# Simulation methods (Department of Mathematics)

**Level: **Doctoral or MA 2nd year**Course Status: ** Elective

**Full description: **

**Brief introduction to the course**

The aim of the course is to give an introduction to simulation methods. The illustrative tasks will be from statistical physics. The main simulation techniques as Monte Carlo method and molecular dynamics will be outlined. Special accelerating techniques will be discussed. Some numerical methods of nonlinear dynamical systems will be treated. Simulation of fractal growth models and cellular automata will be discussed.

After a brief survey on this field, we will introduce cluster counting method and numerical study of percolation models. How to use finite size scaling to learn about the infinite size limit? The Metropolis Monte Carlo algorithm: Ergodicity and detailed balance. Critical slowing down and accelerating algorithms.

**The goals of the course:**

The goal of the course is to improve the students programing skills and make them acquainted with modern simulation methods.

**More detailed display of contents.**

Weekly breakdown of the course:

1 Brief summary of statistical physics

2 Prerequisites

Computer simulation, initial conditions, boundary conditions, characteristic times, averages, rand

3. Percolation and cluster counting

4. Metropolis Monte Carlo

5. Acceleration algorithms

6. Dynamic Monte Carlo

7. Optimization

8. Molecular dynamics

9. Scaling surface dynamics and stochastic partial differential equations

10. Fractal growth

11. Cellular automata

12. Project presentation

*1 Brief summary of statistical physics*

Equilibrium, detailed balance, time and ensemble averages, equilibrium density distributions. Entropy, free energy, fluctuations, linear response. Phase transitions, scaling, universality.

*2 Prerequisites*

Computer simulation as a basic tool in natural sciences, initial conditions, boundary conditions, characteristic times, averages, sampling, random numbers. Finite size scaling.

*3. Percolation and cluster counting*

The percolation model, critical properties. Hoshen-Kopelman algorithm. Calculation of the critical exponents.

*4. Metropolis Monte Carlo*

Hamiltonian systems. Importance sampling. Markov chains with detailed balance. Ergodicity. The Ising model. Simple fluids.

*5. Acceleration algorithms*

Critical slowing down. Bit coding. Cluster algorithms.

*6. Dynamic Monte Carlo*

Different sources of slowing down (hydrodynamic, critical, metastability-related). Diffusion on percolation clusters. Relation to conductivity

*7. Optimization*

NP complete problems, approximate heuristic algorithms.

*8. Molecular dynamics*

Time scales. Efficient algorithms (Verlet, leap-frog, predictor-corrector). Linked cell method. Nosé-Hoover thermostat.

*9. Scaling surface dynamics and stochastic partial differential equations*

Stochastic surface growth. Self-affine scaling. Simple models (Edwards-Wilkinson, Eden, ballistic deposition). The EW and the Kardar-Parisi-Zhang equations. Numerical solution.

*10. Fractal growth*

Diffusion limited aggregation. Efficient algorithm. Randomness and anisotropy. Noise reduction.

*11. Cellular automata*

CA as nonlinear mappings. Categorization. Game of life. hydrodynamic CA. Boltzmann lattice method.

*12 Project presentation*

Such further items as assessment deadlines, office hours, contact details etc are at the discretion of the department or the individual.

Consultation upon agreement.

Janos Kertesz, KerteszJ@ceu.edu

Nador 11, Room 610

The students, who successfully absolve the course will be able to actively use simulation techniques, write their own programs if they meet mathematical problems, where simulation seems to be a helpful approach.

**Assessment:**

The students have to solve independently a problem, which will be handed out in week 8. They have to submit the program (in C, C++, FORTRAN or python) and a pdf file with description of the algorithm and presentation of the results. The solutions will be presented in the last class.

In addition there is an oral examination.

The final grade will be composed 50% from the project and 50% from the examination.