# Introduction to mathematical physics, spring 2011

**Stochastic particle systems**

### Lecturer

### Scope

10 cu.

### Type

Advanced studies

**Description**

Stochastic particle systems (or "Interacting particle systems") are stochastic processes that are widely

used in physics (in non-equilibrium statistical mechanics) and in biology (spatial population dynamics).

In the simplest case (the simple exclusion process) particles perform random jumps on a lattice with

the constraint that no site contains more than one particle. The course provides an introduction to the

mathematical theory of these processes and then discusses in more detail the simple exclusion process

where recently some remarkable connections to combinatorics and random matrices have been obtained.

**Contents**

Review of probability

Markov jump processes and their generators

Particle systems: simple exclusion process (SEP), zero range process

Hydrodynamic limit

One dimensional SEP: random matrices, KPZ equation, universality

### Prerequisites

Basic probabiity

**Lectures**

Weeks 4-9 and 11-18 tuesday 14-16, thursday 14-16 C123.

Easter holiday 21.-27.4.

**First lecture: Thursday January 27!!**

**No lectures on Tuesday April 5 and Thursday April 7!!**

**Next lecture: Thursday April 28!!**

### Lecture notes

Pages 1-59

Pages 60-78

Pages 79-89

Pages 90-98

Pages 100-127

### Exams

The exam is on thursday the 11th of August at 10-14.

### Bibliography

Timo Seppäläinen: Translation Invariant Exclusion Processes http://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf

T. M. Liggett: Interacting particle systems, volume 276 of Grundlehren der Mathematischen Wissenschaften, Springer- Verlag, New York, 1985

H. Spohn. Large scale dynamics of interacting particles. Springer-Verlag, Berlin, 1991

### Registration

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### Exercise groups

Group |
Day |
Time |
Place |
Instructor |
---|---|---|---|---|

1. |
Fri |
14-16 |
C122 |
Christian Webb |

### Exercises

##### Due to Ferrari's course, the first exercise session will be on THURSDAY the 10th at 12.15 in the classroom B120.

Exercise 1 Solutions 1

Exercise 2 Solutions 2

Exercise 3 Solutions 3

Exercise 4 Solutions 4

Exercise 5 Solutions 5

Exercise 6 Solutions 6

Exercise 7

Exercise 8

Exercise 9