## Non-equilibrium stochastic processes

in Physics, Chemistry, and Biology 2022

Course codes: MCC011/TIF106/FIM785

(7.5 credit units)

T. Löfwander

B. Mehlig

**New: ** Link to online forum. Please register here. Don't hesitate to ask any questions concerning course contents, structure, and examination that you may have.

This page contains information only about the part taught by B. Mehlig. Please refer to main course home page on CANVAS for this course.

### Plan

Plan for the part taught by B. Mehlig

Lecture 1: *Kramers problem and first passage times.* Arrhenius law, transition-state theory, Kramers problem: overdamped limit: solution of Fokker-Planck equation with integrating factor. Saddle-point approximation.

Lecture 2: *Correlated random walks*. Random walks vs. correlated random walks. Path-coalescence transition. Order parameter: Lyapunov exponent. Computation of Lyapunov exponent.

Lecture 3: *Turbulent aerosols.* Stokes equation. Relation to correlated random walks in the overdamped limit. Solution in the white-noise limit: mapping onto Kramers escape problem. Exact solution. WKB approximation (large-deviation theory).

Lecture 4: *Turbulent aerosols continued*. Algebraic perturbation theory for Lyapunov exponent: mapping to perturbed quantum harmonic oscillator. Perturbation theory.

Lecture 5: *Population dynamics.* Master equation for one-step processes: Poisson process, random walk. Steady state of one-step processes. Non-linear birth-death processes. Probability of extinction, time to extinction. Large-deviation theory for time to extinction. Time-dependent WKB approximation for Poisson process.

Lecture 6: *Reaction kinetics*. Branching-annihilation reaction, moment equations. Reaction rate for diffusive absorption. Diffusion-limited annihilation reactions. Failure of mean-field theory. Critical dimension. Self-avoiding random walk. Role of fluctuations: single-species reactions in one dimension.

Lecture 7: *Brownian motors and ratchets*. Life at low Reynolds number. Role of diffusion. Example: Brownian particle in asymmetric periodic potential. Modulation of potential or temperature as a function of time: out-of-equilbrium directed transport.

### Examples sheet

1. Path coalescence [pdf]. Email PDF to

Every student must hand in her/his own solution on paper. Same rules as for written exams apply: it is not allowed to copy any material from anywhere unless appropriate reference is given. All figures must have axis labels and captions giving all information necessary to reproduce the figure. Describe your results in words. Always compare with theory. Summarise problems, discuss possible causes.

Maximal number of points: 20.

### Preparation for exam

Sample questions for 2021 home exam [pdf].

### Literature

In addition to lecture notes:

1. Lennart Sjögren, Lecture notes Stochastic processes (Chapters 1, 2, 3, 4, 5, 6, 7, 8, 9,10)

2. N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Third Edition, North-Holland

3. H. A. Kramers, *Brownian motion in a field of force and the diffusion model of chemical reactions*,

Physica** 7** (1940) 284 [pdf]

4. P. L. Krapivsky, S. Redner & E. Ben-Naim, *A kinetic view of statistical physics*, Cambridge University Press (2010)

5. R. D. Astumian & P. Hänggi, *Brownian motors*, Physics Today, November 2002, p. 33

### Further Resources

1. R. Kubo, M. Toda, N. Hashitume, Statistical Physics II (Nonequilibrium Statistical Mechanics), Springer

2. H. Risken, The Fokker Planck equation - methods of solution and applications, Second Edition, Springer

3. E. S. Lander and M. S. Waterman, eds., Calculating the secrets of life, National Academic Press, Washington (1995), On-line version of this book

4. E. S. Allman and J. A. Rhodes, Mathematical Models in Biology, Cambridge University Press (2004)