Lecture 1 Introduction: Diffusion

- Random forces, mean-square displacements, Einstein relation

- Ergodic hypothesis

- Application areas of stochastic processes - Lecture 2 Fundamental concepts

- Short review of probability theory, probabilities and probability densities, moments and cumulants

- Bayes formula

- Most important probability distributions (Binomial distribution, Poisson distribution, normal distribution, power-law distributions)

- Central limit theorem

- Stochastic processes: time-discrete and time-continuous

- Markov processes and transition probabilities

- Gaussian white noise and Wiener processes

- Lecture 3 The Fokker-Planck equation

- Chapman-Kolmogorov equation

- Derivation: weak formulation using smooth test function, Taylor expansion of test function, Kramers-Moyal coefficients as moments of time evolution operator, integration by parts

- Fokker-Planck equation for the Langevin equation with non-multiplicative noise

- Lecture 4 Dynkin equation and Kramer’s escape rate theory

- First passage times for Wiener process

- Role of boundary conditions, first-passage times (FPT), differential equation for FPT

- Spectral representation of the Fokker-Planck operator, eigenfunctions, decay times

- Dynkin equation for first-passage times

- Derivation of Kramer’s rates

- Polya’s theorem

- Diffusion to an absorbing target, electrostatic analogy - Lecture 5 Synchronization of noisy oscillators

- Noisy oscillators, phase correlation function, half-width-at-half-maximum, quality factor

- Adler equation

- Diffusion in tilted potential, phase-slip rates (Stratonovich), giant diffusion

- Stochastic resonance: the periodically driven bistable potential - Lecture 6 Ito versus Stratonovich calculus

- Stochastic differential equations with multiplicative noise

- Explicit versus semi-implicit integration (Euler-Maruyama and Euler-Heun schemes)

- Noise-induced drift

- Example: The 2D- and 3D-rotor and dielectric relaxation

- Noise in chemical reactions: small-number fluctuations - Lecture 7 The fluctuation-dissipation theorem

- Examples: Nyquist noise

- Fluctuation spectra and linear response theory

- Wiener-Khinchine theorem

- Application: Calibrating optical tweezers

- Lecture 8 Link to statistical physics

- Boltzmann distribution

- Derivation of detailed balance

- Position-dependent diffusion - Lecture 9 Statistical testing and decision making

- Hypothesis testing, ROC, significance levels

- Decision theory: Bayesian cost criterion, sequential decision making

- Maximum-Likelihood estimates, the rationale behind least-square fitting

- Bayesian parameter estimation

- Kalman filters: Linear dynamic models, measurement models; Optimal update rule

- Outlook: infotaxis - Lecture 10 Stochastic resonance

- Diffusion in the periodically driven double-well potential

- Signal-to-noise ratio

- Example: Mechano-sensation in crayfish

**Background needed:**- Multivariate calculus

- Elementary probability theory

- Ordinary and partial differential equations

- Basic programming skills

WS 2016/17: Haupt-Seminar: "Statistical Physics of Information:

Information theory in Physics and Biology"

**Abstract:**Information is a key concept in statistical physics, allowing one to give a vivid interpretation of entropy and partial knowledge of microscopic system states. On the other hand, tools from statisticalphysics have proven extremely useful to describe the stochastic dynamics of complex systems, an important example being biological cells that process noisy sensory information. In this seminar, by a series of student's presentation on selected topics, we will introduce both fundamental concepts of information theory as well as show how these concepts are applied in current state biological physics.

**Prerequisites:**Thermodynamics and statistical physics I, multi-variate calculus,basic knowledge of statistics. No prior knowledge of information theory required.

**Readers:**Benjamin Friedrich, Meik Dörpinghaus

**Time:**Every Wednesday 11.10 (3.DS), starting 19.10.

**Room:**APS 2026 (Nöthnitzer Str. 46, Computer Science Building)

**1st meeting with topic assignments:19.10.2016**

**2nd meeting: 26.10.2016: **Introductory lecture on the basics: frequency of occurrence/ probability / likelihood;Bayes'formula;maximum likelihood estimation; Shannon entropy;mutual information;

**Reading list (preliminary):**

- Shannon's theory of information and channel coding theorem (C. E. Shannon, “A mathematical theory of communication,” Bell SystemTechnical Journal, vol. 27, pp. 623–656, 1948.See also textbook [CT] )

- Introduction to Bayesian Decision theory (text book [MC] or [K2])

- Kobayashi Phys. Rev. Lett. 2010: Implementation of Bayesian Decision Making by Intracellular Kinetics

- Zwicker et al. PNAS 2016:Receptor arrays optimized for natural odor statistics

- Berg an Purcell, Biophys. J. 1977: Physics of Chemoreception [combined with Alvarez et al. Trend Cell Biol 2013]

- Andrews et al. PLoS Comp. Biol. 2006:Optimal Noise Filtering in the ChemotacticResponse of Escherichia coli

- Vergassola et al. Nature 2007: Infotaxis

- Brenner et al. Neuron 26, 2000: Adaptive Rescaling Maximizes Information Transmission

- A. Wald, “Sequential tests of statistical hypotheses,” Ann. Math. Statist., vol. 16, no. 2, pp. 117–186, June 1945; see also chapter 6 of [CM] for a pedagogical approach)

- Roldan et al. Phys. Rev. Lett. 2015: Decision Making in the arrow of time

- Montufar et al. Neural Comp. 2011 :Refinements of Universal Approximation Results for Deep

Belief Networks and Restricted Boltzmann Machines

**Textbooks:**

[CT] T. Cover and J. Thomas, Elements of Information Theory, 2nd edition.New York: Wiley & Sons, 2006.

[K1] Kay, Steven M. "Fundamentals of statistical signal processing, volume I: estimation theory." (1993).

[K2] Kay, Steven M. "Fundamentals of statistical signal processing: Detection theory, vol. 2." (1998).

[CM] J. L. Melsa, D. L. Cohn et al., Decision and estimation theory. McGraw- Hill, 1978

**Additional topics (only if time permits):**

- Whiting et al. Sci. Reports 2015: Towards a Physarum learning chip

- Molecular Systems Biology 3, 2007: Towards a theory of biological robustness

- Quinn, Christopher J., et al. "Estimating the directed information to infer causal relationships in ensemble neural spike train recordings."Journal of computational neuroscience30.1 (2011): 17-44.

-Jiao, Jiantao, et al. "Universal estimation of directed information."IEEE Transactions on Information Theory 59.10 (2013): 6220-6242.

- Varian PNAS 2016:Causal inference in economics and marketing

**Format of the seminar: **Each student will present a 30 min presentation of his/her assigned paper.This will be followed by a scientific discussion on the topic. Ideally, each student will have a 1 hour consultation with one of us to help with the preparation of the presentation.Additionally, we will use this seminar as a platform to train professional presentation skills. For that, constructive feedback shall be provided in a friendly atmosphere.

## SS 2016:Nonlinear dynamics and stochastic processes

Nonlinear dynamical systems are studied in many fields of physics, including classical mechanics, thermodynamics, laser physics, and even biological physics and economics. This will be a first course in nonlinear dynamics, with a focus on geometric aspects and applications. In a second part of the lecture, we will give an introduction to stochastic processes and study the dynamics of nonlinear systems in the presence of noise.

**Content:** Stability analysis, bifurcation theory, oscillators and synchronization, pattern formation, introduction to chaos, introduction to stochastic processes, Langevin and Fokker-Planck formalism, application to first passage time problems and Kramer’s escape rate theory.

**Prerequisites:** Ordinary Differential Equations, Multi-variate calculus.

**Reader:** Benjamin Friedrich

**Time:** Every Tuesday 16.40 (6.DS)

**Room:** PHY/B214 (Häckelstrasse 3 on main campus)

**Relevant literature:**

- Strogatz: Nonlinear Dynamics and Chaos, Westview, 2001 (gebraucht ab 20€)
- Risken: The Fokker-Planck Equation, Springer, 1996
- Stratonovich: Topics in the Theory of Random Noise, Martino, 2014
- Ott: Chaos in Dynamical Systems, Cambridge University Press, 2002
- VanKampen: Stochastic Processes in Physics and Chemistry, North-Holland, 2007

**Script:**

- Introdution to Dynamical Systems Theory --> Download PDF
- Linear Stability Analysis --> Download PDF
- Bifurcation theory --> Download PDF
- Oscillations --> Download PDF
- Synchronization --> Download PDF
- Introduction to Chaos Theory --> Download PDF
- Numerics --> Download PDF
- Pattern formation --> Download PDF
- Diffusion and Introduction to Stochastic Processes --> Download PDF
- Nonlinear stochastic dynamcis: Ito vs. Stratonovich -- > Download PDF
- Fokker-Planck formalism and Kramer's escape rate theory-- > Download PDF

## SS 2015: Continuum Mechanics for Biological Physics

Continuum mechanics describes fluids and solids; thus providing an indispensable toolkit for biological physics: Living cells move in fluids or adhere to elastic substrates, while showing both elastic and fluid-like properties themselves.

Content. Fluid dynamics and elasticity theory with emphasis on foundations and examples: Navier-Stokes equation from mass and force balance. Reynolds number. Fundamental solution and multipole gymnastiques. Blood flow. Elastostatics. Contact mechanics. Beam theory. Elastic rods, shells, and solids. Visco-elasticity. Experimental methods in cell rheology. Outlook on non-linear elasticity and cell rheology.

**Reader:** Benjamin Friedrich

**Tutorial:** Alexander Mietke

**Supervision:** Stephan Grilll

**Time for lecture:** Tuesday 14.50-16.20

**Location:** Biotec, Seminar room E06 (map)

**Time for tutorial:** Tuesday 16.40-18.10

**Location for tutorials:** Biotec, Seminar room E06 (map)

**Contact:** benjamin.friedrich@pks.mpg.de / amietke@pks.mpg.de**Script:**

- lecture notes (hydrodynamics)
- lecture notes (elasticity theory)

- Fluid dynamics basics: Navier-Stokes equation
- Viscous stresses, Reynolds number, examples of Stokes flow
- Solving Stokes flow using multipoles
- Swimming at low Reynolds numbers
- Mini-course: turbulence
- Hemodynamics: how your blood flows
- Introduction to elasticity theory
- Elasticity of rods, spherical shells, spheroids
- Contact mechanics
- Visco-elasticity

**Relevant literature:**

- Landau-Lifshitz: Elasticity theory
- Landau-Lifshitz: Fluid mechanics
- Happel-Brenner: Low Reynolds Number Hydrodynamics
- Vogel: Life in moving fluids
- Batchelor: An introduction to fluid mechanics
- Feynman: The Feynman Lectures, vol. 2
- Johnson: Contact Mechanics
- Berg: Random Walks in Biology

WS 2014/2015: Kinematics of Noisy Motion

This will be a hands-on course on the stochastic differential geometry of space curves, linking abstract mathematics and concrete examples. As a warm-up, we will discuss translational and rotational diffusion in the plane. Coupling translations and rotations, we will naturally encounter multiplicative noise, allowing us to discuss the subtleties of Ito versus Stratonovich calculus. We will then see how active translations and rotations can be conveniently described as a path on a Lie group called SE(3). No prior knowledge of Lie groups needed.

**Dates:** 28.11.2014, 05.12.2014, 12.12.2014 Friday, 13:00 - 14:30 Location: WIL-C203

**Script:** click here

## SS 2014: Continuum Mechanics for Biological Physics

Continuum mechanics describes fluids and solids; thus providing an indispensable toolkit for biological physics: Living cells move in fluids or adhere to elastic substrates, while showing both elastic and fluid-like properties themselves.

Content. Fluid dynamics and elasticity theory with emphasis on foundations and examples: Navier-Stokes equation from mass and force balance. Reynolds number. Fundamental solution and multipole gymnastiques. Blood flow. Elastostatics. Contact mechanics. Beam theory. Elastic rods, shells, and solids. Visco-elasticity. Experimental methods in cell rheology. Outlook on non-linear elasticity and cell rheology.

Readers: Benjamin M Friedrich, Elisabeth Fischer-Friedrich

Supervision: Stephan Grill

Time: Friday 9.20

**Location:** Physics building on Main campus, room B214

**Contact:** benjamin.friedrich@pks.mpg.de

**Script:**

- lecture notes (hydrodynamics)
- lecture notes (elasticity theory)

- Fluid dynamics basics: Navier-Stokes equation
- Viscous stresses, Reynolds number, examples of Stokes flow
- Solving Stokes flow using multipoles
- Swimming at low Reynolds numbers
- Mini-course: turbulence
- Hemodynamics: how your blood flows
- Introduction to elasticity theory
- Elasticity of rods, spherical shells, spheroids
- Contact mechanics
- Visco-elasticity

**Relevant literature:**

- Landau-Lifshitz: Elasticity theory
- Landau-Lifshitz: Fluid mechanics
- Happel-Brenner: Low Reynolds Number Hydrodynamics
- Vogel: Life in moving fluids
- Batchelor: An introduction to fluid mechanics
- Feynman: The Feynman Lectures, vol. 2
- Johnson: Contact Mechanics
- Berg: Random Walks in Biology