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# Introduction to Monte Carlo Methods

Lecturer:

Dr. Matthew Dennison

Dates:

January 16 to February 13, 2017

no lecture on January 26

Time:

Mondays: 10:15 - 11:45

Thursdays: 10:15 - 11:45

Place:

Mondays: EW 202 (TU Berlin, EW building)

Thursdays: EW 731 (TU Berlin, EW building)

**Lecture Material:**Please log in to access the lecture slides.

# Course Outline

1. Introduction to the MC method

An introduction to the Monte Carlo method. This will give a history of its development, from Buffon's needle experiment to the first computer simulations. Some uses of Monte Carlo integration will be covered, how and when it should be used, how to make it more efficient etc.**2. MC simulation**

The basics of the method in statistical physics will be covered, followed by the implementation of a basic algorithm:

- trial moves
- acceptance criteria
- boundary conditions, etc.

Considerations such as obeying detailed balance etc. will be discussed. Early papers on MC simulations will be covered.**3. Ensembles other than NVT, anisotropic particles**

The methods for MC simulations in other ensembles (NPT, μVT ....), and when each one should be used (their advantages and disadvantages etc.). Studying phases other than liquids, and simulating anisotropic particles. This will cover some main papers on the isotropic-crystal phase transition in spheres, simulations on ellipsoids/spherocylinders, etc.**4. Predicting phase behaviour**

The problems with forming some phases (solids etc.). Some techniques for looking at close packed systems (floppy box method etc.). How to calculated free energies, chemical potentials and pressures in order to find stable phases, coexistence etc. and some context on phase diagrams etc.

Exploring the hard sphere phase diagram.**5. Examining system properties**

Uses of Monte Carlo simulations for predicting system properties:

- the structural properties of crystals and liquid crystals, etc.
- elastic properties of liquid crystals
- elastic properties of solids

6. Equilibrating complex systemsTechniques for equilibrating glassy systems, metastable phases etc. This will cover several works on MC simulations of glasses, developing MC moves to speed up equilibration (multi-particle moves, rejection free moves) and how these have been used to resolve whether a phase is stable or not.

7. Biased sampling and parallel tempering

Methods for bias / umbrella sampling and parallel tempering, how these can be used for systems with large energy barriers etc.

8. Beyond the Metropolis algorithm

Methods such as kinetic / dynamic Monte Carlo, the Gillespie algorithm, Configurational Bias Monte Carlo (and other methods for simulating polymers).

Applications of the Monte Carlo method outside of statistical physics (finance, biological systems, etc.).