Turbulent flow

Chaos,entropy, dynamics.

Introduction to the fundamental mathematical concepts and problems of classical dynamics.

What are Integrability, Chaos and Entropy? Why has time an arrow? The course is an introduction to classical dynamics and the mathematical problems it has lead to. We will discuss how simple deterministic systems can give rise to chaos and randomness. We will also discuss how chaotic systems with very large number of degrees of freedom can nevertheless give rise to simple statistical description and how dynamical equations which are symmetric with respect to reversing the direction of time can give rise to phenomena that are not any more time symmetric. These are fundamental problems of mathematical physics that were raised by Boltzmann, Poincare and others and which are still under intensive investigation.

The course is the first of introductory courses to mathematical physics. The second one deals with mathematical problems inspired by quantum physics and will be lectured on Spring 2021.

TARGET AUDIENCE

Mathematics students interested in the mathematics inspired by physics or physics students interested in the mathematical foundations of physics.

PREREQUISITES

Mathematics Students: Differential equations, Probability Theory, Measure and Integration are useful.
Physics students: Mathematical Methods for Physics 2.
No physics background is necessary but of course some is useful.

CONTENTS

1. Deterministic dynamics
2. Vector fields and flows
3. Examples: 1d motion, from simple to complex
4. Linearization
5. Invariant manifolds
6. Chaos: Arnold's cat, homoclinic tangle, Smale's solenoid
7. Symbolic dynamics
8. Ergodic theory, Lyapunov exponents
9. Integrable systems
10. Near integrable systems: KAM Theorem
11. Foundations of statistical mechanics, irreversibility and the arrow of time

COURSE MATERIAL

Lecture notes will be posted.

## Antti-Jukka Kupiainen

Published, 1.3.2020 at 10:36

The exam covers the lecture notes chapters 1-10 on the web page.
You are not required to master the more mathematical proofs (like
Picard, stable manifold theorem, ergodic theorems, KAM theorem) but
you are supposed to know the definitions of the basic concepts.
Homework are a good guide to exam problems.

a project.

## Joona Oikarinen

Published, 28.2.2020 at 12:25

The exam takes places in the room CK111.

## Joona Oikarinen

Published, 28.2.2020 at 12:25

The date for the exam is MONDAY 2ND OF MARCH, AT 14:00-16:00 (starts 14:00, not 16:00), room CK111.

If you are unable to attend the exam at this date, contact the teaching assistant via e-mail joona.oikarinen@helsinki.fi and an alternative time will be arranged.

## Joona Oikarinen

Published, 17.2.2020 at 19:45

If you are interested in completing the project (which gives 5 extra credits), contact the lecturer Antti Kupiainen via e-mail antti.kupiainen@helsinki.fi or see him in his office (Room D334, Exactum).

## Joona Oikarinen

Published, 29.1.2020 at 12:29

No exercise class on Monday 3rd of February.

Next exercise class is on Monday 10th of February.

### Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Fri 17.1.2020
12:15 - 14:00
Fri 17.1.2020
14:15 - 16:00
Mon 20.1.2020
12:15 - 14:00
Fri 24.1.2020
12:15 - 14:00
Fri 24.1.2020
14:15 - 16:00
Mon 27.1.2020
12:15 - 14:00
Fri 31.1.2020
12:15 - 14:00
Fri 31.1.2020
14:15 - 16:00
Mon 3.2.2020
12:15 - 14:00
Fri 7.2.2020
12:15 - 14:00
Fri 7.2.2020
14:15 - 16:00
Mon 10.2.2020
12:15 - 14:00
Fri 14.2.2020
12:15 - 14:00
Fri 14.2.2020
14:15 - 16:00
Mon 17.2.2020
12:15 - 14:00
Fri 21.2.2020
12:15 - 14:00
Fri 21.2.2020
14:15 - 16:00
Mon 24.2.2020
12:15 - 14:00
Fri 28.2.2020
12:15 - 14:00
Fri 28.2.2020
14:15 - 16:00

## Other

### Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Master studies

(Varies with the content of the course)

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Lectures and exercise classes