# Mathematical Statistical Physics

## Module PH7009

This module is offered by Ludwig-Maximilians University Munich (LMU). It is available for TUM students only within a joint degree program (e. g. M. Sc. Quantum Science & Technology).

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Basic Information

PH7009 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

- Focus Area Theoretical Quantum Science & Technology in M.Sc. Quantum Science & Technology

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workload | Contact hours | Credits (ECTS) |
---|---|---|

270 h | 90 h | 9 CP |

Responsible coordinator of the module PH7009 is Sabine Jansen.

### Content, Learning Outcome and Preconditions

#### Content

The module covers the following topics: Gibbs measures: DLR conditions, existence, uniqueness (Dobrushin's theorem), phase transitions, absence of spontaneous symmetry breaking in two dimensions. Ising model: high temperature phase, Peierls argument, cluster expansion, Fortuin-Kasteleyn representation, FKG inequality, spontaneous symmetry breaking in continuous models. Non-equilibrium model systems: Exclusion processes, matrix product ansatz, interacting particle systems.

The main goal of this module is to acquire a deeper mathematical and physical understanding of phase transitions and collective phenomena that occur in macroscopic interacting particle systems.

#### Learning Outcome

After successful completion of the module the students are able to:

- apply notions to handle infinite systems in the thermodynamic limit in a rigorous way

The primary goal is an in depth mathematical and physical understanding of macroscopic interacting particle systems their collective phenomena like phase transitions.

#### Preconditions

No prerequisites in addition to the requirements for the Master’s program in Quantum Science and Technology. Familiarity with quantum mechanics is assumed, at the level of an introductory course from a Bachelor degree in physics. Basic notions of probability theory and functional analysis can be helpful.

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

Type | SWS | Title | Lecturer(s) | Dates | Links |
---|---|---|---|---|---|

VO | 4.0 | Mathematical Statistical Physics | Helling, R. Jansen, S. | see LSF at LMU Munich |
current |

UE | 2.0 | Übungen zu Mathematical Statistical Physics | Helling, R. Jansen, S. | see LSF at LMU Munich |
current |

#### Learning and Teaching Methods

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. The main teaching material is presented on the blackboard or by beamer. Lectures are supplemented by weekly problem sets, deepening the understanding of core concepts through concrete calculations. Solutions to the problem sets are discussed in the exercise sessions. Participation in the exercise classes is strongly recommended, since the exercises are aids for acquiring a deeper understanding of the core tools of condensed matter many-body physics and field theory and for practicing to solve typical exam problems.

#### Media

Blackboard presentations, slides.

#### Literature

Standard textbooks on many-body theory, e.g.:

- Sacha Friedli and Yvan Velenik: “Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction”
- Ola Bratteli, Derek Robinson: “Operator Algebras and Quantum Statistical Mechanics”

### Module Exam

#### Description of exams and course work

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational tasks.

For example an assignment in the exam might be:

- Demonstrate the existence of the pressure in a perturbed Ising model
- Show that the set of DLR measures is invariant under local perturbations
- Prove that a KMS state with a symmetry implies that the time evolution has the same symemtry

#### Exam Repetition

The exam may be repeated at the end of the semester.